tag:blogger.com,1999:blog-49784264669153795552018-03-05T11:15:12.401-05:00Formal Methods in Political PhilosophyNow available on Amazon.com as an e-book: AN INTRODUCTION TO THE USE OF FORMAL METHODS IN POLITICAL PHILOSOPHYRobert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.comBlogger39125tag:blogger.com,1999:blog-4978426466915379555.post-58471659391050971862015-10-17T11:42:00.002-04:002015-10-17T11:42:32.206-04:00A QUESTION FOR MY READERSI am delighted to learn that this blog is being read by many people in Russia. Is someone using it as a teaching tool? I woujld be very interested to know who is reading it.<br /><br />Thank you.Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0tag:blogger.com,1999:blog-4978426466915379555.post-88569836046271853622015-04-11T10:53:00.000-04:002015-04-11T10:53:13.223-04:00THE WORD IS GETTING AROUND<span style="font-size: large;">There has been a dramatic spike in the number of daily visits to my other blog, the Formal Methods blog. Turns out, according to Google, that the hits are coming from Russia. Inasmuch as I have long been big in Ukraine, maybe we can conclude that a Ukrainian who reads my blog was captured by Russian forces and taken back to the steppes of Central Russia, where he is spreading the word among others in the gulag.</span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0tag:blogger.com,1999:blog-4978426466915379555.post-85668531652968353222010-07-28T05:30:00.004-04:002010-07-28T05:46:56.307-04:00FINAL INSTALLMENT OF FORMAL METHODS TUTORIALWith this segment, I conclude my discussion of Arrow's General Possibility Theorem. And I think this will also conclude this tutorial on the use and abuse of formal methods in political philosophy. I will be happy to respond to questions, if there are any, but I think enough is enough. Thank you all for staying with me on this, for pointing out errors, and for asking questions. It has been fun for me, revisiting material I have not taught for twenty years or more, and I hope it has been informative and fun for you.<br /><br /> An extremely interesting result concerning the consistency of majority rule was produced by the Australian political scientist Duncan Black. In a book called The Theory of Committees and Elections, published in 1958, Black proved an important theorem about circumstances under which majority rule is guaranteed to produce a transitive social preference ordering. In a moment, I am going to go through the proof in detail, but let me first explain intuitively what Black proved. Ever since the French Revolution, political commentators have adopted a convention derived from the seating arrangement in the National Assembly. In that body, Representatives belonging to each party were seated together, and the groups were arrayed in the meeting hall in such a manner that the most radical party, the Jacobins, sat on the extreme left of the hall, and the most reactionary party, the Monarchists, sat on the right, with the other groups seated between them from left to right according to the degree to which their policies deviated from one extreme or the other. Thus was born the left-right political spectrum with which we are all familiar. [Of course, in the U. S. Senate, there are no Communists and only one Socialist, but, as the reign of George W. Bush shows, there are still plenty of Monarchists.]<br /><br /> The interesting fact, crucial for Black's proof, is that wherever a party locates itself on the spectrum, it tends to prefer the positions of the other parties, either to the left or to the right, less and less the farther away they are seated. So, if an individual identifies himself with a party in the middle, he will prefer that party's positions to those of a party a little bit to the left, and he will prefer the policies of the party a little bit to the left over those of a party farther to the left, and so on. The same is true looking to the right. Notice that since only ordinal preference is assumed, you cannot ask, "Is a party somewhat to the left of you farther from you than a party somewhat to the right of you?" [Make sure you understand why this is true. Ask me if it is not.]<br /><br /> Consider contemporary American politics. If I am a moderate Republican [assuming there still is one], I will prefer my position to that of a conservative Republican, and I will prefer that position to a right wing nut. I will also prefer my position to that of a Blue Dog Democrat [looking to my left rather than to my right], and that position to the position of a Liberal Democrat, and that position in turn to the position of a Socialist [Bernie Sanders?].<br /><br /> This can be summarized very nicely on a graph, along the X-axis of which you lay out the left-right political spectrum, while on the Y-axis you represent the order of your preference. Pretty obviously, the graph you draw will have a single peak -- namely, where your first choice is on the X-axis -- and will fall away on each side, going monotonically lower the farther you get on the X-axis from your location on it. In short, your preference, when graphed in this manner, will be single-peaked. Here is an example of a person's preference order graphed in this manner. For purposes of this example, there are five alternatives, (a, b, c, d, e), and the individual has the following preference order: d > e > c > b > a <br /><br /><br /><br /><a href="http://1.bp.blogspot.com/_lOTQWmy99zo/TE_5VEkTa7I/AAAAAAAAADQ/3eJKnNCDg7U/s1600/duncan+black.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 200px; height: 146px;" src="http://1.bp.blogspot.com/_lOTQWmy99zo/TE_5VEkTa7I/AAAAAAAAADQ/3eJKnNCDg7U/s200/duncan+black.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5498887810441243570" /></a><br /><br />Let us suppose that there is a second person whose preference order is a > b > c > d > e. It is obvious that if we posted this person's preferences on the same graph, the two together would look like this:<br /><br /><a href="http://3.bp.blogspot.com/_lOTQWmy99zo/TE_6MDqFM3I/AAAAAAAAADY/PgVntFa7OGM/s1600/duncan+black+2.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 200px; height: 145px;" src="http://3.bp.blogspot.com/_lOTQWmy99zo/TE_6MDqFM3I/AAAAAAAAADY/PgVntFa7OGM/s200/duncan+black+2.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5498888755089847154" /></a><br /><br />Notice that each of these lines has a single peak. The first individual's line peaks at alternative D; the second's at alternative A. If you do a little experimenting, you will find that if you change the order in which the alternatives are laid out on the X-axis, sometimes both lines are still single peaked, sometimes one remains single peaked and one no longer is. Sometimes neither is single peaked. Fr example, if you change the order slightly so that the alternatives are laid out on the X-axis in the order a B E D C, the first individual's line will still be single peaked, but the second individual's line will now be in the shape of a V with one peak at A and another peak at C. [Try it and see. It is too much trouble for me to draw it and scan it and size it and insert it.]<br /><br /> Suppose now that we have an entire voting population, each with his or her own preference order, and that we plot all of those preference orders on a single graph, a separate line for each person. There might be some way of arranging the alternatives along the X-axis so that everyone's preference order, when plotted on that graph, is single peaked. Then again, there might not be. For example, if you have three people and three alternatives, and if those three people have the preferences that give rise to the Paradox of Majority Rule, then there is no way of arranging the three alternatives along the X-axis so that all three individuals' preferences orders can be plotted on that graph single-peakedly. [Try it and see. Remember that mirror images are equivalent for these purposes, so there are really three possible ways of arranging the alternatives along the X-axis, namely xyz, xzy, and yxz.]<br /><br /> Duncan Black proved that if there is some way of arranging the available alternatives along the X-axis so that everyone's preference order, plotted on that graph, is single peaked, then majority rule is guaranteed to produce a consistent social preference order. Notice, in particular, that if everyone's preferences can be mapped onto the familiar left-right spectrum, with each individual preferring an alternative less and less the farther away it is in either direction from the most preferred alternative, then everyone will on that graph have a single peaked order [because it will peak at the most preferred alternative and fall away monotonically to the right and to the left.]<br /><br /> The proof is fairly simple. It goes like this.<br /><br />Step 1: Assume that there are an odd number of individuals [the proof works for an even number of individuals, but in that case there can be ties, which produces social indifference, which then requires an extra couple of steps in the proof, so I am trying to make this as simple as possible.] Assume that their preferences can be plotted onto a graph so that all of the plots are single-peaked.<br /><br />Step 2: Starting at the left, count peaks [there may be many peaks at the same point, of course, showing that all of those people ranked that alternative as first] and keep counting until you reach one more than half of the total number of peaks, i.e. (n/2 + 1). Assume there are p peaks to the left of that point, q peaks at that point, and r peaks to the right, with (p+q+r) = n. Now, by construction, (p+q) > n/2 and p<n/2. Therefore, it must also be the case that (q+r)>n/2, because if (q+r)<n/2 then p>n/2, which by construction it is not.<br /><br />Step 3: Let us call the alternative with the q peaks alternative x. Clearly, there is a majority of individuals who prefer x to every alternative to the right of x on the graph, because there are p+q individuals whose plots are downward sloping from x as you go to the right, which means they prefer x to everything to the right, and p+q is a majority. But there are q+r individuals who prefer x to everything to the left of x, because their plots are downward sloping as you go to the left, and q+r are a majority. So alternative x is preferred in a pairwise comparison by a majority to every other alternative.<br /><br />Step 4: Remove alternative x from the graph, remove alternative x from everyone's preference order, and then redraw all of the plots. They will all still be single-peaked. Why? Well, there are three possible cases: Either the dot representing the individual's ranking of x was the peak, or it was to the left of the peak, or it was to the right. In each case, when you reconnect the remaining dots, the graph remains single-peaked [try it and see. It is too hard to draw it and scan it and upload it. But it is intuitively obvious.]<br /><br />Step 5: You now have a new set of single-peaked plots on a single graph, so go through Steps 2 and 3 all over again. The winning alternative is preferred to every other remaining alternative, and is of course inferior to the first winner. If you now iterate this process until you run out of alternatives, you are left with a fully transitive social preference established by repeated uses of majority rule.<br /><br /> Black's theorem has considerable real world application, as we have seen, but it of course does not identify necessary and sufficient conditions for majority rule to produce a transitive social preference order. It only identifies a sufficient condition, namely single-peakedness. This means that there are sets of individual preferences that cannot be mapped single-peakedly onto a single graph, and yet which by majority rule produce transitive social preference orders. I leave it to you to construct an example of this.Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com6tag:blogger.com,1999:blog-4978426466915379555.post-55395323229962603102010-07-26T01:55:00.001-04:002010-07-26T01:55:14.252-04:00COLLECTIVE C HOICE FIFTH INSTALLMENT<span xmlns=''><p><span style='text-decoration:underline'>Proof of Arrow's Theorem</span><br /> </p><p>Step 1. By Condition P, there is at least one decisive set for each ordered pair, namely the set of all the individuals. From all the decisive sets, choose a smallest decisive set, V, and let it be decisive for some ordered pair (x,y). What I mean is this: Consider each set of individuals that is decisive for some ordered pair or other. Since there is a finite number of individuals, each of these sets must have some finite number of individuals in it. And the sets may have very different numbers of individuals in them. But one or more of them must be the smallest set. So arbitrarily choose one of the smallest, call it V, and label the pair of alternatives over which it is decisive (x,y).<br /></p><p>Step 2: By Condition P, V cannot be empty. [Go back and look at Condition P and make sure you see why this is so. It is not hard]. Furthermore, by Lemma 3, V cannot have only one member [because Lemma 3 proved that no single individual, i, can be decisive for any ordered pair (x,y) ]. Therefore, V must have at least two members.<br /></p><p>Step 3: Partition the individuals 1, 2, ......, n in the following way:<br /></p><p> The set of all individuals<br /></p><p> ---------------------------------------------------------------<br /></p><p> | |<br /></p><p> V V<sub>3</sub><br /> </p><p> ---------------------<br /></p><p> V<sub>1</sub> V<sub>2</sub><br /> </p><p> Where V<sub>1</sub> = a set containing exactly one individual in V<br /></p><p> V<sub>2</sub> = the set of all members of V except the one individual in V<sub>1</sub><br /> </p><p> V<sub>3</sub> = the rest of the individuals, if there are any.<br /></p><p> Is this clear? V is a smallest decisive set. It must have at least two individuals in it. So it can be divided into V<sub>1</sub> containing just one individual, and V<sub>2</sub> containing the rest of V. V<sub>3</sub> is then everyone else, if there is anyone else not in the smallest decisive set V.<br /></p><p>Step 4: Now let the individuals in the society have the following rankings of three alternatives, x, y, and z. [And now you will see how this is an extension of the original Paradox of Majority Rule with which we began.]<br /></p><p> V<sub>1</sub>: x > y and y > z<br /></p><p> V<sub>2</sub>: z> x and x> y<br /></p><p> V<sub>3</sub>: y>z and z>x<br /></p><p>[You see? This is one of those circular sets of preference orders: xyz, zxy, yzx]<br /></p><p> V<sub>1</sub> is non-empty, by construction.<br /></p><p> V<sub>2</sub> is non-empty, by the previous argument.<br /></p><p> V<sub>3</sub> may be empty.<br /></p><p>Step 5: a) By hypothesis, V is decisive for x against y. But V is the union of V<sub>1</sub> and V<sub>2</sub>, and xP<sub>i</sub>y for all i in V<sub>1</sub> and V<sub>2</sub>. Therefore, xPy. [i.e., the society prefers x to y.]<br /></p><p> b) For all i in the union of V<sub>1</sub> and V<sub>3</sub>, yP<sub>i</sub>z. For all j in V<sub>2</sub>, zP<sub>j</sub>y. If zPy, then V<sub>2</sub> is decisive for (x,y). But by construction, V<sub>2</sub> is too small to be decisive for anything against anything, because V<sub>2</sub> is one individual smaller than a smallest decisive set, V. Therefore not zPy. Hence, yRz [see the definitions of P and R].<br /></p><p> c) Therefore xPz by Lemma 1(f) [go back and look at it].<br /></p><p> d) But xP<sub>1</sub>z and zP<sub>i</sub>x for all i not in V<sub>1</sub>, so it cannot be that xPz, because that would make V<sub>1</sub> decisive for (x,z), which contradicts Lemma 3. Therefore, not xPz.<br /></p><p>Step 6: The conclusion of Step 5d) contradicts Step 5c). Thus, we have derived a contradiction from the assumption that there is a Social Welfare Function that satisfies Conditions 1', 3, P, and 5. Therefore, there is no SWF that satisfies the four Conditions. Quod erat demonstrandum. <br /></p><p> OK. Everybody, take a deep breath. This is a lot to absorb. Arrow's Theorem is a major result, and it deserves to be studied carefully. Go back and re-read what I have written and make sure you understand every step. It is not obscure. It is just a little complicated. If you have questions, post them as a comment to this blog and I will answer them.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com2tag:blogger.com,1999:blog-4978426466915379555.post-87669086866917178642010-07-23T04:53:00.001-04:002010-07-23T04:53:23.444-04:00COLLECTIVE CHOICE FOURTH INSTALLMENT<span xmlns=''><p> <strong>[end of Arrow third installment]</strong><br /> </p><p>Proof of Lemma 3: Assume xDy for i [i,e,, i is decisive for x against y]<br /></p><p> The proof now proceeds in two stages. First, for an environment [x,y,z], constructed by adding some randomly chosen third element z to x and y, we show that i is a dictator over [x,y, z].<br /></p><p> Then we show how to extend this result step by step to the conclusion that i is a dictator over the entire environment S of admissible alternatives.<br /></p><p>First Stage: Proof that i is a dictator over the environment [x,y,z]<br /></p><p>(step i) Construct a set of individual orderings over [x,y,z] as follows.<br /></p><p> R<sub>i</sub>: x > y > z [i.e., individual i's ordering of the three]<br /></p><p> All the other R<sub>j</sub>: yP<sub>j</sub>x yP<sub>j</sub>z R<sub>j</sub>[x,z] unspecified<br /></p><p> In other words, we will prove something that is true regardless of how everyone other than i ranks x against z.<br /></p><p>(step ii) xP<sub>i</sub>y by construction. But, by hypothesis xDy for i. Therefore xPy<br /></p><p> In words, i is assumed to strongly prefer x to y, and since by hypothesis i is decisive for x against y, the society also strongly prefers x to y.<br /></p><p>(step iii) For all i, yP<sub>i</sub>z, by construction Therefore, yPz, by Condition P, and xPz by Lemma 1(c). In words, since everyone strongly prefers y to z, so does the society. And since the society strongly prefers x to y and y to z, it strongly prefers x to z [since Axiom II, which is used to prove Lemma 1(c), stipulates that the SWF is transitive.]<br /></p><p>(step iv) So xPz when xP<sub>i</sub>z, regardless of how anyone else ranks x and z. [check the construction of the individual orderings in step (i) ]<br /></p><p>(step v) Hence xḎz for i, which is to say that i dictates over the ordered pair (x,z)<br /></p><p>(step vi) Now consider (y,z) and assume the following set of individual orderings:<br /></p><p> R<sub>i</sub>: y > x > z<br /></p><p> All the other R<sub>j</sub>: yP<sub>j</sub>x zP<sub>j</sub>x and R<sub>j</sub>[y,z] unspecified.<br /></p><p>(step vii) yP<sub>i</sub>x for all i. Therefore yPx by Condition P<br /></p><p>(step viii) xḎz for i, by (v). Hence xPz.<br /></p><p>(step ix) So yPz by Lemma 1(c). Thus yḎz for i.<br /></p><p> In words, we have now shown that i dictates over the ordered pair (y,z). Let us take a minute to review what is going on here. We are trying to prove that if i is decisive for a single ordered pair, (x,y), then i is a dictator over an environment consisting of x, y, and some randomly chosen z. If we can show that i is a dictator for every ordered pair in the environment [x,y,z] then we shall have shown that i is a dictator over that environment. There are six ordered pairs that can be selected from the environment, namely (x,y), (x,z), (y,x), (y,z), (z,x), and (z,y). So we must establish that i dictates over every single one of these ordered pairs. We have already established that i dictates over (x,z) in step (v) and over (y,z) in step (ix).<br /></p><p>(step x) We can now extend this argument to the other four ordered pairs that can be selected from the environment [x,y,z]. In particular, let us do this for the ordered pair (y,x). Construct the following set of orderings:<br /></p><p> R<sub>i</sub>: y > z > x<br /></p><p> All the other R<sub>j</sub>: zP<sub>j</sub>y zP<sub>j</sub>x R<sub>j</sub>[x,y] unspecified.<br /></p><p>(step xi) zP<sub>i</sub>x for all i. Hence zPx by Condition P<br /></p><p>(step xii) yḎz for i by (step ix). Hence yPz<br /></p><p>(step xiii) So yPx by Lemma 1(c). Thus yḎx for i.<br /></p><p> So we have proved [or can do so, by just iterating these steps a few more times] that i dictates over every ordered pair in [x,y,z], and therefore i is a dictator over the environment [x,y,z]. So much for Stage One of the proof of Lemma 3. Now, take a deep breath, review what has just happened to make sure you understand it, and we will continue to:<br /></p><p>Stage Two: The extension of our result to the entire environment, S, of available alternatives. Keep in mind that S, however large it may be, is finite.<br /></p><p> Assume xDy for i [our initial assumption -- just repeating for clarity] and also assume the result of Stage One. Now consider any ordered pair of alternatives (z,w) selected from the environment S. There are just seven possibilities.<br /></p><p> 1. x = z w is a third alternative<br /></p><p> 2. x = w z is a third alternative<br /></p><p> 3. y = z w is a third alternative<br /></p><p> 4. y = w z is a third alternative<br /></p><p> 5. x = z y = w<br /></p><p> 6. y = z x = w<br /></p><p> 7. Neither z nor w is either x or y<br /></p><p>Case 1: We have an environment consisting of three alternatives: [x=z, y, w]. Stage One shows that if xDy for i, then x=zḎw for i.<br /></p><p>Case 2, 3, 4: Similarly<br /></p><p>Case 5: Trivial<br /></p><p>Case 6: Add any other element v to form the environment [x=w, y=z, v]. From x=wDy=z for i, it follows that y=zḎx=w for i. [In words, just in case you are getting lost: In the case in which y is element z and x is element w, from the fact that i is decisive for x against y, which is to say for w against z, , it follows that i dictates over y and x, which is to say over z and w. This is just a recap of Stage One.