Pedagogical Note: This exposition is intended for people unfamiliar with the material. I am trying to explains things slowly and clearly, without inside baseball allusions to sophisticated interpretations. I want each step to be completely clear to anyone who is following along. Maybe at the end I can discuss some of the trickier mathematics. I do not believe that makes any difference to the applications of this material in law, political science, etc.

Part II. The Elements of Rational Choice Theory

Gamblers have always known that in deciding how to place your bets, it is essential to take into account both how much you can win or lose and how likely you are to win or lose. One bet may offer a chance for an enormous payoff [a national lottery, say] but very little chance of winning, while another bet offers a pretty good chance of a small gain. "Enormous" and "small" are not much help, and neither are "little" and "pretty good." How should you evaluate the relative attraction of two bets? How should you decide what is a reasonable entry fee for playing a gambling game?

For a very long time, one standard answer has been to calculate what is called the mathematical expectation of a gamble. This is the size of the possible gain discounted, or multiplied by, the probability of winning. If a gamble pays winners $100 and players have a 10% chance of winning, then the mathematical expectation of the gamble is (100)(.1) = $10. A prudent gambler will pay no more than ten dollars to play that game.

What exactly does it mean to say that the mathematical expectation of the game is $10? Well, one thing it does not mean is that there is any chance at all of winning exactly ten dollars. If you play this game, there are only two possible outcomes: either you will win one hundred dollars or you will just lose your entry fee. The standard answer is that if you play the game over and over again, your average winnings will tend to cluster more and more tightly around ten dollars. This is the sort of calculation that casinos and casino gamblers make. The games are all set up so that over the long run, the House tends to average a small gain per play. That is why casinos make money.

There are two problems with this explanation, both of which play an extremely important role in the application of this formalism to political theory, legal theory, military strategy, and so forth. The first problem is that not all of life is a casino, and in some situations we are not presented with the realistic possibility of "playing" over and over again. Think nuclear war. The second problem is that very long runs of losses are possible, even in a game with an attractive mathematical expectation, and there is always the chance, as you play again and again, that you will run out of money before things "average out."

Are there any other rules someone might propose for making "rational" choices? Indeed there are, but before we talk about any of them, we need to do a good deal more technical preparation.

Notice that to carry out a calculation of mathematical expectation, there are two things you must know: the precise value of each outcome, be it a win or a loss, and the precise probability of each possible outcome. If you do know both of these things, then you are said to be making a decision under risk. If you do not know the probabilities of the several outcomes, but you do know the precise value of each possible outcome, then you are said to be making a decision under uncertainty. Game Theory analyses decision under uncertainty. The Original Position in Rawls' theory is a situation of decision under uncertainty. Maximization of expected utility presupposes decision under risk.

If you are not sure what all the possible outcomes of a choice are, or you do know what they are but do not know what value you place upon them, then these theories have nothing to tell you about how you should make decisions, even though that is a pretty good description of the fix we usually find ourselves in in the real world. As my favorite Rational Choice Theorist, Donald Rumsfeld, liked to say, during his glory days when the Iraq invasion was going the way he wanted it to, there are the knowns and the known unknowns, and then there are the unknown unknowns.

At this point, enter Nicolaus Bernoulli [1687-1759], cousin of the Bernoulli whose work helps to explain why airplanes fly. It seems that in St. Petersburg, which was, in the 18th century, a popular watering hole for European aristocrats, a game was being played by skilled and knowledgeable gamblers whose betting behavior seemed to contradict the well-known rule of maximizing mathematical expectation.

The gamble was this: a fair coin is tossed again and again until it comes up heads. You receive, as your winnings, a number of ducats equal to 2 raised to the (n-1) power, where n is the number of the toss on which heads appears. [Ducats because you couldn't spend rubles anywhere but in Russia, and there are only so many sets of nested Russian dolls you can buy]. So if the coin comes up heads on the first toss, you get 1 ducat. [Why? Because in this case, n = 1, and 1 - 1 = 0, and 2 to the zero power is 1] if the coin does not come up heads until the second toss, then you get 2 ducats. [Why? Because in this case n = 2, and 2 - 1 = 1, and 2 to the first power is 2.] If heads comes up on the third toss, you get 4 ducats, on the fourth toss 8 ducats, and so on ad infinitum. The question for the gamblers was: How much should I be willing to pay to enter the game? Now the classic answer, well understood was: Pay any amount up to the mathematical expectation of the game. So all that was necessary was to calculate the mathematical expectation of the game.

You calculate the mathematical expectation of the game by taking the payoff of each possible outcome [heads on the first toss, heads not until the second toss, heads not until the third toss, etc.], discounting that payoff by the probability of that outcome, and then adding up all th discounted payoffs. That is the mathematical expectation of the gamble.