<br /></p><p>case 7: This is the only potentially problematic one case, and it needs a little explaining. We are starting from the assumption that i is decisive for x against y, and we want to show that i is a dictator over some totally different of alternatives z and w, so we are going to creep up on that conclusion, as it were. First we will add one of those two other alternatives, z, to the two alternatives x and y to form the environment [x,y,z]. From Stage One, if xDy for i then xḎz for i. But trivially, since xḎz for i, it follows that xDz for i. [The point is that if i dictates over x and z, then of course i is decisive for x against z].<br /></p><p> Now add w to x and z to form the environment [x,z,w]. Since xDz for i, it follows that zḎw for i, by Stage One. In words, if i is decisive for x against z, then in the environment [x,z,w], i dictates over z and w. This follows from Stage One. What this shows is just how powerful Lemma 3 really is.<br /></p><p> Thus we have demonstrated that xDy for i implies zḎw for i, for all z and w in S. In other words, if i is decisive for some ordered pair (x,y), then i is a dictator over S. But Condition 5 stipulates that no individual may be a dictator. Therefore:<br /></p><p> An acceptable Social Welfare Function does not permit any individual to be decisive for even a single ordered pair of alternatives in the environment S of available alternatives.<br /></p><p> Can we all say Ta-Da? This is the heavy lifting in Arrow's theorem. Using this Lemma, we can now fairly quickly prove that there is no SWF satisfying Axioms I and II and all four Conditions, 1', 3, P, and D.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com3tag:blogger.com,1999:blog-4978426466915379555.post-17433928625716419872010-07-21T06:40:00.002-04:002010-07-23T04:48:43.750-04:00COLLECTIVE CHOICE THEORY THIRD INSTALLMENT<span xmlns=''><p>This is really a devastating theorem. Basically, it says that there is no voting mechanism that gets around the Paradox of Majority Rule. The proof proceeds as follows. First Arrow states a set of little results about the relations R, I, and P. You are already familiar with them. They are trivial, as we shall see. Then he proves a little Lemma about the choice function. Then he proves a big important Lemma that is really the guts of the theorem. Finally, he uses the Lemmas to prove what is essentially an extension of the Paradox of Majority Rule, and he is done. We are going to go through this slowly and carefully. Let us start with the two little lemmas. Lemma 1 and Lemmas 2.<br /></p><p>Lemma 1: (a) For all x, xRx<br /></p><p> (b) If xPy then xRy<br /></p><p> (c) If xPy and yPz then xPz<br /></p><p> (d) If xIy and yIz then xIz<br /></p><p> (e) For all x and y, either xRy or yPx<br /></p><p> (f) If xPy and yRz then xPz<br /></p><p> These all follow immediately from the definitions of R, I, and P, the assumptions of transitivity and completeness, and truth functional logic. Arrow includes them as an omnibus Lemma because at one point or another in his proof he will appeal to one or another of them. You should work through all the little proofs as an exercise. I will go through just one to show you what they look like.<br /></p><p> (e) xRy or yRx [completeness]<br /></p><p> So if not xRy, then yRx.<br /></p><p> But the definition of yPx is yRx and not xRy<br /></p><p> Therefore, either xRy or yPx<br /></p><p>Lemma 2: xPy if and only if x is the sole element of C([x,y])<br /></p><p> If you review the definition of the Choice set, you will see that this Lemma is intuitively obvious. It says that in the little environment, S, consisting of nothing but x and y, if xPy, then x is the only element in the Choice set, C(S). Since this is a bi-conditional [if and only if], we have to prove it in each direction.<br /></p><p> a. Assume xPy. Then xRy, by Lemma 1(b). [See, this is why he put those little things in Lemma 1]. Furthermore, xRx, by Lemma 1(a). So x is in C([x,y]), because it is at least as good [i.e., R] as each of the elements of S, namely x and y. But if xPy then not yRx. Therefore, y is not in C([x,y]). So x is the sole element of C([x,y]).<br /></p><p> b. Assume x is the sole element of C([x,y]). Since y is not in C([x,y]), not yRx. Therefore xPy.<br /></p><p>Lemma 3: If an individual, i, is decisive for some ordered pair (x,y) then i is a dictator.<br /></p><p> This is a rather surprising and very important Lemma. It is the key to the proof of Arrow's theorem, and shows us just how powerful the apparently innocuous Four Conditions really are. To understand the Lemma, you must first know what is meant by an ordered pair and then you must be given three definitions, including one for the notion of "decisive."<br /></p><p> Easy stuff first. An ordered pair is a pair in a specified order. An ordered pair is indicated by curved parentheses. Thus, the ordered pair (x,y) is the pair [x,y] in the order first x then y. As we shall see, to say that individual is decisive for some ordered pair (x,y) is to say that i can, speaking informally, make the society choose x over y regardless of what anyone else thinks. But a person might be decisive for x over y and not be decisive for y over x. We shall see in a moment how all this works out. Now let us turn to the three definitions that Arrow is going to make use of in the proof of Lemma 3.<br /></p><p> Definition 1: "A set of individuals V is decisive for (x,y)" =<sub>df</sub> "if xP<sub>i</sub>y for all i in V and yP<sub>j</sub>x for all j not in V, then xPy"<br /></p><p> In other words, to say that a set of individuals V is decisive for the ordered pair (x,y) is to say that if everyone in V strongly prefers x to y, and everyone not in V strongly prefers y to x, then the society will strongly prefer x to y. Under majority rule, for example, any set of individuals V that has at least one more than half of all the individuals in the society in it is decisive for every ordered pair of alternatives (x,y). <br /></p><p> Definition 2: "xḎy for i" or "i dictates over (x,y)" =<sub>df</sub> "If xP<sub>i</sub>y then xPy"<br /></p><p> In words, we say that individual i dictates over the ordered pair (x,y) if whenever individual i strongly prefers x to y, so does the society regardless of how everyone else ranks x and y. [Notice that the capital letter D has a little line underneath it.]<br /></p><p> Definition 3: "xDy for i" or "i is decisive for (x,y)" =<sub>df</sub> "If xP<sub>i</sub>y, and for all j not equal to i, yP<sub>j</sub>x, then xPy."<br /></p><p> In words, i is said to be decisive for the ordered pair (x,y) if when i strongly prefers x to y and everyone else strongly prefers y to x, the society prefers x to y. [Notice that in this definition, the capital letter D does not have a little line underneath it.]<br /></p><p> Ok. Now we are ready to state and prove the crucial Lemma 3.<br /></p><p>Lemma 3: If xDy for i, then zḎw for i, for all z,w in S<br /></p><p> In words, what this says is that if any individual, i, is decisive for some ordered pair (x,y) then that individual i is a dictator [i.e., dictates over any ordered pair (z,w) chosen from S]. This is an astonishing result. It says that if the Social Welfare Function allows someone to compel the society to follow her ranking of some ordered pair, no matter what, against the opposition of everyone else, then the Social Welfare Function makes her an absolute dictator. [<em>L'ėtat c'est moi</em>]. Here is the proof. It is going to take a while, so settle down. In order to make this manageable, I must use the various symbols we have defined. Let me review them here, so that I do not need to keep repeating myself.<br /></p><p> An ordered pair is indicated by curved parentheses: (x,y), as opposed to a non-ordered pair, which is indicated by brackets: [x,y].<br /></p><p> xḎy for i, which is D with a line under it, means "i dictates over (x,y)" (an ordered pair)<br /></p><p> xDy for i, which is D with no line under it, means "i is decisive for (x,y)"<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com3tag:blogger.com,1999:blog-4978426466915379555.post-26902244548218443922010-07-19T03:06:00.001-04:002010-07-19T03:06:12.958-04:00COLLECTIVE CHOICE THEORY SECOND INSTALLMENT<span xmlns=''><p>From here on, I am going to break the exposition into short bits, because this is hard, and I do not want to lose anyone. My apologies to those of you who are having no trouble following it. <br /></p><p> First of all, notice that Arrow assumes only ordinal preference. This means that there is no way in the proof to take account of intensity of preference, only order of preference. Let me give an example to make this clearer. In 1992, George H. W. Bush, Bill Clinton, and H. Ross Perot ran for the Presidency. There were some devoted followers of Perot who were crazy about him, and almost indifferent between Bush and Clinton, whom they viewed as both beltway politicians. Let us suppose that one of these supporters ranked Perot first, way ahead of the other two, and gave the edge slightly to Bush over Clinton, perhaps because Bush was a Republican. A second Perot supporter might have been rather unhappy with the choices offered that year, but preferred Perot slightly over Bush, while hating Clinton passionately. From Arrow's perspective which is that of ordinal preference, these two voters had identical preferences, namely Perot > Bush > Clinton, and an Arrovian SWF would treat the two individual preference orders as interchangeable. <br /></p><p> Now, there are many ways in which citizens in America can give expression to the intensity of their preferences, as political scientists are fond of pointing out. One is simply by bothering to vote. Voter enthusiasm, in a nation half of whose eligible voters routinely fail to go to the polls, is a major determinant of the outcome of elections. A second way is by contributing to campaigns, volunteering for campaign work, and so forth. Yet another way is through a vast array of voluntary organizations dedicated to pursuing some issue agenda or advantaging some economic or regional group. <em>None of this can find expression in the sort of Social Welfare Function Arrow has defined.</em> This is a very important limitation on the method of collective decision that we call voting. Now, there are voting schemes that allow voters to give expression to the intensity of their preferences [such as giving each voter a number of votes, which he or she can spread around among many candidates or concentrate entirely on one candidate], but these too are ruled out by Arrow, who only allows the SWF to take account of individual ordinal preferences.<br /></p><p> The second thing to note is that the requirement of completeness placed upon the SWF rules out partial orderings, such as those established by Pareto-Preference. It is often the case that every individual in the society prefers some alternative x to some other alternative y, and if there are a number of such cases, a robust partial ordering might be established that, while not complete, nevertheless allows the society to rank a sizeable number of the available alternatives. This option too is ruled out by Arrow's two axioms. These observations have the virtue of helping us to understand just how restricting a collective decision-making apparatus like majority rule is.<br /></p><p> We are now ready to state the four conditions that Arrow defines as somehow capturing the spirit of majoritarian democracy. Arrow's theorem will simply be the proposition that there is no Social Welfare Function, defined as he has in the materials above, that is compatible with all four conditions. In the original form of the proof, the conditions were, as you might expect, called Conditions 1, 2, 3, 4, and 5. In the revised version, which I shall be setting forth here, they are called Conditions 1' [a revised version of Condition 1], Condition 3 [which also is sometimes called the Independence of Irrelevant Alternatives], Condition P [for Pareto], and Condition 5. Here they are. I will tell you now that Condition 3 is the kinky one.<br /></p><p><span style='text-decoration:underline'>Condition 1'</span>: All logically possible rankings of the alternative social states are permitted. This is a really interesting condition. What it says, formally speaking, is that each individual may order the alternatives, x, y, z, ... in any consistent way. What it rules out, not so obviously, is any religious or cultural or other constraint on preference. For example, if among the alternatives are various dietary rules, or rules governing abortions, or rules governing dress, nothing is ruled in or ruled out. The individuals are free to rank alternatives in any consistent manner.<br /></p><p><span style='text-decoration:underline'>Condition 3</span>: Let R<sub>1</sub>, R<sub>2</sub>, ......, R<sub>n</sub> and R<sub>1</sub>', R<sub>2</sub>', .... R<sub>n</sub>' be two sets of individual orderings of the entire set of alternatives x, y, z, .... and let C(S) and C'(S) be the corresponding social choice functions. If, for all individuals i and all alternatives x and y in a given environment S, xR<sub>i</sub>y if and only if xR<sub>i</sub>'y, then C(S) and C'(S) are the same.<br /></p><p> OK, this is confusing, so let us go through it slowly step by step and figure out what it means. To get to the punch line first, this condition says that the society's eventual identification of best elements in an environment is going to be determined solely by the rankings by the individuals of the alternatives in that environment, and not by the rankings by the individuals of alternatives not in the environment. [Remember, the Environment, S, is a subset of all the possible alternatives.] Now, take the condition one phrase at a time. First of all, suppose we have two different sets of individual rankings of all the alternatives. The first set of rankings is the R<sub>i</sub> [there are as many rankings in the set as there are individuals -- namely, the first individual's ranking, R<sub>1</sub>, the second individual's ranking, R<sub>2</sub>, and so forth.] The second set of rankings is the R<sub>i</sub>', which may be different from the first set. <br /></p><p> Now, separate out some subset of alternatives, which we will call the Environment S, and focusing only on the alternatives in S, take a look at the way in which the individuals rank those alternatives, ignoring how they rank any of the alternatives left out of S. If the two sets of individual orderings, R<sub>i</sub> and R<sub>i</sub>', are exactly the same for the alternatives in S, then when the Social Welfare Function cranks out a social ranking, R, based on the individual orderings R<sub>i</sub> and a social ranking, R', based on the individual orderings R<sub>i</sub>', Condition 3 stipulates that the set of best elements [The Social Choice set] will be the same for R and for R'.<br /></p><p> Whew, that still isn't very clear, is it? So let us ask the obvious question: What would this Condition rule out? Here is the answer, in the form of an elaborate example. Just follow along.<br /></p><p> Suppose that in the 1992 presidential election, there are just three voters, whom we shall call 1, 2, and 3. Also, suppose there are a total of four eligible candidates: George H. W. Bush, Bill Clinton, H. Ross Perot, and me. Now suppose there are two alternative sets of the rankings of these four candidates by individuals 1, 2, and 3.<br /></p><p>R<sub>i</sub>: Individual 1: Wolff > Clinton > Bush > Perot<br /></p><p> Individual 2: Bush > Perot > Wolff > Clinton<br /></p><p> Individual 3: Wolff > Clinton > Bush > Perot<br /></p><p>R<sub>i</sub>': Individual 1: Clinton > Bush > Perot > Wolff<br /></p><p> Individual 2: Bush > Perot > Clinton > Wolff<br /></p><p> Individual 3: Clinton > Bush > Perot > Wolff<br /></p><p> The crucial thing to notice about these two alternative sets of rankings is that they are identical with regard to the environment S = (Bush, Clinton, Perot). The only difference between the two sets is that in the second set, Wolff has been moved to the bottom of everyone's list. [The voters find out I am an anarchist.]<br /></p><p> Now let us consider the following Social Welfare Function: For each individual ranking, assign 10 points to the first choice, 7 points to the second choice, 3 points to the third choice, and 2 points to the fourth choice. Then, for any Environment, S, selected from the totality of available alternatives, determine the social ranking by adding up all of the points awarded to each alternative by the individual rankings. Got it?<br /></p><p> Go ahead and carry out that exercise. If you do, you will find that for the first set of rankings, the R<sub>i</sub>, and for the Environment S = (Bush, Clinton, Perot). the SWF gives 16 points to Clinton, 16 points to Bush, and 11 points to Perot. So, C(S), the society's decision as to which candidates are at the top, is (Clinton, Bush), because they each have the same number of points, namely 16. But if you now carry out the same process with regard to the second set of individual rankings, the R<sub>i</sub>', and the same Environment S, you will discover that the SWF assigns 23 points to Clinton, 24 points to Bush, and 13 points to Perot, which means that C'(S) is (Bush). So the social choice in the Environment S has changed, despite the fact that the relative rankings of the elements in S have not changed, because of a change in the rankings of an element not in S, namely Wolff. And this is just what Condition 3 rules out. It says that the Social Welfare Function cannot be one that could produce a result like this.<br /></p><p> All of us are familiar with this sort of problem from sports meets or the Olympics. When we are trying to decide which team or country has done best, we have to find some way to add up Gold medals and Silver medal and Bronze medals, and maybe fourth and fifth places as well. And, as we all know, you get different results, depending on how many points you award for each type of medal. Arrow's Condition 3 rules out SWFs like that.<br /></p><p><span style='text-decoration:underline'>Condition P</span>: If xP<sub>i</sub>y for all i, then xPy. This just says that if everyone strongly prefers x to y, so does the society. This is a very weak constraint on the SWF.<br /></p><p><span style='text-decoration:underline'>Condition 5</span>: The Social Welfare Function is not dictatorial.<br /></p><p> Remember the definition of "dictatorial" above. This rules out "l'état c'est moi" as a Social Welfare Function. <br /></p><p> So, we have the definitions, etc., and we have the four Conditions that Arrow imposes on a Social Welfare Function. Remember that a Social Welfare Function is <em>defined</em> as a mapping that produces a social ranking that satisfies Axioms I and II. Now Arrow is ready to state his theorem. It is quite simple:<br /></p><p> <strong>There is no Social Welfare Function that satisfies the four Conditions</strong>.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0tag:blogger.com,1999:blog-4978426466915379555.post-15904700396898369202010-07-16T05:54:00.003-04:002010-07-17T11:34:45.903-04:00COLLECTIVE CHOICE THEORY FIRST INSTALLMENT<span xmlns=''><p style='text-align: center'><span style='font-size:14pt'><strong>Part Four: Collective Choice Theory</strong><br /> </span></p><p> Collective Choice Theory is the theory of how one selects a rule to go from a set of individual preference orders over alternatives available to a society of those individuals to a collective or social preference order over those same alternatives. [Or, as they say in the trade, how to "map a set of individual preference orders onto a social preference order."] There is a long history of debates about how to make social or collective decisions, going back at least two and a half millennia in the West. The simplest answer is to identify one person in the society and stipulate that his or her preference order will <strong>be</strong> the social preference order. <em>L'etat, c'est moi,</em> as Louis XIV is reputed to have said. A variant of this solution is the ancient Athenian practice of rotating political positions. One can also choose a person by lot whose preferences will thereupon become the social preference. A quite different method is that used by the old Polish parliament, which consisted of all the aristocrats in the country [there were quite a few, the entry conditions for being considered an aristocrat being low]. Since each of them thought of himself as answerable only to God, they imposed a condition of unanimity on themselves. If as few as one Polish aristocrat objected to a statute, it did not become law.<br /></p><p> These rules for mapping individual preference orders onto a social preference order, unattractive as they may be on other grounds, all have one very attractive feature in common: They guarantee that if all of the individual preference orders are <strong>ordinal orderings</strong>, which is to say if each of them is complete, reflexive, and transitive [you see, I told you we would use that stuff], then the social preference order will also be an <strong>ordinal ordering</strong>, and that is something you really, really want. You want it to be complete, so that it will tell you in each case how to choose. And you want it to be transitive, so that you do not get into a situation where your <strong>Collective Choice Rule</strong> tells the society to choose a over b, b over c, and c over a.<br /></p><p> To sum it all up in a phrase, the aim of Collective Choice Theory is to find a way of mapping <strong>minimally</strong><br /> <strong>rational individual preferences</strong> onto a <strong>minimally rational social preference.