O.K. There is a 1/2 chance that heads will turn up on the first toss, and (n-1) in this case is 0. 2 to the 0 power is 1. So, discounting the value of the payoff, one ducat, by the odds, one-half, the value to a gambler of heads turning up on the first toss is 1/2 ducat. There is a 1/4 chance heads won't turn up until the second toss. If that happens, then (n-1) is (2-1) or 1. So the payoff is 2 ducats, and the odds of getting it are 1/4, so the expected value is again 1/2 ducat. A little thought or a little experimentation will you show that every payoff of this endless series of terms is 1/2 ducat. And the sum of an infinite number of 1/2s is infinite. In other words, the MATHEMATICAL EXPECTATION of the gamble is INFINITE!

So according to the rule that everybody accepted, namely evaluating a gamble as equal to its mathematical expectation, a gambler should be willing to pay any amount of money he can lay his hands on to enter the game. Or, as Bernoulli rather quaintly puts it, a gambler should refuse to accept any amount of money that another gambler offers him for his ticket to play the game. But that is crazy! As Bernoulli observed in St Petersburg, experienced gamblers would not dream of doing anything like that. How to explain this apparently irrational behavior?

Bernoulli's answer was that the gamblers, contrary to popular opinion, were not trying to maximize their money winnings. Instead, they were trying to maximize the utility they got from their money winnings. I am going to continue with Bernoulli's analysis, and put off for a few paragraphs a deeper look at his solution to the puzzle, but let us be clear right here what he is saying. According to Bernoulli, the gamblers know what the mathematical expectation of the gamble is, in money, and they also know how much "utility," whatever that is, those amounts of money will give them. We will come back to this very soon. Let us continue. Drawing on the theory of logarithms, which had been around for about a century, thanks to John Napier, Bernoulli decided that there was a logarithmic relationship between the amount of money a gambler won or lost and the amount of utility he got from that money. Indeed, Bernoulli claimed to know the formula. It was, he said:

u = b log (a + D)/a

Where a is the amount of one's fortune before the gamble, D is one's winnings from a toss of the coin, u is the utility the gambler gains from the winnings, D, and b is a constant to be determined empirically -- by observation, one supposes. Bernoulli then demonstrated mathematically that if a pauper was given an entry ticket to the game [a pauper is someone with no previous fortune, so D = 0], the rational thing for him to do would be to sell that ticket for as little as 2 ducats. For an ordinary gambler, who presumably already has some money in his pockets, the ticket would not be worth even that much.

There is a whole lot to say about this little story, so settle down. First of all, where on earth did Bernoulli get that formula from? The answer is pretty clear. He invented it, because he knew how to manipulate it mathematically to get the answer he wanted. For our story, it doesn't really matter, because that formula is going to disappear pretty soon, and never reappear in this blog. I just put it in because once in your life you should know what Bernoulli said. Pretty clearly, he decided that was the shape of the utility function because he knew how to solve that formula for D = 0. Here are the really important things that can be said:

(i) If you plot that formula on a graph whose x axis is the amount of money won and whose y axis is the amount of utility gained from that money, you will find that the line rises sharply at first and then bends more and more toward the horizontal, so that as it goes out infinitely to the right [representing longer and longer streaks of all tails before a head shows up], it approaches the horizontal asymptotically. So the total value of the entire gamble in utility approaches some finite amount. Each additional half ducat one gains by yet another tails yields less additional utility than the half ducat gained by the previous appearance of tails. This is called having declining marginal utility for money [or at least for ducats], and if you can grasp this idea, you can understand most of modern Economics.

For all manner of purposes, economists routinely assume declining marginal utility for money. The idea is that although the next dollar is probably worth a good deal to you if you are of modest income, if you keep getting dollars, after a while each additional one will be worth less and less to you. If people's utility functions are nice and regular like this -- continuous, monotonically increasing, with declining marginality -- then we can use the calculus to do all sorts of nifty things that make economists feel really good about themselves.

But the fact is that people's utility functions, assuming they have them, are much less conveniently smooth than that. They are lumpy, they go backwards, and in all sorts of ways are not amenable to the simple manipulations that the calculus allows us when dealing with what are called continuous functions. Leonard Savage and Milton Friedman, a long time ago, published a famous little paper in which they pointed out that since the same people often buy both lottery tickets and life insurance [in the first case paying for the privilege of risk and in the second case paying to avoid risk], their utility functions cannot exhibit monotonic declining marginal utility.

(ii) The second point is equally important, but not usually mentioned. So long as you are talking about the money payoffs, you are in the realm of the public, the objective, the easily measurable. But once you shift to talking about utility, all of this easy publicity disappears. Each person has his or her own utility function relating money to utility, and there is no reason at all to suppose that any two people have identical utility functions. Furthermore, for all manner of well known philosophical reasons, it is impossible to compare one person's utility with another person's utility [basically because you cannot see into another person's mind]. And since a person's utility function, even if it is a cardinal function, is invariant under a linear transformation, there is no way of knowing whether one person's units of utility are bigger or smaller than another person's units, nor is there any way of knowing the relation between the zero point of one person's scale of utility and the zero point of another person's scale. Both of those [scale and size of unity] are arbitrary. It is exactly as though you found two thermometers using different scales of temperature, and there was no way of sticking them into the same bucket of water to discover the conversion formula. [Now you begin to see why I went through that technical stuff earlier]. Everyone has had the experience of thinking that some guy is a sissy because he cannot stand a little pain, even though his wife went through childbirth without an epideural. But suppose he replies that he is more sensitive to pain, and suffers more from something that others find bearable. Not even modern neurophysiology can determine whether that is true or false. Recall the aristocrats who claimed that peasants have courser sensibilities and therefore suffer less from sleeping on rocks than a princess does from having a pea under her mattress. The appropriate response to that claim is not Rational Choice Theory. It is the guillotine. For all of these reasons, it never makes sense to try to add together the utilities of two different people, unless some very special conditions are present [see discussion below of the concept of a zero sum game.]