</strong><br /> </p><p> For the past several hundred years, everybody's favorite candidate for a Collective Choice Rule has been <strong>majority rule</strong>. This is a rule that says that the social preference between any two alternatives is to be decided by a vote of all those empowered to decide, with the alternative gaining a majority of the votes being preferred over the alternative gaining a minority of the votes. Should two alternatives, in a pairwise comparison, gain exactly the same number of votes, then the society is to be <strong>indifferent</strong> between the two. <br /></p><p> Enter the Marquis de Condorcet, who published an essay in 1785 called [in English] <em>Essay on the Application of Analysis to the Probability of Majority Decisions</em>. In this essay, Condorcet presented an example of a situation in which a group of voters, each of whom has perfectly rational preferences over a set of alternatives, will, by the application of majority rule, arrive at an <strong>inconsistent</strong> group or social preference. This is, to put it as mildly as I can, a tad embarrassing. Indeed, it calls into question the legitimacy of majority rule, which lies at the heart of every variant of democratic theory that had been put forward at that time, or indeed has been put forward since.<br /></p><p> Let us take a moment to set out the example and examine it. In its simplest form, it involves three voters, whom we shall call X, Y, and Z, and three alternatives, which we shall call a, b, and c. We may suppose that a, b, and c are three different tax plans, say. Let us now assume that the three voters have the following preferences over the set of alternatives S = (a, b, c).<br /></p><p> X prefers a to b and b to c. Since X is minimally rational, he also prefers a to c.<br /></p><p> Y prefers b to c and c to a. Since she is also minimally rational, she prefers b to a.<br /></p><p> Z prefers c to a and a to b. As rational as X and Y, she naturally prefers c to b.<br /></p><p> Now they take a series of pairwise votes to determine the collective or social preference order among the three alternatives. When they vote for a or b, X and Z vote for a, Y votes for b. Alternative a wins. When they vote for b or c, X and Y vote for b, Z votes for c, alternative b wins. Now, if the social ordering is to be <strong>transitive</strong>, then the society must prefer a to c. What happens when X, Y, and Z choose between a and c? X prefers a to c. But Y and Z both prefer c to a. So the society must, by majority rule, prefer c to a. Whoops. The society's preference order violates transitivity.<br /></p><p> And that is the whole story. The selection of a social or collective preference order by majority rule cannot guarantee the transitivity of the social preference order, and therefore does not even meet the most minimal test of rationality. There are, of course, lots and lots of sets of individual preference orders that generate a consistent social preference order when Majority Rule is applied to them. The problem is that here is at least one, and actually many more, that are turned by Majority Rule into an inconsistent preference order.<br /></p><p> If you have never encountered this paradox before [the so-called <strong>paradox of majority rule</strong>], you may be inclined to think that it is a trick or a scam or an illusion. Alas, not so. It is just as it appears. Majority Rule really is capable of generating an inconsistent social preference ordering.<br /></p><p> All of this was well known in the eighteenth century, and was, as we shall see later on, the subject of some imaginative elaboration by none other than the Reverend Dodgson, better known as Lewis Carroll. Enter now the young, brilliant economist Kenneth Arrow in the middle of the twentieth century. Coming out of a tradition of economic theorizing called Social Welfare Economics, to which a number of major figures, such as Abram Bergson, had contributed, Arrow conceived the idea of analyzing the underlying structure of the old Paradox of Majority Rule and generalizing it. The result, which he presented in his doctoral dissertation no less, was The General Possibility Theorem. Arrow published the theorem in 1951 in a monograph entitled <em>Social Choice and Individual Values</em>. <br /></p><p> Another great economist and fellow Nobel Prize winner, Amartya Sen, in 1970 published <em>Collective Choice and Social Welfare</em>, in which he generalized and extended Arrow's work in astonishing ways. Sen's book is difficult, but it is simply beautiful, and deeply satisfying. I strongly urge you, if you have a taste for this sort of thing, to tackle it. Sen has written widely and brilliantly on a host of extremely important social problems, including economic inequality, famine, and the demographic imbalance between men and women in the People's Republic of China. His little series of Radcliffe Lectures, published in 1973 as <em>On Economic Inequality</em>, is the finest use of formal methods to illuminate and analyze a social problem of which I am aware. It is a perfect example of the <em>proper</em> use of formal methods in social philosophy, and as such deserves your attention.<br /></p><p> In <em>Collective Choice and Social Welfare</em>, Sen gives a simpler and more elegant proof of Arrow's General Possibility Theorem. Nevertheless, I have chosen in this blog to expound Arrow's original proof. Let me explain why. It often happens that the first appearance of an important new theorem is somewhat clumsy, valid no doubt, but longer and more complicated than necessary. Later theorists refine it and simplify it until what took many pages can be demonstrated quickly in a few lines. Sometimes, this development is unambiguously better, but at other times, the original proof, clumsy though it may be, reveals the central idea more perspicuously than the later simplifications do. I find this to be true in the case of Arrow's theorem. Sen's simplification serves several purposes, not the least of which is to set things up formally for his extremely important extension and elaboration of Arrow's work. Therefore, I urge you to look at it, once you have worked with me through Arrow's original proof.<br /></p><p> Now let us begin. This is going to take a while, so settle down. Before we get into the weeds, let me try to explain in general terms what Arrow is doing. He asks, in effect, what are the underlying general assumptions of majoritarian decision making? What is it about voting with majority rule that appeals to us? He identifies five conditions or presuppositions [later reduced and simplified to four] that capture the logic of majority rule in a general way, and then shows that <em>no </em>way of making collective decisions that satisfies all four of them guarantees that the resulting social or collective choice will be consistent. This way of thinking about the problem accomplishes three things simultaneously. First, it unpacks majority rule voting into its component parts so that we can look at it and understand it better. Second, it generalizes the Paradox of Majority Rule so that we realize we cannot avoid it simply by tweaking Majority Rule a bit [for example by requiring a two-thirds majority.] And finally, it allows us to see just exactly what Majority Rule does <em>not</em> do -- in other words, it gives us insight into what would be totally different ways of making collective decisions.<br /></p><p> We start with a series of assumptions, definitions, and notational conventions, some of which are already familiar to you from the opening segments of this general tutorial. This is going to be tedious, but learning these up now will make it infinitely easier to follow the proof. Here they are:<br /></p><p>(a) We start with a set of mutually exclusive alternatives, x, y, z, ..... These may be all of the possible candidates in an election [i,.e., every single person who is eligible to hold office under the rules governing the election], every possible tax scheme that might come before Congress, all of the various possible decisions a City Council might take concerning zoning regulations, and so forth. The point of the phrase "mutually exclusive" is to rule out, for example, "Obama" and "Obama or Clinton" as two of the available alternatives.<br /></p><p>(b) On any give occasion when a decision is to be made, there is a subset, S, of the available alternatives, which will be called The Environment. This might be, for example, the relatively small number of people who have stated publicly that they would like to be elected to that office, or all the people who have formed campaign committees, or all the people who survive the primary season and are on the final ballot. Each of these is a subset of all the people eligible to hold the office [not necessarily a proper subset -- i.e., not necessarily smaller than the total set of alternatives. All that is required is that S be included in the set of all alternatives, not that it be smaller than that set].<br /></p><p>(c) There is a set of individuals ["voters"], identified by numerical subscripts, 1, 2, 3, 4, ....<br /></p><p>(d) Each individual is assumed to have a <strong>complete, transitive ranking</strong> of the entire set of alternatives, which we indicate using the notation introduced earlier -- the binary relations R, I, and P. Just to review, xR<sub>i</sub>y means that individual i considers alternative x to be as good as or better than alternative y. xP<sub>i</sub>y and xI<sub>i</sub>y are derived from R in the way indicated in the opening segments of this tutorial. What we are aiming for, of course, is a collective or social ranking, and that is indicated by the same letters, R, P, and I without the subscripts. So xPy means that the <em>society</em> prefers s to y. The whole point of this exercise is to start with complete, transitive <strong>individual</strong> rankings of the alternatives and then see whether there is any way of going from the individual rankings to a social ranking that satisfies certain conditions [see below] and results in a <strong>social </strong>ranking that is complete and transitive.<br /></p><p>(e) R<sub>i</sub> all by itself refers to individual i's ranking of the entire set of alternatives, x, y, z, .... Correspondingly, R all by itself refers to the society's ranking of the entire set of alternatives.<br /></p><p>(f) We shall have occasion to refer to different possible rankings, by an individual i, of the set of alternatives. We will indicate these different rankings by superscripts. So, for example, R<sub>i</sub> is one ranking by individual i of the entire set of alternatives. R<sub>i</sub>' is a second ranking. R<sub>i</sub>'' is a third ranking. And R<sub>i</sub><sup>*</sup> is a fourth ranking. A ranking R<sub>i </sub>can be thought of either as a list showing the way individual i ranks the alternatives, including ties [indifference], or as a set of all the ordered pairs (x,y) such that xR<sub>i</sub>y. <br /></p><p>(g) A Social Welfare Function [ an SWF ] is a function that maps sets of individual rankings onto a social ranking. Such a mapping function qualifies as an SWF just in case <strong>both</strong> the individual rankings, the R<sub>i</sub>, <strong>and</strong> the social or collective ranking, R, satisfy Axioms I and II below -- which is to say, just in case the rankings, both individual and social, are <strong>complete</strong> and <strong>transitive</strong>.<br /></p><p>(h) A Social Welfare Function is said to be Dictatorial if there is some individual i such that, for all x and y, xP<sub>i</sub>y implies xPy regardless of the orderings of all of the individuals other than i. Thus, in particular, to say that an SWF is dictatorial is to say that there is some individual who can impose his or her will on the society with regard to the choice between any pair, x and y, even if everyone else in the society has the opposite preference as between those two alternatives.<br /></p><p> (i) Finally, we define something called a Social Choice Function [ symbolized as C(S).] C(S) is the set of all alternatives x in the Environment S such that for every y in S, xRy. In other words, C(S) is the set of top alternatives or best alternatives in S. Quite often, C(S) will contain only one alternative, the one that the society prefers over all the others. But it may include more than one if the society is indifferent as among several best alternatives.<br /></p><p> Those are the nine definitions and stipulations. The key new ones that we have not met before are S, the set of available alternatives, R, the social ranking, SWF, a Social Welfare Function, and C(S), the Choice Function. Now Arrow lays down two Axioms governing the social ordering, R. These are:<br /></p><p>Axiom I: For all x and y, xRy or yRx [Completeness]<br /></p><p>Axiom II: For all x, y, and z, if xRy and yRz then xRz. [Transitivity]<br /></p><p> O.K. So much for the preliminary throat clearing. I want you to go over these definitions and stipulations until you are comfortable with them. The proof is going to be a formal argument couched in terms of these symbols and appealing to these assumptions and axioms. You will find it impossible to follow if you do not have a solid grasp on these preliminary definitions and so forth. While you are doing that, I want to talk for a bit about several important points that are implicit in what we have just laid down, but may not be obvious.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com8tag:blogger.com,1999:blog-4978426466915379555.post-57155713669974000312010-07-14T05:47:00.001-04:002010-07-14T05:47:44.909-04:00Rawls Last Installment<span xmlns=''><p>The first thing an individual in the Original Position must do when confronted with a choice of basic organizational rules for society is to decide how well or badly off she is, or was, before entering the Hall of Justice. [I shall simply stipulate that our representative person is female, but of the course the person does not know this, or indeed anything else of a particular nature about him/her self.] Since Rawls says that she is rationally self-interested, and is prepared to enter into the bargaining game because she believes that a satisfactory outcome will be to her advantage, she clearly needs to know what her baseline situation is. Otherwise, she cannot make a judgment as to whether a proposed rule will make her better off. Remember: she not only does not know who in particular she is or where in her society she is situated. She also does not know what stage of history she is located in.<br /></p><p> Faced with the necessity of stipulating a pre-bargain baseline [defined, we may suppose, simply by some specified amount of Primary Goods -- this whole thing just gets hopelessly complicated if we try to flesh out her situation in any more realistic manner], she really has only three options. For each possible stage of history in which she might be located, she can either adopt the premise that she is the worst off representative person in that society; or she can adopt the premise that she is the best off representative individual; or she can carry out an expected utility calculation, assigning some level of Primary Goods and some probability to every representative position in the society, and then multiplying the two and summing the results, In this third case, she will say to herself something like this: "There are seven representative positions in the society; fifteen percent of the people are in the first, ten percent in the second, etc etc. The first position has so and so much of the Primary Goods assigned to it, the second such and such amount, and so forth; with no more information than that I am one of the people in the society, I conclude that I have a fifteen percent chance of being in the first position, a ten percent chance in the second position, and so forth. Assuming that I know what my cardinal utility function is for Primary Goods, I can now carry out my expected utility calculation."<br /></p><p> Sigh. I told you this was going to be messy. I am pretty sure, from correspondence I had with Jack, that he is aware of a good deal of this, but I do not think he ever fully appreciated how deeply it undercut his central claim that he was advancing a <em>theorem</em>. At this point, Rawls says that a rational person, recognizing how important the choice is that she is about to make, will adopt an extremely <em>conservative</em> way of evaluating alternatives. What does this mean?<br /></p><p> Well, the first thing it means is assuming that outside the Hall of Justice, in the real world, she is one of the persons occupying the least advantaged representative position in society. Why is this conservative? Because if she assumes that she is in fact well off in the real world, she will be correspondingly less willing to make a deal, and this threatens to leave her utterly disadvantaged should the optimistic assumption about herself prove false. She must protect herself against the chance that she is one of society's poor, and the best way to do this is to agree to inequalities of any sort only if they work to the advantage of those least well off.<br /></p><p> But reasoning in this fashion, she might be tempted to carry out some sort of expected utility calculation and opt for a set of principles that maximizes the <em>average</em> utility that each representative person will enjoy. To be sure, that can be risky, since a higher average overall might be compatible with a <em>lower</em> utility to the least well off. In an expected utility calculation, that risk might be compensated for by a chance at a very much higher payoff to the better off representative positions.<br /></p><p> Rawls now argues that the rational individual under the Veil of Ignorance will reject expected utility calculations and instead opt for the extremely conservative, and also extremely controversial, "maximin" rule proposed by von Neuman. On page 163 of my book [see the chapter to which I have linked], I quote Rawls' reasons for adopting this rule. Here is what he says: "There are three chief features of situations that give plausibility to this unusual rule... The situation is one in which a knowledge of likelihoods is impossible or at best extremely insecure...The person choosing has a conception of the good such that he cares very little, if anything, for what he might gain above the minimum stipend that he can, in fact, be sure of by following the maximin rule. It is not worthwhile for him to take a chance for the sake of a further advantage, especially when it may turn out that he loses much that is important to him.... The rejected alternatives have outcomes that one can hardly accept. The situation involves grave risks." [All four passages from Rawls, p. 154]<br /></p><p> In my book, I have given a formal analysis of these claims, complete with nifty diagrams, but I want here to step back and try to get a sense of what Rawls is really talking about. Remember, first of all, that Rawls is <em>not</em> talking about the quantity of Primary Goods that the various principles of justice offer as possibilities, but rather about the <em>utility</em> that the utility function of the individual under the Veil of Ignorance associates with these various amounts of Primary Goods. The distinction is essential for understanding what Rawls is saying.<br /></p><p> Concretely, Rawls is claiming that the rational individual under the Veil of Ignorance will say to herself: "If I opt for a system of social organization that holds out the possibility of vast wealth for a few, but that fails to protect those at the bottom from absolute penury, I am risking ending up in a disastrous situation, one that "involves grave risks." But all I stand to gain is the chance at one of the top spots, even though I "care very little, if anything, for what [I] might gain above the minimum stipend that [I] can, in fact, be sure of by following the maximin rule."<br /></p><p> Fair warning: I am now going to say something that is mean-spirited and snarky, but I really do not know how else to get at what is going on in this argument. I apologize if I offend anyone. Here goes:<br /></p><p> What sort of person says to himself or herself what the individual in the Original Position, according to Rawls, says? Not just a <em>rational</em> person. There is nothing formally irrational about being willing to risk utter penury for a chance at fabulous wealth. That is just a matter of having a utility function of a particular shape[one that is, over a certain range, monotonically increasing rather than decreasing.] Would Gordon Gekko think this way? [If there is anyone who does not recognize the name, Gordon Gekko is the main character of the 1987 film, <em>Wall Street</em>, starring Michael Douglas. If you haven't seen it, by all means get it from NetFlix.] Of course not. But Gordon Gekko is not formally irrational. He just places a very high value on vast wealth and has a very high tolerance for risk. What about Picasso? I think not. If you offered Picasso a chance at artistic immortality, with penury and misery as the alternative if he turned out not to have real talent, I think he would have grabbed the chance with both hands. In fact, of course, he did.<br /></p><p> No, the sort of person who would reason as Rawls thinks the individual in the Original Position would is a tenured professor -- someone who has a comfortable albeit modest lifestyle that is absolutely assured against any risks, someone who has perhaps turned down other careers offering much larger rewards but also "involving grave risks." In short, the sort of person who would reason as Rawls thinks the individual in the Original Position would is ... John Rawls.<br /></p><p> Strip away all the talk about theorems, all the lovely filigree of philosophical elaboration, all the Reflective Equilibrium and Strains of Commitment and allusions to Game Theory, and you have a simple <em>apologia pro vita sua</em>. <br /></p><p> If the Representative Individual in the Original Position is an academic at a good American university or college that offers life tenure and a comfortable middle class life, then I think it is quite likely that he or she would opt for Rawls' two principles. They guarantee a continuation of that pleasant life style, combined with a virtuous but really cost free concern for the poor downtrodden denizens of the Inner City [the least well off representative individuals].<br /></p><p> Now, that is just about as mean-spirited as I have ever been in print [though not, I am afraid, in person], but what else can one conclude if one takes Rawls' theory seriously and tries to think through what it really means?<br /></p><p> The time has come to step back from the details of Rawls' discussion and try to get some perspective on what is, when all is said and done, the most important contribution to political philosophy of the past hundred years and more. I observed at the beginning of these remarks that Rawls offered his very new theory at a time when Anglo-American Ethical Theory was mired in an antinomy -- a several decades long face off between Intuitionism and Utilitarianism. Rawls invited us to get past that stalled historical moment by making use of ideas drawn from Game Theory [and also from neo-classical economics, but that is another matter.] If he had simply offered his Two Principles as an alternative to, or perhaps more accurately as a fusion of the best parts of, Intuitionism and Utilitarianism, there is no question that his proposal would have commanded considerable attention. The elegance of his discussion of Utilitarianism and the interesting and suggestive detail of the fully elaborated version of his proposal would, I am sure, have generated a lively discussion among philosophers, political theorists, and others.<br /></p><p> But what made Rawls' theory stand out as deserving of what constitutional lawyers call heightened scrutiny was his claim to be able to establish his two principles as the solution of a bargaining game. Now, even if this thesis could be sustained, it would still be open to readers to reject Rawls' claim that the solution of such a game ought to be considered <em>the principles of social justice</em>. But a genuine proof of Rawls' theorem would have vaulted his theory to an entirely unique status in ethical and political theory. Such a theorem would have taken its place beside Kenneth Arrow's General Possibility Theorem as a major result of formal analysis. [I remain convinced, in the absence of any textual or anecdotal evidence whatsoever, that this is exactly what Rawls dreamed of accomplishing.] This is why, both in my book and in these blog posts, I have focused almost exclusively on the logical status of the theorem that Rawls adumbrates in "Justice as Fairness," and continues to allude to as a theorem, albeit in a hedged manner, in Ðistributive Justice" and <em>A Theory of Justice</em>.<br /></p><p> I think I have demonstrated that the theorem is not valid, either in its original or in its revised form, or, more precisely, that it can only be made plausible by so many <em>ad hoc</em> adjustments, presuppositions, and qualifications that it loses its grip on our attention. I also think it is clear that the theory, as Rawls sets it forth in his book, covertly valorizes, without adequate argument, one particular substantive vision of the good society -- a vision some components of which I share, but for which Rawls fails to offer an argument.<br /></p><p> Well, this is twenty-four pages about Rawls, which is enough, I think, for this blog. I will turn my attention next to the single most important formal result in the application of formal methods to political philosophy: The General Possibility Theorem of Kenneth Arrow. My tone will change dramatically, as you will discover. No sniping or snarking, no <em>ad hominem</em> arguments. Arrow's result, like von Neuman's Fundamental Theorem, is a genuine triumph, and I shall do my best in expounding it to make its logical structure clear.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com4tag:blogger.com,1999:blog-4978426466915379555.post-70849429017021415702010-07-12T07:06:00.001-04:002010-07-12T07:06:23.224-04:00RAWLS FOURTH INSTALLMENT<span xmlns=''><p>Why the change? Pretty clearly, it is required by the special conditions imposed by the Veil of Ignorance. Since the parties in the Original Position now do not know who they are, they are pretty well forced <em>either</em> to carry out expected utility calculations over the entire range of positions in the society, some one of which each of them will actually turn out to occupy when they leave the Hall of Justice and regain their knowledge of who they are, <em>or else</em> to adopt the conservative assumption that they occupy the lowest position, and bargain to improve its allocation of Primary Goods. As we shall see, this is Rawls' version of the conservative rule proposed by von Neuman, to maximize one's security level.<br /></p><p> All this goes by so quickly in Rawls' exposition of the mature form of his theory that unless you are paying real close attention, you may not notice how wildly implausible, or even downright impossible, it all is. This is the form of the theory that everyone is familiar with, but people usually do not have any coherent idea why Rawls has made all of these very powerful stipulations. The reason, as I have indicated, is that each element of the final theory is designed to meet an objection to an earlier form of the theory.<br /></p><p> So where does all of this revision leave us? Well, first of all, something odd has happened along the way, as Rawls has altered his description of the choice situation to meet and overcome the difficulties with the first formulation. The original idea was that the parties to be governed by the agreed upon foundational rules would confront one another and bargain. The parties were assumed to be rationally self-interested, but with differing interests and desires. Rawls' central idea was that if to this premise of rational self-interest we added only one additional premise -- the willingness of the parties to abide by a set of rules arising from the bargain, the willingness to take that one step beyond self-interest to something resembling what is involved in having a morality -- then we could prove that the one and only set of rules on which they would self-interestedly settle would be his "two principles."<br /></p><p> But if we think about it for a moment, we will realize that after the revisions, Rawls no longer has a Bargaining Game that looks anything like this. Since the players have been stripped of any individuating features that might distinguish them from one another [such as differences in tastes or talents, or indeed even differences in which stage of human history they happen to be located in], there are no rational grounds on which any two of them could reason differently from one another about the choice of the basic structure and rules of the society in which they will find themselves when they emerge from the Veil of Ignorance. In short, what began as a problem in Bargaining Theory has morphed into a problem in the Theory of Rational Choice. [This is one of the reasons why Rawls tended to move toward what he himself called the Kantian Interpretation" of his theory. But that really does get us too far afield.]<br /></p><p> Before addressing the central question, viz., are these two principles thus revised, the solution to the Bargaining Game, thus altered, there is one subsidiary matter I should like to take up. Rawls says that although the parties in the Original Position under the Veil of Ignorance have temporarily forgotten who they are, what their specific desires are, and where they are located in history and in the structure of their society, they do retain a knowledge of the "general facts about human society." Each of these individuals, Rawls elaborates, understands "political affairs and the principles of economic theory," as well as "the basis of social organization and the laws of human psychology." In my book, <em>Understanding Rawls</em>, I argued that this is an epistemologically impossible state of affairs. There is not time or room here to repeat what I have said there [another shameless plug for my book :) ], but I think it is worth indicating the line of argument that I develop there.<br /></p><p> The first thing to be clear about is that under the Veil of Ignorance, the individuals in the Original Position do not even know in which stage of human history they are located. This fact leads Rawls into an extremely interesting discussion about the appropriate rate of savings that should be chosen as part of the basic socio-economic structure being negotiated. Debates about the social rate of savings are familiar to economists, but have been virtually absent from the political philosophy literature. It is greatly to Rawls' credit that he recognized this and introduced the subject into his theory. For those who are unacquainted with the discussion, the central issue is this: The capital required for future economic activity [seed corn, machinery, Research and Development, and their monetary equivalents] must be obtained from current production by somehow imposing limits on consumption -- eat all the corn this season, and there is no seed for next season's crop. Simple prudence dictates that people this year save for next year. But what shall we say about the responsibility of people in this generation to save for generations as yet unborn? A high, self-denying rate of social savings, such as that now being enforced by the Chinese government, will make possible an explosion of production in future generations, to the manifest benefit of those who are then alive. But that future production will come at the expense of this generation, which will have to deny itself some measure of present consumption. <br /></p><p> From the perspective of Rawls' theory, the question becomes: Under the Veil of Ignorance, what rate of savings will rationally self-interested individuals choose to impose upon themselves once they emerge from the Veil and discover which generation of their society's evolution they are actually located in? I encourage readers interested in this subject to take a look at Rawls' discussion.<br /></p><p> But getting back to the epistemological issue, the individuals in the Original Position are presumed to know the general facts of nature, society, economy, and human psychology, and even to know the broad outlines of the historical evolution of societies, but not to know where in that evolutionary process they are themselves located. Rawls clearly thinks it is possible for someone to be in this particular epistemological position. I think it is not. Why not?<br /></p><p> First of all, the individuals in the Original Position are blocked from accessing certain individuating facts about themselves, but they have not lost their powers of reason. To put the point simplistically, if the Veil has enabled them to retain the knowledge that All men are mammals and the knowledge that All mammals are animals, then their unimpaired powers of reason will allow them to conclude that All men are animals. Somewhat more to the point, if they know the standard theorems concerning the relation of supply to demand in the determination of price in a capitalist economy based on the production of commodities for sale in the marketplace, then they will be able to infer that their society has undergone the transition from Feudal to Capitalist social relations of production, <em>because until such a transition has taken place, individuals do not even possess the concepts that are employed in the formulation of those economic laws.</em> What is more, if, as I believe, capitalist social relations of production systematically mystify the underlying structure of exploitation on which capitalist profit rests, so that people mistakenly but inevitably perceive those relations as the expression of eternal and immutable economic laws, then only someone enmeshed in a capitalist society and economy will make the mistake of thinking that there <em>are</em> "laws of supply and demand."<br /></p><p> Now, maybe I am right about that, and maybe I am wrong. But by building these assumptions into the structure of the bargaining game from which he hopes to extract <strong>the</strong> principles of justice, Rawls has begged all of the questions that might be raised by someone like me ["begged" in the proper use of that term -- i.e., assumed what is to be proved]. This is one more example of my general claim that the misuse of formal methods allows authors to present their ideologically laden assumptions as value-neutral elements of a formal analysis.<br /></p><p> Let us now return to the central question: Would the individuals situated under the Veil of Ignorance in the Original Position coordinate on Rawls' Two Principles of Justice as revised in <em>A Theory of Justice</em>? This question is much more difficult to answer now than it was with regard to the first form of the theory. Even to make the question determinate enough to grapple with it we must make a considerable number of assumptions and specifications with regard to matters that Rawls either does not discuss or else leaves up in the air.<br /></p><p> At this point, in order to make this manageable, I must ask you to consult the chapter from my book, a link to which was posted earlier on this blog. I will discuss the problems in general terms, and leave it to each of you to read my detailed analysis in that chapter.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com2tag:blogger.com,1999:blog-4978426466915379555.post-319353536627661232010-07-09T05:50:00.001-04:002010-07-09T05:50:30.766-04:00RAWLS THIRD INSTALLMENT<span xmlns=''><p>The major changes to the bargaining game introduced in "Distributive Justice" and carried over into <em>A Theory of Justice</em> are rather dramatic. They are four in number:<br /></p><p> (1) Rawls introduces the famous Veil of Ignorance, which is a brilliant literary device designed to capture what we mean when we say that judges must be "disinterested" -- i.e., that they must make decisions without taking into consideration their own interests or situation. This takes care of the problem that some participants in the Bargaining Game are, and know they are, among the more talented members of society, while others are, and know they are, among the less talented. But this fix comes with its own problems. <br /></p><p> Stripped of any knowledge of who they are, the players in the Bargaining Game now have no reason at all to bargain, or indeed to do anything else. This will be clear if we keep in mind the analogy to judges. Judges, in their judicial capacity, are not supposed to have an agenda [think Confirmation Hearings for Supreme Court nominees and all the blather about "activist justices" who "make law from the bench."] So if the individuals under the veil of ignorance know nothing at all about who they are, they can have no ends, no purposes, save perhaps the goal to render decisions fairly when they are presented with cases. Hence these individuals have no reason to bargain about anything at all. The self-interest Rawls has equipped them with is vacuous. Rawls is thus forced to endow the individuals in the Original Position with some sorts of purposes, specific enough to make them care what they get from the agreement being hammered out, but not so specific as to recreate the problems that invalidated the first version of his theory.<br /></p><p> (2) Rawls' solution is to state that the individuals in the Original Position know that they have Life Plans. And since it wouldn't do to allow them to have the life plan of a religious hermit, for example [because then they would not care how much stuff they got out of the bargain, or even whether they had civil rights and protections], Rawls stipulates that the Life Plans posited for the individuals in the Original Position require for their accomplishment certain rights and abilities, powers and endowments [i.e., some stuff]. Bu this creates a new raft of problems, of an especially intractable sort, because the sorts of rights and abilities, powers and endowments required for the fulfillment of one Life Plan may be quite different from those required for the fulfillment of a different Life Plan. If your Life Plan is to become a Professor of Philosophy, then access to higher education for all those sufficiently talented is going to loom large in your budget of things you are bargaining for. But if your Life Plan is to become a champion surfer, higher education may drop way down in your list of desiderata. <br /></p><p> [Notice, by the way, that built into Rawls' conception of Life Plans is a particular substantive historical, and cultural, and social conception of individual personality and the good life. I do not have time to go into this in the detail that it deserves, so I will simply refer you to the classic discussion in Karl Mannheim's <em>Ideology and Utopia</em>, Chapter IV, of the chiliastic, liberal-humanitarian, conservative, and socialist-communist forms of the Utopian Mentality and the orientation to time itself. Rawls, to put it in shorthand, assumes that everyone in the Original Position has a liberal-humanitarian orientation toward time, which is simply part of the larger fact that his entire theory is an ideological rationalization of capitalist social democracy. But I digress.]<br /></p><p> To get over the problem posed by the diversity of possible Life Plans, so that he can get back to his theorem, Rawls now makes another assumption, and this one is, technically speaking, a whopper.<br /></p><p> (3) since they do not know which particular Life Plans they have, Rawls asserts [he never offers anything remotely like an argument for it] that one can create an index of the heterogeneous basket of basic stuff that anyone would need to pursue whatever Life Plan he or she might turn out to have, something Rawls calls an Index of Primary Goods He then simply assumes that everyone in the Original Position has positive, albeit declining, marginal utility for the Index of Primary Goods. He assumes this, not because it is plausible [he never offers any arguments for any of this], but because unless he assumes it he won't have a hope of invoking the tools and techniques of modern economic theory with which he thinks he can prove his "theorem."<br /></p><p> The problem of constructing an index of a heterogeneous assortment of rights and abilities, powers, and endowments is completely insoluble, as any honest economist will tell you. Indices like the Consumer Price Index or the Dow Jones Index are hopelessly flawed, and there is no way to fix them. Some of you may be quite familiar with this problem, others may have never heard of it before. I guess maybe I ought to say a word or two about the subject, even though it will slow us down some. The Consumer Price Index is constructed by putting together a list ["a market basket"] of consumer goods in specific proportions that is supposed to reflect the way most Americans spend their household income in a specified time, say a month: So much housing, so many pounds of potatoes, so many lamb chops, so many visits to the doctor, etc. Samples are then taken, and averaged out, of what these items are selling for in various stores, doctors' offices, etc, around the country. The same market basket of goods and services is then priced a month later. If the total cost of the market basket has increased by 1%, then it is said that there has been a 1% increase in the CPI, or that there has been a month to month inflation rate of 1%.<br /></p><p> There are a gazillion problems with this index, all of which are totally unfixable. For example, suppose you have a thirty year fixed rate mortgage on your house. In that case, if housing costs rise dramatically [as they did in the 70's and 80's, for example], the Consumer Price Index, reflecting that rise, may shoot up dramatically. But you won't actually experience that rise, because a major component of <em>your</em> market basket of goods, namely housing, is fixed. The same sort of problem arises with regard to every element in the "market basket." The dramatic rise in health care costs will not affect you so long as you are healthy, but will affect you if you have a special needs child. if you are a vegetarian, a steady decline in the price of meat won't have any effect on your pocketbook. When I was a young man living in Cambridge, Massachusetts, lobster was cheap at the supermarket and steak was expensive. Now, when I shop for dinner, steak is cheaper per pound than all but the least expensive fish. Once again, Rawls unconsciously [I think] builds intro his supposedly universal theory the assumptions, presuppositions, and tastes of a particular social and economic class, which happens to be his.<br /></p><p> (4) The last major change is a revision in the statement of the so-called "Difference Principle." Instead of inequalities working to everyone's advantage, they must now work to the benefit of the least advantaged members of society. This is a very significant change However, Rawls introduces it <em>not </em>by saying that he has changed his mind, or has been compelled by the logic of the bargaining game to alter the theorem, <em>but rather</em> by saying that this is a more reasonable "interpretation" of the original form of the principle. I confess that I find this rather weird and creepy. Rawls treats his own two principles, <em>which he made up</em>, as though they had been inscribed on tablets by The Lord God Himself, and thus required us to interpret them rather than change them. This is one of the strangest features of Rawls' entire discussion.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com2tag:blogger.com,1999:blog-4978426466915379555.post-58547080312121043622010-07-07T05:35:00.001-04:002010-07-07T05:35:38.217-04:00APPLICATIONS RAWLS SECOND INSTALLMENT<span xmlns=''><p>The second principle says that " inequalities are arbitrary unless it is reasonable to expect that they will work out for everyone's advantage." Remember that in the original version of Rawls' principles, it is inequalities in a <em>practice</em> that are being referenced -- differential salaries, for example, or differences in the perks associated with one of the roles in the practice. The core idea is that economic inequalities may actually result in an increase in total social output, for example by attracting especially talented people to positions demanding highly skilled workers. Inequalities may also motivate young people to acquire time consuming and demanding skills whose deployment in more highly compensated positions will once again increase total output. Higher wages are required to attract the talented workers or to persuade them to spend the time and money acquiring the skills. If there is something left over from the increased output after the skilled workers have received sufficient additional compensation to motivate them, then the remainder -- what we might call an "inequality surplus" --can be spread around to the rest of the society, making everyone better off than he or she would have been in a society that enforces equal compensation at the price of a universally lowered standard of living.<br /></p><p> This is really the core idea in Rawls' entire theory of social justice. It is, we may note, the standard Sociological rationale for the extreme inequality of modern capitalist society. I like to think of it as the Brain Surgery argument -- to wit, "If you have to have brain surgery, do you want to be operated on by someone who is paid no more than a burger flipper at Wendy's?" The idea roughly is this: If everyone were paid the same wage -- say something above what is now Minimum Wage -- no one would have any particular motivation to swap the job of burger flipper or ditch digger or garbage collector for the job of corporate executive or brain surgeon or professor, assuming that those jobs had been stripped of their various perks as well as of their higher salaries [corner offices, people who call you "sir" or "ma'am," and so forth.] But society needs talented corporate executives and well-trained brain surgeons and professors. Otherwise the Gross Domestic Product will be sub-optimal and everyone will suffer. So efficiency demands that we slap big salaries on those jobs to motivate some of the more promising burger flippers and garbage collectors to trade in their jobs for brain surgery. If we manage things skillfully, we will pay them just enough to drag them away from their spatulas and garbage trucks and into the executive suites and hospitals. Those salary increases will come out of the GDP, of course, but there will be enough left over to raise the pay of all the remaining burger flippers and garbage collectors. So, assuming no one is envious, and resents the higher salaries of the brain surgeons even though the productivity of the brain surgeons is raising the wages of the burger flippers, everyone will be in favor of this inequality. [This is the reason for the non-envy clause in Rawls' theory, in case you ever wondered.]<br /></p><p> I have spelled this out at length because thus explicated, it is so wildly implausible. As a description of what motivates people in a modern capitalist society to pursue one career path rather than another it is so tone-deaf sociologically and psychologically as to sound like a Jon Stewart send-up. But that is not the focus of these remarks, so let us leave that to one side. <br /></p><p> One of the purposes of the Difference Principle, so called, is supposed to be to allow us to adjudicate complaints against a scheme of unequal compensation by showing that the inequality works to everyone's advantage. But "to everyone's advantage" is, grammatically speaking, a comparative rather than a superlative. To everyone's advantage compared to what? This is a good deal harder to answer than it might seem at first glance.<br /></p><p> Presumably, the answer is that the scheme works to everyone's advantage compared to the same social system with <em>equal </em>compensation, if indeed one can even imagine <em>the same</em> social system without the inequalities. [Would a corporation be the same institution if everyone made the same wage? Would a hospital be?] But there are almost certainly a number of alternative schemes of unequal wages, each of which generates an Inequality Surplus adequate to make everyone better off, <strong>and each of which makes some positions better off and some positions worse off than those positions would be under a different inequality scheme generating a surplus</strong>. If that is so, considerations of Pareto Comparability do not permit us to say which scheme is preferable to which, even though each of them is preferable to the society without inequality.<br /></p><p> Assuming [what is in fact false] that all of these problems with the Two Principles can be solved, we are left with the central question: Would a group of rationally self-interested individuals faced with the circumstances of justice necessarily coordinate on the Two Principles? [By the way, for those who are not up to speed on all of this, the "circumstances of justice" are these: First, that the members of the society have something to gain from cooperation, if they can only agree; and Second, that their pre-agreement assets and powers are sufficiently equal to motivate them to seek common agreement. This is all standard Social Contract stuff, straight out of Hume, Hobbes, etc.] In short, we are faced with a proposed theorem in Bargaining Theory.<br /></p><p> As Rawls conceives the Bargaining Game, this is a multi-party game with full communication among players who are assumed to have cardinal utility functions invariant under affine transformations. Rawls never tells us how the game is played, nor does he even seem to think that he needs to do so. That is one of the odd things about his invocation of "theorems" in Game Theory. We are left to try to imagine for ourselves how the game would actually be played. I think we are meant to imagine something like this: The players sit in a circle in such a way that what each says is heard, and is known to be heard, by all. One player starts, and proposes a rule. [Say, the old Bill Cosby rule from early Sesame Street -- "All for one and everything for myself."] The next player either accepts the first player's rule, which of course she won't, or proposes a new rule. They go around the circle again and again, proposing foundational rules, until they succeed in making one complete circle during which everyone agrees to the same rule. That is then the solution to the game. Rawls says that after a bit "they will settle on [his] two principles." Is he right?<br /></p><p> Alas, no. There are two problems, one procedural, the other substantive. The procedural problem is that the bargaining game has no termination rule. There is no reason for the first player ["Everything for myself"] ever to stop proposing that rule. There is presumably some very, very small, but <em>not</em> zero, probability that sooner or later [probably later] the rest of the players will get tired or drop the ball and agree. Since there are no costs in the game associated with continuing to play it, none of the players has any incentive to "be reasonable."<br /></p><p> You can fix this glitch, of course, by imposing a time limit on the game. But that gives an asymmetrical advantage to the last player whose turn leaves just enough time to go around the circle once. When that moment is reached, the lucky player can propose a rule that makes everyone better off than having no rule at all, but advantages that player [such as "People with naturally curly hair get first dibs on all the nifty jobs," said by someone who has naturally curly hair]. Now, this is silly, right? But when you claim to be proving a theorem that is necessary, that is the sort of thing you have to take into account. This is an example of what I mean by wrapping yourself in the impressive language of Game Theory to make what you are doing sound impressive, while not actually engaging in Game Theoretic arguments. As Rawls says in "Justice as Fairness," "there remain certain details to be filled in, and various alternatives to be ruled out." Indeed.<br /></p><p> The second problem is substantive. The two principles proposed by Rawls would not win unanimous agreement from the players. The problem is this: While the players are faced by the circumstances of justice, and hence are roughly comparable in their powers and endowments, there are nevertheless significant differences among them in natural talents and abilities. Some of them will fare much better than others in a society in which "the positions ... are open to all." What is more, despite these differences in native intelligence or ability, all of them know these facts. Now, if you are one of the most talented members of society, you are going to be in favor of a structure of inequality in which you know quite well that you will be one of those ending up in the favored positions. But if you are not one of the talented, you will conduct an expected utility calculation and come to the conclusion that you might be better off with a system in which the better paid jobs are distributed by lot to anyone who meets certain minimum requirements [for example, a college degree, even with a low GPA]. The imposition of minimum requirements will suffice to generate <em>some</em> Inequality Surplus, but the allocation of the favored jobs by lot will work to the advantage of the less talented members of the society, who will thus have a shot at the higher paid jobs, something they would never have if those jobs were allocated strictly on the basis of a fair competition. [As Senator Roman Hruska of Nebraska said in 1970, defending the Supreme Court nomination of G. Harrold Carswell from the charge of mediocrity, "There are a lot of mediocre judges and people and lawyers. They are entitled to a little representation, aren't they?"]<br /></p><p> So Rawls' "theorem" is no theorem at all. I pointed this out in an article I published in <em>The</em><br /> <em>Journal of Philosophy</em>, and Jack's face fell when I told him about it at an annual APA meeting. But I went on to say that his subsequent essay, "Distributive Justice," changed his theory in a way that met all of my objections, and his face brightened. "Oh, that's all right then," he said before wandering off.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com2tag:blogger.com,1999:blog-4978426466915379555.post-46017697360216311752010-07-05T08:05:00.001-04:002010-07-05T08:05:43.477-04:00Rawls' A THEORY OF JUSTICE First Installment<span xmlns=''><p style='text-align: center'><span style='font-size:14pt'><strong>Part IV Applications -- Rawls<br /></strong></span></p><p> We come now to what might plausibly be considered the real payoff for all the technical thrashing about we have been engaged in: an extended analysis of the core argument in John Rawls' famous book, <em>A Theory of Justice</em>. Rawls' <em>hauptwerk</em> is widely considered the most important contribution to English language political theory of the past century, and is arguably the most influential work of philosophy written in the English language during that time. It is worth our while, therefore, to take the time to look at his central argument carefully and in detail. Thirty-three years ago I wrote a book-length examination of <em>A Theory of Justice</em>, called <em>Understanding Rawls</em>, published by Princeton University Press. Much of what I say here overlaps with what I said in that book, but my focus here is more narrowly on Rawls' attempt to apply Bargaining Theory to his subject. Those interested in a somewhat broader discussion are invited to hunt up my book and take a look. I am going to assume that everyone reading these words has some familiarity with Rawls' theory. <br /></p><p> The core of Rawls' work is a simple and rather lovely idea. In the middle of the twentieth century, Anglo-American ethical theory was stuck in what Kant, two centuries earlier, had called an antinomy. Utilitarians and intuitionists were locked in a death struggle, with each side more than capable of exposing the weaknesses of the other, but each unable to defend itself against the other's crushing arguments. Rawls had the idea that the conflict might be resolved by combining an old tradition of political philosophy -- social contract theory -- with a brand new field of mathematics and economics, Game Theory. Early in the development of the theory that eventually found its full-scale exposition in <em>A Theory of Justice</em>, Rawls claimed that he could prove a <em>theorem</em> in Bargaining Theory, and that the proof of that theorem would constitute a justification for the pair of principles which, he said, were or ought to be the foundation of a just society.<br /></p><p> This was a very bold claim, and had Rawls been able to fulfill its promise, it would have been a monumental achievement. As we shall see, Rawls very early recognized that the original version of the theorem was unprovable, and indeed false. In response to this realization, he made sweeping changes to his theory, resulting in the distinctive form that the theory takes in <em>A Theory of Justice,</em> but unfortunately, the revised theory is not more defensible than the original. Rawls himself seems to have realized this fact, for while repeating the language of "theorem" and "proof," he very considerably backs away from the strong claims that he made in the earliest published version of his theory.<br /></p><p> Before we begin the detailed examination of the argument, let me take just a moment to explain why I believe it is appropriate to bring the tools and insights of Game Theory to bear on <em>A Theory of Justice</em>. That is, after all, not the customary manner in which we engage with the arguments of Hobbes, Locke, Rousseau, Mill, Kant, or any of the other great figures of the Western tradition of democratic political theory. Quite simply, the reason is this: A great part of the plausibility of Rawls' theses derives from his claim that they can be grounded in a formal argument of Bargaining Theory. Absent that claim, the reader is left simply to contemplate Rawls' political theory and consider whether he or she likes it. The <em>argument</em> for the theory is, when all is said and done, the claim that the two principles would be chosen by rationally self-interested individuals situated in what Rawls eventually came to call the Original Position. If that is simply not true, then it is hard to see what other justification Rawls has for his theory.<br /></p><p> It is actually rather difficult to figure out exactly what Rawls' Two Principles <em>mean</em>, and the only way I can see to grapple with them is to take Rawls at his word that they are the solution to a bargaining game, and then see how we might so construe them. In this case, as we shall see, the formal machinery of Game Theory is quite helpful in guiding us to turn Rawls' non-technical language into something precise enough to be subjected to analysis.<br /></p><p> It will be useful for our purposes to begin with the earliest statement of Rawls' theory, as it appeared in an article entitled "Justice as Fairness," published in 1962 in an important collective volume of essays called <em>Philosophy, Politics, and Society, Second Series</em>, edited by Peter Laslett and W. G. Runciman. Two passages from the essay will set things up for the first stage of my analysis.<br /></p><p> "The conception of justice which I want to develop," Rawls writes, "may be stated in the form of two principles as follows: first, each person participating in a practice, or affected by it, has an equal right to the most extensive liberty compatible with a like liberty for all; and second, inequalities are arbitrary unless it is reasonable to expect that they will work out for everyone's advantage, and provided the positions and offices to which they attach, or from which they may be gained, are open to all."<br /></p><p> These principles, Rawls says, would be agreed upon unanimously in a deliberation that he characterizes roughly in the way that "state of nature" political theorists describe the agreement on the Social Contract that constitutes a nation. Although he acknowledges that his remarks "are not offered as a rigorous proof" that persons engaged in this deliberation would agree on the two principles, such a proof requiring "a more elaborate and formal argument," nevertheless, he goes on to say:<br /></p><p> "[T]he proposition I seek to establish is a necessary one, that is, it is intended as a theorem: namely, that when mutually self-interested and rational persons confront one another in typical circumstances of justice, and when they are required by a procedure expressing the constraints of having a morality to jointly acknowledge principles by which their claims on the design of their common practice are to be judged, they will settle on these two principles as restrictions governing the assignment of rights and duties, and thereby accept them as limiting their rights against one another."<br /></p><p> I think it is patently clear that Rawls in these passages is laying claim to the rigor and demonstrative power of Game Theory, at least in its somewhat looser form as Bargaining Theory. He seeks to show that his proposition is a <em>necessary</em> one; he asserts that it is intended as a <em>theorem</em>. Throughout the long evolution and transformation of his theory, Rawls never gave up this claim, for all that he also never came close to providing an argument for it. It is, I believe, the heart and soul of his entire enterprise. Without it, he has nothing but a rather affecting, albeit extremely murky, expression of his personal preferences in social organization. <br /></p><p> What can Game Theory tell us about Rawls' claim? There are two questions that we must try to answer: What do his two principles mean? and Is his assertion a theorem that can be proved with necessity? <br /></p><p> First, a problem: In the original statement of the principles, Rawls states his first principle thus: "each person participating in a practice, or affected by it, has an equal right to the most extensive liberty compatible with a like liberty for all. " In <em>A Theory of Justice</em>, however, the reference to practices is omitted, and instead we get "Each person is to have an equal right to the most extensive basic liberty compatible with a similar liberty for others." [p. 60] Eventually, this is tweaked a bit, and becomes "Each person is to have an equal right to the most extensive total system of equal basic liberties compatible with a similar system of liberty for all." [p. 302] The original formulation is thus intended to apply to practices, such as marriage, or capitalism, or the military, or the judicial system. The final formulation applies to nothing less than the total organization of a society. The shift, as we shall see, makes it forbiddingly difficult to figure out what the principle actually means.<br /></p><p> The first principle, in all of its variants, uses the phrase "the most extensive." That implies that one can rank alternative arrangements of a practice, or, alternative sets of fundamental or constitutional arrangements, in order of the degree of liberty that they embody or promise or make possible or guarantee. But as the term "liberty" is ordinarily used in the context of debates about political systems, it refers to a wide variety of institutional arrangements or guarantees that vary along multiple dimensions. The right to trial by a jury of one's peers, we may suppose, is a form of liberty. So is the right of all adults to vote periodically in elections to select the members of the government. Is a system of government with the first but not the second a more extensive or a less extensive liberty than a system of government with the second but not the first? One might reply, it does not matter, because a system of government with both is superior to either. Hence it would be Pareto preferred to either. But suppose there are liberties that in some of their forms are incompatible, in the sense that guaranteeing one interferes with guaranteeing the other. <br /></p><p> For example [if I may be a trifle facetious], imagine a society consisting solely of authors and literary critics [these days, what with the internet, it does seem as though everyone is either an author or a critic, and sometimes both]. Now authors wish to be free of what they consider unfounded attacks on their writings, and critics wish to be free of what they consider unjustified limitations on their critiques. So the critics will prefer the American system of libel law, which gives very broad latitude to critics, and authors will prefer the British system, which favors those supposedly libeled [unless, of course, the authors wish, in their writings, to say nasty things about the critics, in which case the situation is more complex.] There is no way in this situation simultaneously to maximize the liberty of both groups, since each extension of the liberty of one will be experienced as a loss of liberty by the other. Consequently, as the authors and critics gather to bargain on the founding principles of liberty in their society, they will find it impossible to achieve unanimity on Rawls' first principle, because every attempt to spell out what it means will have them at loggerheads. This problem is not at all trivial, for all that my example may make it seem so. It is referred to in economics as the Indexing Problem, and it will crop again in a different guise in the final version of Rawls' theory.<br /></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com7tag:blogger.com,1999:blog-4978426466915379555.post-44223079038535240912010-07-02T05:44:00.004-04:002010-07-02T05:48:45.964-04:00CHANGE OF PLANI have decided not to go on to the Free Rider Problem, but instead to launch into a lengthy and detailed analysis of John Rawls' famous book, <em></em>A Theory of Justice<em></em>. There are two reasons for this decision: First, when I started to write my discussion of the Free Rider Problem, I found that I had very little useful to say about it that had not already been said in my article on Elster. Second, since an analysis of Rawls is, in some sense, the real payoff for everything we have been doing, I thought I ought to get to it right away. If anyone is interested, after I finish talking about Rawls, I will be happy to answer questions about or expand on my treatment of Elster.Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0tag:blogger.com,1999:blog-4978426466915379555.post-8276213449631229772010-07-02T05:44:00.001-04:002010-07-02T05:44:08.273-04:00The Prisoner's Dilemma Second Installment<span xmlns=''><p>What would happen in the real world? I suggest something like this might happen: A examines the outcome matrix and says to herself: "Look, there is no difference to speak of between a 40 year sentence and a sentence of 40 years and a day. I am going to count on my partner to be sensible, and go for the one day sentence. The very worst that can happen is that I will have a day tacked onto the end of forty years, if I am still alive then, but I have a good shot here at getting off all but scot free."<br /></p><p> Now, from a Game Theoretic point of view, this is not interesting at all. What is the point of introducing outcome matrices and payoff matrices and dominant strategies and Pareto sub-optimal outcomes if, when it gets right down to it, we are going to go into all the messy details of who the players are, what their relation to one another is, what history they have with one another, and all the rest of it? I thought Game Theory was going to enable me to analyze the situation without any of that stuff. <br /></p><p> This is a point of such importance that I need to talk about it for a bit. A very long time ago, Aristotle and Pythagoras and some other smart Greeks [and also some really smart Egyptians, but I don't want to get into the whole <em>Black Athena</em> thing] discovered that in some situations, one can successfully abstract from the details of a problem and still carry out a valid process of reasoning about it by attending only to certain formal or structural features of the situation. One can, for example, carry out long, complex chains of reasoning about shapes and sizes and spatial relationships without any reference to the materials in which these shapes and sizes and relationships are embedded. Now, this was not obvious on the face of it, when they made this historic discovery. You could not get very far reasoning about crops, after all, if you failed to take notice of which crop you were talking about, nor could you say much of interest about metalworking in abstraction from the particular metal in question. But if you know that all human beings are mortals, and you know that all Athenians are human beings, then you <em>can</em> draw the conclusion that All Athenians are mortal, just by attending to the formal syntactic structure of your two premises, namely that All A are B and All B are C, from which it follows, regardless of the details of the story you are telling, that All A are C.<br /></p><p> Formal reasoning of this sort is beguiling, both because it is extremely powerful and because it can be engaged in by people who do not actually know much about the way the world works. There is also a lot of not very sublimated erotic and aggressive energy expressed here. Not for nothing do mathematicians speak about ramming an argument through. Oh well. That could lead us in rather hairy directions.<br /></p><p> Once all of this has gained wide acceptance and has been brought to its present height of complexity and sophistication, everyone wants to get in on the act. I mean, who wants to talk about the psychological profiles of accused individuals enmeshed in the complexities of the criminal justice system when you can slap a 2 x 2 matrix on the page and carry out abstract calculations about dominant strategies? How cool is that? This is the reason why philosophers, who have long since learned that logicians have the highest status in their profession, put backwards E's on the page and talk about "for all x" rather than "everyone."<br /></p><p> The little story called The Prisoner's Dilemma ignores just about every fact about a real <em>Law and Order</em> type situation that could possibly be relevant to thinking about it. Let us look at just a few of the things that are assumed away.<br /></p><p>1. The situation is treated as a two person game. But there are obviously many more than two people involved. First of all, there are the cops who are putting the squeeze on the prisoners. In the real world, they are an important part of the situation, and real prisoners will try, quite rationally, to figure out whatever they can about the cops that will help them make their decision. Furthermore, in the American justice system, the prisoners will have lawyers. So at a bare minimum, this is a five person game [one cop, two prisoners, two lawyers]. <br /></p><p>2. To force the story into a 2 x 2 matrix, one must suppose that each player has only two strategies. Recall what I said about how extraordinarily simple a game must be to offer only two strategies to each player. In the real world, there will be an arraignment, and there will be some jockeying over venue and date of trial and which judge is going to hear the case and whether to opt for a jury trial or go for a bench trial. Lots of moves, therefore lots of strategies, therefore no 2 x 2 matrix.<br /></p><p>3. To make the story fit the matrix ["the punishment fit the crime"], we must abstract from every important fact about the two criminals, including sex, race, religion, personal relationship, past history with the criminal justice system, and so on and on, and then we must assume, against all plausibility, that each criminal will rank the outcomes purely on the basis of the length of the jail sentence to himself or herself.<br /></p><p> Now, if we could, by doing all of this, draw conclusions whose validity is totally independent of all the details we have abstracted from, just as the validity of geometric calculation is independent of the color of the shapes whose area we are computing, then we would indeed have a very powerful tool for the analysis of economic, political, legal, and military problems. It would be a tool that could both help us to predict how people <em>will</em> act and also enable us to prescribe how rational individuals <em>should</em> act. But in fact, what remains when we have stripped away all the detail necessary to reduce a complex situation to a 2 x 2 matrix is a structure that neither assists in prediction nor guides us in prescription.<br /></p><p> If we focus simply on the formal structure of a two person game with two pure strategies for each player, it is obvious that there are 24 different orders in which each player can rank the four outcomes, setting to one side for the moment the possibility of indifference. How do I arrive at this number? Simple. A [or B] has four choices for the number one spot in the ranking. For each of these, there are three possibilities for the number two spot. There are then two ways of choosing among the remaining two outcomes for the number three spot, at which point the remaining outcome is ranked number four. 4 x 3 x 2 x 1 = 24. Since A's rankings are logically independent of B's rankings, there are 24 x 24 = 576 possible combinations of rankings by A and B of the outcomes of the four possible strategy pairs. The Prisoner's Dilemma is simply one of those 576, to which a story has been attached. <br /></p><p> People enamored of this sort of thing have thought up little stories for some of the other possible pairs of rankings. [The following examples come from the pages of Baird, Gertner, and Picker, mentioned earlier]. For example, the following pair has had attached to it a story about The Battle of the Sexes [now fallen into disfavor for reasons of political correctness]:<br /></p><p> A: O21 > O12 > O22 > O11<br /></p><p> B: O21 > O12 > O11 > O22<br /></p><p>Another pair of preference orders has a story about collective bargaining attached to it:<br /></p><p> A: O21 > O11 > O12 > O22<br /></p><p> B: O12 > O11 > O21 > O22<br /></p><p> If we allow for indifference, then there are lots more possible pairs of preference orders. Here is one that has a story attached to it called The Stag Hunt:<br /></p><p> A: O11 > O21 = O22 > O12<br /></p><p> B: O11 > O12 = O22 > O21<br /></p><p> I have no doubt that with sufficient time and imagination, one could think up many more stories to attach to yet other pairs of ordinal rankings of the four outcomes in a game with two pure strategies for each player. None of these little preference structures really models, in a useful way, relations between men and women, or collective bargaining, or stag hunts [since matching pennies really is a game, with all the simplifications and rules and such that characterize games, there is no reason at all why a Game Theoretic analysis should not be useful in understanding it, but one doesn't often encounter real world situations, even in Las Vegas casinos, where people are engaged in matching pennies.]<br /></p><p> What is the upshot of this rather bilious discussion of The Prisoner's Dilemma? Put simply, it is this: The abstractions and simplifications required to transform a real situation of choice, deliberation, conflict, and cooperation into a two-person game suitable for Game Theoretic analysis fail to identify formal or structural features of the situation that are, at one and the same time, essential to the nature of the situation and independent of the facts or characteristics that have been set aside in the process of simplification. That, after all, is what does happen when we reduce an informal argument to a syllogism. Consequently, anything we can infer from the formal syllogistic structure of the argument must hold true for the full argument, once the content we have abstracted from is reintroduced. <br /></p><p> Just to make sure this point is clear: Suppose I come upon a text in which the author tries to establish that some Republicans are honorable. She begins, we may suppose, by noting that all Republicans are Americans, and then offers evidence to support that claim the some Americans are honorable, whereupon he concludes that some Republicans are honorable. When we convert this to syllogistic form, it becomes: All A are B. Some B are C. Therefore, Some A are C. Thus separated from its content, the argument is quickly seen to be invalid [although, let us remember, that fact does <strong>not</strong> imply that the conclusion is false, only that it has not been established by the argument. Fair is fair.] The As that are B may not be among the Bs that are C. [Venn diagrams, anyone?] In this case, the abstraction required to convert the informal argument into syllogistic form succeeds in identifying a formal structure of the original argument. Hence the formal analysis is valid.<br /></p><p> But in the case of the Prisoner's Dilemma, essential elements of the original situation must be simplified away, removing aspects of the situation that are structurally essential to it. The result is not to lay bare the underlying formal structure of the original situation, but rather to substitute for the original situation another, simpler situation that can be exhibited in appropriate Game Theoretic form. The reasoning concerning this new situation is correct, but there is no reason to suppose that it applies as well to the original situation. <br /></p><p> Conclusion: Be not beguiled by 2 x 2 matrices.</p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com4tag:blogger.com,1999:blog-4978426466915379555.post-20534612318063679212010-06-30T07:03:00.001-04:002010-06-30T07:03:51.802-04:00APPLICATIONS: PRISONER'S DILEMMA, FIRST INSTALLMENT<span xmlns=''><p><span style='font-size:14pt'><strong>Part Four</strong></span><span style='font-size:1pt'><br /> </span></p><p style='text-align: center'><span style='font-size:14pt'><strong>Applications<br /></strong></span></p><p style='text-align: center'><br /> </p><p> The time has come to put all of this formal stuff to use. In the second major part of this tutorial, I shall examine a number of attempts to apply the materials of Game Theory and Rational Choice Theory to substantive issues in political theory, economics, military strategy, and the law. My message will in the main be negative. I shall argue, again and again, that authors attempting to gain rigor or clarity or insight by the use of these methods actually misuse them, failing to understand them correctly or failing to understand the scope and nature of the simplifications and abstractions that are required before the materials of Game Theory and Rational Choice Theory can be properly applied.<br /></p><p><br /> </p><p> I have asked you to read two essays and a chapter of a book, all by me, and all available by clicking on the links provided in the blog post of June 2, 2010. In order to move things along and keep this tutorial to a manageable size, I am going to rely on you to do that reading, so that I can refer to it without summarizing it or repeating what I have said in those texts.<br /></p><p><br /> </p><p> My order of discussion will be as follows:<br /></p><p><br /> </p><p> 1. A discussion of the Prisoner's Dilemma<br /></p><p> 2. A discussion of the Free Rider Problem<br /></p><p> 3. An extended and very detailed analysis of the central thesis of John Rawls' <em>A Theory of justice</em>.<br /></p><p> 4. A brief discussion of certain arguments in Robert Nozick's <em>Anarchy, State, and Utopia</em>.<br /></p><p> 5. A discussion of some of the applications of Game Theory and Rational Choice Theory in <em>Game Theory and the Law</em> by Baird, Gertner, and Picker.<br /></p><p> 6. A discussion of the role played by Game Theory in the debates about military strategy and deterrence policy in the United States in the first twenty years following World War II. In connection with this portion of the discussion, I will make available the text of a book I wrote in 1962 but was never able to get published.<br /></p><p><br /> </p><p> Assuming anyone is still with me after all of that, I will entertain suggestions of how we might usefully keep this tutorial going. Alternatively, I can go back to playing Spider Solitaire on my computer. :)<br /></p><p><br /> </p><p style='text-align: center'><span style='text-decoration:underline'><strong>The Prisoner's Dilemma<br /></strong></span></p><p style='text-align: center'><br /> </p><p> The Prisoner's Dilemma is a little story told about a 2 x 2 matrix. For those who are unfamiliar with the story [assuming someone fitting that description is reading these words], here is the statement of the "dilemma" on Wikipedia:<br /></p><p><br /> </p><p>"Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated the prisoners, visit each of them to offer the same deal. If one testifies for the prosecution against the other (<em>defects</em>) and the other remains silent (<em>cooperates</em>), the defector goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?"<br /></p><p><br /> </p><p> The following matrix is taken to represent the situation. <br /></p><p> <br /><table border='0' style='border-collapse:collapse'><colgroup><col style='width:213px'/><col style='width:213px'/><col style='width:213px'/></colgroup><tbody valign='top'><tr><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'> </td><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>B1 cooperate</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>B2 defect</span></p></td></tr><tr><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A1 cooperate</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>6 months, 6 months</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>10 years, Go free</span></p></td></tr><tr><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A2 defect [</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>Go free, 10 years</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>5 years, 5 years</span></p></td></tr></tbody></table><p><br /> </p><p> The problem supposedly posed by this little story is that when each player acts rationally, selecting a strategy solely by considerations of what we have called dominance [A2 dominates A1 as a strategy; B2 dominates B1 as a strategy], the result is an outcome that <em>both</em> players consider sub-optimal. The outcome of the strategy pair [A1,B1], namely six months for each, is preferred by both players to the outcome of the strategy pair [A2,B2], which results in each player serving five years, <em>but the players fail to coordinate on this strategy pair</em><br /> <em>even though both players are aware of the contents of the matrix and can see that they would be mutually better off if only they would cooperate.<br /></em></p><p><br /> </p><p> For reasons that are beyond me, this fact about the matrix, and the little story associated with it, is considered by many people to reveal some deep structural flaw in the theory of rational decision making, akin to the so-called "paradox of democracy" in Collective Choice Theory. Military strategists, legal theorists, political philosophers, and economists profess to find Prisoner's Dilemma type situations throughout the universe, and some, like Jon Elster [as we shall see when we come to the Free Rider Problem] believe that it calls into question the very possibility of collective action.<br /></p><p><br /> </p><p> There is a good deal to be said about the Prisoner's Dilemma, from a formal point of view, so let us get to it. [Inasmuch as there are two prisoners, it ought to be called The Prisoners' Dilemma, but never mind.] The first problem is that everyone who discusses the subject confuses an outcome matrix with a payoff matrix. In the game being discussed here, there are two players, each of whom has two pure strategies. There are no chance elements or "moves by nature" [such as tosses of a coin, spins of a wheel, or rolls of a pair of dice]. Let us use the notation O11 to denote the outcome that results when player A plays her strategy 1 and player B plays his strategy 1. O12 will mean the outcome when A plays her strategy 1 and B plays his strategy 2, and so forth. There are thus four possible outcomes: O11, O12, O21, O22.<br /></p><p><br /> </p><p> In this case, O11 is "A serves six months and B serves six months." O12 is "A serves 10 years and B goes free," and so forth. Thus, the <em>Outcome Matrix</em> for the game looks like this:<br /></p><p><br /> <table border='0' style='border-collapse:collapse'><colgroup><col style='width:213px'/><col style='width:213px'/><col style='width:213px'/></colgroup><tbody valign='top'><tr><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'> </td><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>B1</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>B2</span></p></td></tr><tr><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A1</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A serves six months and B serves six months</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A serves ten years and B goes free</span></p></td></tr><tr><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A2</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A goes free and B serves ten years</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A serves 5 years and B serves five years</span></p></td></tr></tbody></table><p> <br /> </p><p> Notice that instead of putting a comma between A's sentence and B's sentence, I put the word "and." That is a fact of the most profound importance, believe it or not. <em>The totality of both sentences, and anything else that results from the playing of those two strategies, is the outcome.</em> Once the outcome matrix is defined by the rules of the game, each player defines an ordinal preference ranking of the four outcomes. The players are assumed to be rational -- which in the context of Game Theory means two things: First, each has a complete, transitive preference order over the four outcomes; and Second, each makes choices on the basis of that ordering, always choosing the alternative ranked higher in the preference ordering over an alternative ranked lower.<br /></p><p><br /> </p><p> Nothing in Rational Choice Theory dictates in which order the two players in our little game will rank the alternatives. A might hate B's guts so much that she is willing to do some time herself if it will put B in jail. Alternatively, she might love him so much that she will do anything to see him go free. A and B might be sister and brother, or they might be co-religionists, or they might be sworn comrades in a struggle against tyranny. [They might even be fellow protesters arrested in an anti-apartheid demonstration at Harvard's Fogg Art Museum -- see my other blog for a story about how that turned out.] <br /></p><p><br /> </p><p> "But you are missing the whole point," someone might protest. "Game Theory allows us to analyze situations independently of all these considerations. That is its power." To which I reply, "No, you are missing the real point, which is that in order to apply the formal models of Game Theory, you must set aside virtually everything that might actually influence the outcome of a real world situation. How much insight into any legal, political, military, or economic situation can you hope to gain when you have set to one side everything that determines the outcome of such situations in real life?"<br /></p><p><br /> </p><p> In practice, of course, everyone assumes that A ranks the outcomes as follows: O21 > O11 > O22 > O12. B is assumed to rank the outcomes O12 > O11 > O22 > O21. With those assumptions, since only ordinal preference is assumed in this game, the <em>payoff matrix</em> of the game can then be constructed, and here it is:<br /></p><p><br /> <table border='0' style='border-collapse:collapse'><colgroup><col style='width:213px'/><col style='width:213px'/><col style='width:213px'/></colgroup><tbody valign='top'><tr><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'> </td><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>B1</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>B2</span></p></td></tr><tr><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A1</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>second, second</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>fourth, first</span></p></td></tr><tr><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A2</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>first, fourth</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>third, third</span></p></td></tr></tbody></table><p> <br /></p><p> [Notice, by the way, that this is not a game with strictly opposed preference orders, because both A and B prefer O11 to O22. With strictly opposed preference orders, you cannot get a Pareto sub-optimal outcome from a pair of dominant strategies -- for extra credit, prove that. :) ]<br /></p><p><br /> </p><p> That payoff matrix contains the totality of the information relevant to a game theoretic analysis. Nothing else. But what about those jail terms? Those are part of the outcome matrix, not the payoff matrix. The payoff matrix gives the utility of each outcome to each player, and with an ordinal ranking, the only utility information we have is that a player ranks one of the outcomes first, second, third, or fourth [or is indifferent between two or more of them, of course, but let us try to keep this simple.] But ten years versus going scot free, and all that? That is just part of the little story that is told to perk up the spirits of readers who are made nervous by mathematics. We all know that when you are introducing kindergarteners to geometry, it may help to color the triangles red and blue and put little happy faces on the circles and turn the squares into SpongeBob SquarePants. But eventually, the kids must learn that none of that has anything to do with the proofs of the theorems. The Pythagorean Theorem is just as valid for white triangles as for red ones.<br /></p><p><br /> </p><p> To see how beguiled we can be by irrelevant stories, consider the following outcome matrix, derived from a variant of the story we have been dealing with:<br /></p><p><br /> <table border='0' style='border-collapse:collapse'><colgroup><col style='width:213px'/><col style='width:213px'/><col style='width:213px'/></colgroup><tbody valign='top'><tr><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'> </td><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>B1</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: solid black 0.5pt; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>B2</span></p></td></tr><tr><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A1</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A serves one day and B serves one day</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A serves 40 years and a day and B goes free</span></p></td></tr><tr><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: solid black 0.5pt; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A2</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A goes free and B serves 40 years and a day</span></p></td><td style='padding-left: 7px; padding-right: 7px; border-top: none; border-left: none; border-bottom: solid black 0.5pt; border-right: solid black 0.5pt'><p><span style='font-family:Times New Roman; font-size:12pt'>A serves 40 years and B serves 40 years</span></p></td></tr></tbody></table><p><br /> </p><p> In this variant, if both criminals keep their mouths shut, they go free after only one night in jail. If they both rat, they spend forty years in jail. If one rats and the other doesn't, the squealer goes free today and the other serves 40 years and a day. Both criminals know this, of course, because the premise of the game is that this is Decision Under Uncertainty, meaning that they know the content of the outcome matrix and of the payoff matrix but not the choice made by the other player. The structure of the payoff matrix associated with this outcome matrix is supposed to be identical with that associated with the original story, namely: For A, O21 > O11 > O22 > O12, and for B, O12 >O11 >O22 > O21, because the premise of the little example is that each player rates the outcomes solely on the basis of the length of his or her sentence, <em>regardless of how long or short that is</em>. It is therefore still the case that O11 is preferred by both players to O22, and it is still the case that <strong>IF</strong> each player's preference order is determined <strong>solely</strong> by a consideration of that player's sentencing possibilities [and that each player prefers less time in jail to more], and that each player chooses a strategy <strong>solely</strong> by attending to considerations of dominance, then the two of them will end up with a Pareto sub-optimal result. But how likely is all of that to occur in the real world? I suggest the answer is, not likely at all. For the upshot of the game to remain the same, we must assume two things, neither of which is even remotely plausible in any but the most bizarre circumstances: First, that each player is perfectly prepared to condemn his or her partner in crime to a sentence of 40 years and a day just to have a chance at reducing a one day sentence to zero; and second, that the two of them, faced with this extraordinary outcome matrix, cannot coordinate on the Pareto Preferred Outcome without the benefit of communication.<br /></p></p></p></p></p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com3tag:blogger.com,1999:blog-4978426466915379555.post-65398019534177810662010-06-28T06:09:00.002-04:002010-06-28T06:16:58.634-04:00THE SCRIBNER'S DILEMMAWelcome back from Spring Break. As you know, I was planning to resume my discussion of the use and abuse of formal models with a discussion of the Prisoner's Dilemma. However, yesterday, as I was writing the subsequent discussion, of the Free Rider Problem, I hit a random key and everything disappeared into cyberpurgatory, from which I have been unable to retrieve it. I tried descending into the cyber underworld, but despite the helpful suggestions of several readers, I simply could not find my lost file. Perhaps the cause was the 103 degree heat here in Chapel Hill, or maybe it was the fact that I had posted a comment on my other blog critical of America's Afghan policy.<br /><br />At any rate, after a troubled night of sleep, I am refreshed, and ready to recreate the fourteen pages that I lost. On Wednesday, this tutorial will resume. As they say in airliners when you have been sitting on the tarmac for three hours, Thank you for your patience [as though you had a choice.]Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0tag:blogger.com,1999:blog-4978426466915379555.post-42205946858979115652010-06-27T11:19:00.002-04:002010-06-27T11:25:59.028-04:00SCREWEDI have just lost the next two posts of this blog -- one on the Prisoner's Dilemma, the other on the Free Rider Problem. I was cruising along, writing in WORD [Windows Vista] when I hit a key [I don't know which one] and everything in the file disappeared. I have done everything I can think of -- I went online and found official Microsoft instructions for finding lost files. No luck. It simply does not seem to be anywhere, even though my version of WORD is set to make an automatic backup every three minutes.<br /><br />If anyone has any ideas, please let me know. I am too old for this! I am going back to a pen and a pad with a carbon sheet under every page. I just do not know whether I can rewrite the thirteen pages or so that I have just lost.Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com3tag:blogger.com,1999:blog-4978426466915379555.post-11360491781233199412010-06-25T13:58:00.002-04:002010-06-25T14:06:55.027-04:00HOME AT LASTWell, I am home again, after a totally successful Paris trip, to give my sister a smashing eightieth birthday. Twenty-two people gathered for a champagne reception followed by a dinner cruise on the Seine. I got to spend time with my sons, my grandchildren, my daughter in law, my sister, my nephew and niece and grandnephews and grandnieces, and my Parisian cousins.<br /><br />Now, it is back to work. If you have not read the three essays I set as homework during the break, see below the links, posted on June 2nd. On Monday, I will resume three times a week installments of this tutorial. I will start with a discussion of The Prisoners' Dilemma, then move on to The Free Rider Problem, as discussed in the essay about Jon Elster. After that will come an extended discussion of Rawls, then a brief discussion of Robert Nozick, then some examination of the use of Game Theory in legal theory, and after that perhaps a discussion of the use of Game Theory and Rational choice Theory in nuclear deterrence and military strategy discussions.<br /><br />If anyone is still with me after all of that, we shall see what else remains to be said.Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0tag:blogger.com,1999:blog-4978426466915379555.post-39115558588699977162010-06-15T02:22:00.002-04:002010-06-15T02:24:49.739-04:00MID-VACATION REMINDERI am still in Paris, preparing for my sister's eightieth birthday bash, but I shall return to this tutorial on June 25th with a discussion of The Prisoner's Dilemma. I hope you will stay with me, and will do the reading I suggested in my last post before the Spring Break.Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0tag:blogger.com,1999:blog-4978426466915379555.post-78025361708491841552010-06-02T05:40:00.003-04:002010-06-02T05:53:23.986-04:00READING ASSIGNMENTS DURING SPRING BREAKI shall be in Paris until June 24th. While I am there, I would like you to read three selections from my writings. When I return, I will begin a lengthy discussion of these and other writings by various authors, applying all the technical materials we have been studying on this blog.<br /><br />I would like you to read <a href="http://www.people.umass.edu/rwolff/elster.pdf">this</a> essay about Jon Elster<br /><br />Then, I would like you to read <a href="http://www.people.umass.edu/rwolff/Rawls chapter.pdf">this</a> selection from my book on Rawls<br /><br />Finally, I would like you to read <a href="http://www.people.umass.edu/rwolff/nozickplus.pdf">this</a> essay about Robert Nozick's ANARCHY, STATE, AND UTOPIA.<br /><br /><br />All three authors make extended use of the Formal Methods we havwe been studying.<br /><br />I hope I see you back here when I return.Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com2tag:blogger.com,1999:blog-4978426466915379555.post-65311059986664372472010-06-02T05:39:00.002-04:002010-06-02T09:04:50.595-04:00Last Installment Before Spring Break<span xmlns=''><p>Once again, let us pause to catch our breath. We arrived at this magnificent theorem by making a series of very powerful constraining and simplifying assumptions. Let us just list some of them:<br /></p><p><br /> </p><p> (0) We began by talking about games.<br /></p><p> (1) We limited ourselves to two person games<br /></p><p> (2) We limited ourselves to players whose preferences satisfy the six powerful Axioms from which we can deduce that their preferences can be represented by cardinal utility functions.<br /></p><p> (3) We limited ourselves to players with strictly competitive preferences<br /></p><p> (4) We allowed for mixed strategies.<br /></p><p> (5) We accepted mathematical expectation as a rational way of calculating the value of a strategy involving elements of risk.<br /></p><p> (6) We adopted von Neuman's extremely conservative rule of choice of strategies -- maximizing the security levels.<br /></p><p> (7) We assumed no pre-play communication between the players.<br /></p><p> (8) We assumed perfect knowledge by both players of the information required to construct the payoff matrix or payoff space.<br /></p><p><br /> </p><p> Every one of these assumptions can be altered or dropped. When that happens, a vast array of possibilities open up. No really powerful theorems can be proved about any of those possibilities, but lots and lots can be said. Here is how I am going to proceed. First, I am going to discuss each of these assumptions briefly and sketch the sorts of possibilities that open up when we drop it or alter it. After that, I will gather up everything we have learned and apply it to a number of specific texts in which Game Theory concepts are used. I will offer a discussion of the so-called Prisoner's Dilemma, a full scale analysis of John Rawls' central claim in <em>A Theory of Justice</em>, a critique of Robert Nozick's <em>Anarchy, State, and Utopia</em>, a detailed critique of a book by Jon Elster called <em>Making Sense of Marx</em>, a critique of the use made of Game Theory by nuclear deterrence strategists, and some remarks on the use of Game Theory concepts in writings by legal theorists. By then, you ought to be able to carry out this sort of critique yourselves whenever you encounter Game Theoretic or Rational Choice notions in your field of specialization.<br /></p><p><br /> </p><p> Now let me say something about each of the nine assumptions listed above.<br /></p><p><br /> </p><p><span style='text-decoration:underline'><strong>(0) The Modeling of Real Situations as Games<br /></strong></span></p><p><br /> </p><p> I identify this as assumption zero because it is so fundamental to the entire intellectual enterprise that it is easy to forget what a powerful simplification and idealization it is. Games are activities <strong>defined by</strong> rules. Imagine yourself watching two people playing chess, not knowing what chess is, but knowing only that a game is being played in the area. How would you describe what you are watching? Which of the things you see are appropriately included in the game and which are extraneous? Which characteristics of the various objects and people in the neighborhood are part of, or relevant to, the game? Is gender relevant? Is race relevant? Is the dog sitting by the table part of the game? Are the troubled sighs of one of the persons a part of the game? How do you know when the game begins and when it ends? Is the clothing of the persons in the area relevant? Are all of the people in the area part of the game, or only some of them? Indeed, are any of them part of the game? You cannot answer any of these questions easily without alluding to the rules of the game of chess. Once you acquaint yourself with the rules of chess, all of these questions have easy answers.<br /></p><p> Now imagine yourself watching a war. Not one of the questions I raised in the previous paragraph has an obvious answer with regard to a war. When does a war start and when does it end? Are the economic activities taking place in the vicinity of the fighting part of the war or not? Who are the participants in a war? States that have formally declared war on one another, other nearby states, private individuals? And so forth. War is not a game. I don't mean that in the usual sense -- that it is serious, that people get killed, etc. I mean it in the Game Theory sense. War is not an activity defined by a set of rules with reference to which those questions can be answered. Neither is market exchange, contrary to what you might imagine, nor is love, nor indeed is politics. There are many <strong>descriptive</strong> generalizations you can make about war, market exchange, love, and politics, but no statements that are <strong>determinative</strong> or <strong>definitive</strong> of those human activities. When you apply the concepts of Game Theory to any one of them, you are covertly importing into your discussion all the powerful simplifications and rule-governed stipulations that permit us to identify an activity as a game. Whenever you read an author who uses the concepts of Game Theory [move, payoff, strategy, zero sum, Prisoners' Dilemma, etc] in talking about some political or military or legal or economic situation, think about that.<br /></p><p><br /> </p><p><span style='text-decoration:underline'><strong>(1) Games with more than two persons:<br /></strong></span></p><p><br /> </p><p> As soon as we open things up to allow for more than two players in a game, everything gets very complicated. First of all, with three or more players, no meaning can be given to the concept of opposed preference orders. We can still make the assumption of cardinal utility functions if we wish, because that is an assumption about an individual player's preference structure, and has no reference to any particular game. With three or more players, it also becomes difficult to represent the game by means of a payoff matrix. Not impossible -- we can always define an n-dimension matrix -- just very difficult either to visualize or to employ as a heuristic device for analyzing a game. That is why writers who invoke the concepts or the language of Game Theory will sometimes reduce a complex social situation to "a player and everyone else," in effect trying to turn a multi-player game into a two player game. That is almost always a bad idea, because in order to treat a group of people as one player, you must abstract from precisely the intra-party dynamics that you usually want to analyze.<br /></p><p> Multi-player games also for the first time introduce the possibility of coalitions of players. Coalitions may either be overt and explicit, as when several players agree to work together, or they may be tacit, as when players who are not communicating overtly with one another begin to adjust their behavior to one another in reciprocal ways for cooperative ends. Once we allow for coalitions, we encounter the possibility of defections of one or more parties from a coalition, and that leads to the possibility that two players or groups of players will bid for the allegiance of a player by offering adjustments in the payoff schedule, or side payments. <br /></p><p> All of this sounds very enticing and interesting, and I can just imagine some of you salivating and saying to yourselves, "Yeah, yeah, now he is getting to the good stuff." But I want to issue a caution. The appeal of Game Theory to social scientists, philosophers, and others, is that it offers a powerful analytical structure. That power is achieved, as I have labored to show you, by making a series of very precise, constraining simplifications and assumptions. As soon as you start relaxing those assumptions and simplifications, you rapidly lose the power of the analytical framework. <strong>You cannot have your cake and eat it too.</strong> By the time you have loosened things up enough so that you can fit your own concerns and problems into the Game Theory conceptual framework, you will almost certainly have lost the rigor and power you were lusting after, and you are probably better off using your ordinary powers of analysis and reason. Otherwise, you are just tricking your argument out in a costume, in effect wearing the garb of a Jedi knight and carrying a toy light saber to impress your children. <br /></p><p><br /> </p><p><span style='text-decoration:underline'><strong>(2) Abrogating one of the Six Axioms<br /></strong></span></p><p><br /> </p><p> The six Axioms laid down by von Neuman conjointly permit us to represent a player's preferences by means of a cardinal utility function. There are various ways in which we might ease those axioms. One is to assume only an ordinal preference structure. As we have seen, that is sufficient for solving some two-person games, and it might be sufficient for usefully analyzing some multi-party games. We may need no more than the knowledge of the order in which individuals rank alternatives. All majority rule voting systems, for example, require only ordinal preference orders, a fact that is important when considering the so-called "paradox of majority rule." <br /></p><p><br /> </p><p> The assumption of completeness is very powerful and potentially covertly biased in favor of one or another ideological position, a fact that I will try to show you when we come to talk about Nozick's work. In effect, the assumption of completeness serves the purpose of transforming all relationships into market exchanges, with results that are very consequential and, at least for some of us, baleful.<br /></p><p><br /> </p><p> Transitivity is also a powerful assumption, and some authors, most notably Rawls, have chosen to deny it in certain argumentative contexts. Recall my brief discussion of Lexicographic orders. When Rawls says that the First Principle of Justice is "lexically prior" to the Difference Principle, he is denying transitivity. He is also, as we shall see, making an extremely implausible claim. Whether he understood that is an interesting question.<br /></p><p><br /> </p><p> One of the trickiest thickets to negotiate is the relationship between money and utility. Because the Axioms we must posit in order to represent a player's preferences by a cardinal utility function are so daunting, those who like to invoke the impressive looking formalism of Game Theory almost always just give up and treat the money payoffs in a game [or a game like situation] as equivalent to the players' utilities. This is wrong, and some folks seem to know that it is wrong, but they almost never get further than just making some casual assumption of declining marginal utility for money. The issue of aversion to risk is usually ignored, or botched.<br /></p><p><br /> </p><p> To give you one quick example of the tendency of writers to ignore the complexity of the six Axioms, here is the entry in the end-of-volume Glossary for "von Neuman-Morgenstern Expected Utility Theory," in <em>Game Theory and the Law</em> by Douglas G. Baird, Robert H. Gertner, and Randal C. Picker:<br /></p><p><br /> </p><p> "Von Neuman and Morgenstern proved that, when individuals make choices under uncertainty in a way that meets a few plausible consistency conditions, one can always assign a utility function to outcomes so that the decisions people make are the ones they would make if they were maximizing expected utility. This theory justifies our assumption throughout the text that we can establish <em>payoffs</em> for all <em>strategy</em> combinations, even when they are mixed, and that individuals will choose a <em>strategy</em> based on whether it will lead to the highest expected <em>payoff</em>."<br /></p><p><br /> </p><p> Now that you have sweated through my <strong>informal</strong> explanation of each of the six Axioms, I leave it to you whether they are correctly characterized as "a few plausible consistency conditions."<br /></p><p><br /> </p><p><span style='text-decoration:underline'><strong>(3) Relaxing the Assumption of Strictly Competitive Preferences<br /></strong></span></p><p><br /> </p><p> As I have already pointed out, there are a great many two-party situations [like two people negotiating over the price of a house] in which the parties do not have strictly opposed preference orders. This is manifestly true in nuclear deterrence strategy situations in which it is in the interest of both parties to avoid one outcome -- namely mutually destructive all out war. <br /></p><p> <br /> </p><p> In addition to games that are partly competitive and partly cooperative, we can also consider totally cooperative games, sometimes called "coordination games." Here is one example. In his book, <em>The Strategy of Conflict</em>, Schelling cites a coordination game he invented to try out on his Harvard classes. He divided his class into pairs of students, and told them that without consultation, they were to try to coordinate on a time and place where they would meet. Each member of the pair was to write a time and place on a slip of paper, and then the two of them would read the slips together. "Winning" meant both students choosing the same time and place. An impressive proportion of the pairs, Schelling reported, won the game by coordinating on "Harvard Square at noon when classes let out." Obviously, their success in coordinating involved their bringing to the game all manner of information that would be considered extraneous in a competitive game, such as the fact that both players are Harvard students. Some time after reading this, I was chatting with a Harvard couple I knew, and I decided to try the game out on them. When I opened the first piece of paper, my heart sank. The young man had written, "4:30 p.m., The Coffee Connection." "Oh Lord," I thought, "he didn't understand the game at all." Then I looked at the young lady's piece of paper. It read, "4:30 p.m., The Coffee Connection." It seems that is where they met every day for coffee. Schelling wins again!<br /></p><p>Not much in the way of theorems, but a great deal in the way of insight, can be gained from analyzing these situations, as Schelling has shown. <br /></p><p><br /> </p><p><span style='text-decoration:underline'><strong>(4) Mixed Strategies<br /></strong></span></p><p><br /> </p><p> The subject of mixed strategies has an interesting history. During the Second World War, the Allies struggled with the problem of defending the huge trans-Atlantic convoys of military supply ships going from the United States to England against then terrible depredations of the Nazi wolf packs of u-boats. The best defense was Allied airplanes capable of spotting u-boats from the air and bombing them, but the question was, What routes should the planes fly? If the planes, day after day, flew the same routes, the u-boats learned their patterns and maneuvered to avoid them. There was also the constant threat of espionage, of the secret anti-u-boat routes being stolen. The Allied planners finally figured out that a mixed strategy of routes determined by a lottery rather than by decision of the High Command held out the most promising hope of success.<br /></p><p><br /> </p><p> Generally speaking, however, mixed strategies are a bit of <em>arcana</em> perfect for proving a powerful mathematical theorem but not much use in choosing a plan of action.<br /></p><p><br /> </p><p><span style='text-decoration:underline'><strong>(5)-(6) Calculation of Mathematical Expectation versus Maximization of Security Levels<br /></strong></span></p><p><br /> </p><p> We have already discussed at some length the limitations of maximization of expected utility as a criterion of rationality of decision making. von Neuman and Morgenstern reject it in favor of the much more conservative rule of maximizing one's security level. We have also seen that this rule of decision making does not allow for risk aversion [or a taste for risk], unless we totally change the set over which preferences are expressed, so that they become compound lotteries over even total future prospects rather than Outcomes in any ordinary sense. As we have also seen, maximization of expected utility rules out lexicographic preference orders, and when I come to talk about the application of this methodology to nuclear strategy and deterrence policy, I will argue that the assumption of non-lexicographic preference orders covertly constitutes an argument for a nuclear strategy favoring the Air Force or the Army rather than the Navy in the inside-the-Beltway budget battles.<br /></p><p><br /> </p><p><span style='text-decoration:underline'><strong>(7) Pre-Play Communication<br /></strong></span></p><p><br /> </p><p> Once we permit pre-play communication, all manner of fascinating possibilities open up. As we might expect, situations with pre-play communication and non-strictly opposed preference orders are among the richest fields for discussion and at the same time allow for the least in the way of rigorous argument or proof. In the hands of an author with a good imagination and a sense of humor, this can be lots of fun, but virtually everything that can be said about such situations can be said without calling them games and drawing imposing looking 2 x 2 payoff matrices. For example, as any hotshot deal maker in the business world knows, when you are engaged in a negotiation, it is sometimes very useful to make yourself deliberately unreachable as the clock ticks on toward the deadline for a deal. If a deal must be struck by noon on Tuesday, and if both parties want to reach agreement somewhere in the bargaining space defined by the largest amount of money the first party is willing to pay and the smallest amount the second party is willing to accept, it is tactically smart for the buyer to make a lowball offer within that space, and then be unavailable until noon Tuesday [somewhere without cell phone coverage, in the ICU of a hospital, on an airplane.] The seller must then accept the offer or lose the sale. Since by hypothesis the seller is willing, albeit reluctant, to sell at that price, she will accept rather than lose the sale. If the seller sees this coming, she can in turn give binding instructions to her agent to accept no offer unless there is the possibility of a counteroffer before the deadline. Then she can make <em>herself</em> unavailable. And so forth. This is the stuff of upscale yuppie prime time tv shows. It just sounds more impressive when you call it Game Theory.<br /></p><p><br /> </p><p><span style='text-decoration:underline'><strong>(8) Perfect Information<br /></strong></span></p><p><br /> </p><p> The general subject of perfect and imperfect information has been so much discussed in economics of late that I need not say anything here. Suffice it to note that formal Game Theory assumes perfect information of the payoff matrix, which embodies both the rules of the game and players' preference structures. Games do allow for imperfect information, of course. Poker players do not know one another's cards, for example. But that is a different matter, built into the rules of the game.<br /></p><p> </p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com3tag:blogger.com,1999:blog-4978426466915379555.post-36651074120606538792010-05-30T20:10:00.002-04:002010-05-30T20:15:47.739-04:00SPRING BREAKOn Wednesday, I will post the last installment [big boffo ending] before going on a three week vacation to Paris. Before I go, I will post links to three things I have written -- one on Rawls, one on Nozick, and one on Elster -- that I would very much like you to read before I return. When I come back, I will start a lengthy discussion of the ways in which these formal materials are used in Political Theory, Deterrence Theory and Military Strategy, and in Legal Theory. That is the real payoff of everything we have been doing so far, and I hope you will stay with me as we go through that material.Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0tag:blogger.com,1999:blog-4978426466915379555.post-58684301492755028892010-05-30T17:59:00.001-04:002010-05-30T17:59:11.314-04:00Eleventh Installment<span xmlns=''><p> Now let us introduce the concept of a Mixed Strategy. All along, we have been working with two-person games whose rules allow for only a finite number of moves [with cut-off points like the rules that limit a chess game]. The Game Tree for such a game, however complex, defines a finite number of strategies for each player, even though that number may, as we have seen, be large even for very simple games. With a finite number of strategies for each player, we can convert the extensive form of the game to the normal form by constructing a payoff matrix with a finite number of rows representing A's strategies and a finite number of columns representing B's strategies. [From a mathematician's point of view, it doesn't matter how big the number of rows and columns, so long as they are finite in number]. But as we have seen, even with strictly opposed preferences, a game in which both players seek to maximize their security level may not have a stable equilibrium solution. Now von Neuman takes the final step by introducing the concept of a mixed strategy.<br /></p><p> When A gives her instructions to the Referee, she need not specify one of her pure strategies. Instead, she can define a probability distribution over her pure strategies and instruct the Referee to construct a Lottery that embodies that distribution. Just to be clear, this is a Lottery in which the "prizes" are strategies, not outcomes. Before leaving for her appointment, A tells the Referee to spin the wheel that has been constructed and play whichever strategy comes up. This is a real spin of the wheel. Neither A nor the referee can know which pure strategy will be played until the wheel has been spun. B can do the same thing, of course.<br /></p><p> Each Lottery, or probability distribution over the set of pure strategies, is a mixed strategy, and quite obviously there are an infinite number of them. With an infinite number of mixed strategies for A, and an infinite number for B, there are of course also an infinite number of <strong>mixed strategy pairs</strong>, which is to say pairs each of which consists of one mixed strategy for A and one mixed strategy for B. Notice that a pure strategy now becomes simply a mixed strategy in which all the probability weights but one are zero.<br /></p><p> For any mixed strategy pair, A can calculate the value to her of those mixed strategies being played against one another, although it is obviously tedious to do so. She says to herself: Suppose I play mixed strategy MA1 and B plays mixed strategy MB1. MA1 offers a .3 probability of my playing pure strategy A1, and MB1 offers B a .2 probability of his playing pure strategy B1, so I will calculate the payoff to me of A1 played against B1 and then discount that payoff, or multiply it, by (.4)(.1) = .04. Then I will do the same for A2 against B1, etc etc etc. Then I will add up all the bits of payoffs, and that is the value to me of the mixed strategy pair( MA1, MB1). I hope by now this is clear and reasonably obvious.<br /></p><p> However, we can no longer construct a payoff matrix, because there are infinitely many mixed strategies for each player. Instead, we need a space of points that represent the payoffs to A of each of the infinite pairs of mixed strategies. Since we are dealing now with strictly competitive zero-sum games, we do not need to represent the payoff to B. Under the normalization we have chosen, that is simply 1 minus the payoff to A.<br /></p><p> At this point I must do what mathematicians call "waving their hands." That is, instead of giving rigorous definitions and proofs [not that I have been doing that so far], I must simply wave my hands and describe informally what is going on, and hope that you have enough mathematical intuition to grasp what I am saying. To represent all of the mixed strategy pairs and their payoffs to A, we are going to need a space with (n-1) + (m-1) + 1 dimensions. The first (n-1) dimensions will represent the probability weights being given to A's n pure strategies. [(n-1) because the weights must add up to 1, so once you have specified (n-1) of them, the last one is implicit.] The next (m-1) dimensions represent the probability weights being given to B's m pure strategies. The last dimension, which you can think of intuitively as the height of a point off of a hyperplane, represents A's payoff. We only need to represent A's payoff because this is a zero-sum game, and B's payoff is just the negative of A's. Obviously, we are only interested in the part of the space that runs, on each axis, from 0 to 1, because both the probability weights and the payoffs all run from 0 to 1 inclusive.<br /></p><p> It is said that the great Russian English mathematician Besicovitch could visualize objects in n-dimensional vector space. If he wanted to know whether something was true, he would "look" at the object in the space and rotate it, examining it until it was obvious to him what its properties were. Then he would take out pen and paper and go through the tedium of proving what he already knew was true. I suspect the same must have been true of von Neuman. Well, God knows it isn't true of me, so I must just soldier on, trying to connect the dots.<br /></p><p> You and I can get some visual sense of what such a space would be like by thinking of the simplest case, in which A and B each have only two strategies. In that nice simple case, the number of dimensions we need is (2-1) + (2-1) + 1, or 3. And most of us can imagine a three-dimensional system of axes. Just think of a three dimensional graph with an x-axis and a y-axis forming a plane or bottom, and a z-axis sticking up. The infinity of A's mixed strategies can be represented as points along the x-axis running from the origin to plus 1. The origin represents the mixed strategy with zero weight given to A1, and therefore a weight of 1 given to A2. In other words, it represents the pure strategy A2. Any point along the line represents some mixture of A1 and A2. The point 1 on the x-axis represents the pure strategy A1. Same thing on the y-axis for B's strategies B1, B2, and the mixtures of them. We thus have a square bounded by the points (0,0), (1,0), (0,1), and (1,1). The z-axis measures the payoff to A for each point in that square, and that set of points between 0 and 1 in height that together form a surface over the square. <br /></p><p> Now [here goes the hand-waving], the function mapping mixed strategy pairs onto payoffs is a continuous one, because a tiny change in the assignment of probability weights results in a tiny change in the payoff. [I hope that is obvious. If it isn't, take my word for it -- which is a great thing for a philosopher to say who is trying to explain some mathematics, I know, but I have my limits!]<br /></p><p> OK. Got that in your mind's eye? Now, let us recall that von Neuman offered a "solution" of strictly competitive games in terms of something called security levels and equilibrium pairs of strategies. Suppose that somewhere in that space of payoffs, there is a point that represents the payoff to a pair of equilibrium mixed strategies. What would that mean and what would it look like?<br /></p><p> Well, what it would mean is this: First of all, if B holds to his mixed-strategy choice, any movement A makes back and forth along the x-axis is going to be worse for her. [That is what it means to maximize your security level]. Visually, that means that as she moves back and forth along the x-axis, the point in space representing the payoff to her goes down. What is more, because of continuity, it goes down smoothly. A little movement one way or the other produces a little move down of the payoff point. A bigger move one way or the other produces a bigger move down. For B, the whole situation is reversed, because B's payoffs are equal to the negative of A's payoff. [Zero-sum game]. So, if A holds to her mixed strategy choice, any movement B makes along the y-axis will push the payoff point up [up for B is bad, because the payoff point is A's payoff, and B's is the negative of that.]<br /></p><p> Now, what does this region of the payoff surface look like? Well, the point we are focusing on has the property that if you move back or forth in one dimension the surface falls off, and if you move back or forth in the other dimension, the surface climbs. Have you ever watched a Western movie? Have you ever ridden a horse? Can you see that the surface looks like a saddle? Especially a Western saddle, which has a pommel to grab onto and a nice high back to the seat. The point right under you on the saddle, the point you are sitting on, is a "saddle point." It has the property that if you run your finger from that point side to side [along the y-axis], your finger goes down, and if you run your finger from that point front or back, your finger goes up.<br /></p><p> Now we know what an equilibrium point looks like, at least in three dimensional space, for the case in which A and B each have two pure strategies. Exactly the analogous thing would be true of a hyperplane in hyperspace [you can get your light sabers at the desk before we go into warp speed]. So, we can say that <strong>If there is a saddle point in the space representing a two person zero sum mixed strategy game, then that point will occur at the intersection of an equilibrium pair of mixed strategies, and in that sense will be a Solution to the game</strong>.<br /></p><p> So, are there such points? Now comes the boffo ending for which all of this has been a preparation. John von Neuman proved the following theorem: <strong>Every two person zero sum mixed strategy game has a solution. That solution is represented by a saddle point in the n-dimensional vector space representing the normal form of the game.</strong> I really think von Neuman must have seen this in one exquisite flash of mathematical intuition, and then just cranked out a proof of a proposition he could just see is true. I am not going to go through the proof [I am not completely crazy], but having brought you this far, I think I owe it to you to just tell you the idea underlying it.<br /></p><p> In a nutshell, here it is. von Neuman defines a continuous transformation or mapping of the strategy space onto itself. He then proves that the transformation has this neat property: a point is mapped by the transformation onto itself [i.e., it remains invariant under the transformation] if and only if that point is a saddle point, and is thus a solution to the game. He then appeals to a famous theorem proved by the great Dutch mathematician L. E. J. Brower, which states that every continuous transformation of a compact space onto itself has at least one fixed point, which is to say a point that the transformation maps onto itself. [Hence, this is known as the Fixed Point Theorem.] <strong>Ta Da!</strong> [Can you believe that when I taught this stuff to my philosophy graduate students at UMass, I not only proved von Neuman's theorem, I even proved Brouwer's theorem? Ah, there were giants in the earth in those days, as the Good Book says.]<br /></p><p> And that is it, folks. That is the high point of formal Game Theory. There is a vast amount more to say, and I am going to say a good deal of it, but the subject never again rises to this level of formal elegance or power. Notice, before we move on, one important fact. von Neuman proves that every zero-sum two person mixed strategy game has a solution. But since Brower's theorem just tells you there exists a fixed point, and doesn't tell you how to find it, in general Game Theory cannot tell us how to solve even this limited category of games. [If there is anyone out there who has ever been involved with Linear Programming, every Linear Programming problem is equivalent to a zero-sum mixed strategy two person game, so that is why, in a certain sense of why, you also cannot be sure of solving a Linear Programming problem.] Oh yes, one final point, which we have already encountered in a simpler form. if there are two saddle points, they are equivalent, in the sense that they give the same payoffs to A and to B.</p></span>Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com2tag:blogger.com,1999:blog-4978426466915379555.post-88816574763976622882010-05-28T05:06:00.001-04:002010-05-28T05:08:22.984-04:00SHAMELESS COMMERCE DEPARTMENTSince I am about to leave for Paris, I thought I would take the opportunity to do some advertising. Six years ago, Susie and I bought a little one room ground floor studio apartment on the Left Bank in the old part of Paris, half a block from the Seine, catty-corner from Notre Dame. We rent it out when we are not there, regularly advertising in the back pages of the NEW YORK REVIEW OF BOOKS. The ad reads " Paris - Blue State Special." We will be there in June, and it is rented for much of the summer, but starting at the beginning of September, it is mostly open. If anyone is interested,send me an email. Where else in Paris can you find the complete German language edition of the collected works of Marx and Engels, not to speak of a shelf of books on Kant and a copy of every version of every book I have ever published?Robert Paul Wolffhttp://www.blogger.com/profile/11970360952872431856noreply@blogger.com0