(iii) The third point is that Bernoulli gives us no way to figure out what someone's utility function is, and neither do most of the people who followed after him and adopted the practice of talking about utility functions. When we get to Game Theory, I will go through the rather complex set of premises that Howard Raiffa and Duncan Luce lay down in their invaluable book, Games and Decisions, from which one can deduce that someone has a cardinal utility function invariant under linear transformations. You will see that it is a huge leap of faith to suppose that people have cardinal utility functions. It is even a leap of faith to suppose that they have utility functions at all.

(iv) But the big problem is, Bernoulli does not tell us what utility is, and neither do any of the people who follow him. This is a huge subject, and I can only scratch the surface of it. Here goes. The word "utility" means "usefulness," which immediately raises the question, useful for what? Intuitively, we do not think of pleasure useful. It is, as we say, an end, not a means. David Hume, in his great work A Treatise of Human Nature, speaks of things that are "useful or agreeable to ourselves or others," clearly implying that a thing that is useful is useful for getting something else that is agreeable.

In the modern discussions of what is called "utility theory," or "the theory of expected utility," this distinction is simply ignored, and utility is treated as somehow equivalent to pleasure. This confusion or unclarity was made worse by the ethical and political theorists -- James Mill, John Stuart Mill, Jeremy Bentham -- who asserted that an act is right only insofar as it produces the greatest happiness for the greatest number, and then called that view Utilitarianism.

(v) If you put these various comments together, here is the sort of problem you arrive at: Rational Choice Theorists take it for granted that the rational thing to do in any situation is to choose the alternative that maximizes expected utility. This presupposes four things, not one of which they can plausibly argue for: First, that we know all the possible outcomes in a situation and their probabilities; Second, that each of us has, and has access to, a cardinal utility function invariant under linear transformations that takes as its argument an outcome and has as its value the measurable quantum of utility that outcome will yield; Third, that we know what quality, experience, or state of mind we are referring to when we speak of a quantum of utility; and finally, that we should choose in accordance with the principle of the maximization of expected utility even in situations in which there is no realistic opportunity to repeat the choice endlessly many times so as to generate a series of outcomes. The elegance of the mathematics seduces rational choice theorists and others into sliding past all these serious issues so that they can get to the fun stuff of playing with the mathematics.

## Monday, May 10, 2010

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Because of the difficulties of defining and measuring "utility," as you point out, some economists have resorted to "revealed preferences." Here judgments of preferences can be inferred, assuming that what you chose you actually preferred, and so rankings are made among various commodity bundles on that basis. But I'm not sure that revealed preferences can result in "continuous differentiable functions."

ReplyDeleteThere are a number of problems with revealed preferences, and they are not just mathematical. The principal problem is that the theory ceases to be a guide to rational choice and becomes a tautological description of actual choice. See my earlier comments about Sophie's Choice. I will have more to say about this down the line.

ReplyDeleteThe requirement that we know all the outcomes and their probabilities looks like an idealizing requirement more suitable to a normative theory than an explanatory one. So if one uses rational choice theory to explain economic behavior, I don't see why one should stick to that requirement.

ReplyDeleteNormative theories like ideal observer theories require full information, but then they are vulnerable to the conditional fallacy. Actual agents, precisely because they lack full information, can have good reasons to gather more information, or to satisfice rather than maximize, but ideal agents can have no such reasons.

(I still don't understand Bernoulli's formula, but probably it's just my numerical illiteracy. From your explanation it's not clear whether D = winnings or previous fortune. But I take it that the key point is the introduction of a notion of expected utility that can diverge from monetary value and has decreasing marginal utility with increasing consumption.)

On your final point about maximizing expected utility in one-shot gambles... couldn't the utility from one's risk preference/aversion (instead of mathematical expectation) get the job done?

Boram Lee, I don't understand the last question. What job would it get done, and how?

ReplyDeleteThanks, most fun post so far in my opinion. Two questions:

ReplyDelete1. Diminishing marginal utility: Some economists will point out that, if you really want to, you can accommodate any data (even lottery tickets and life insurance) without giving up diminishing marginal utility. But this is, on Popperian grounds, a weakness of the theory. But let's say the economist went another route by agreeing that individual utility functions will be irregular, but that this will wash out when aggregated. As prices reflect massive aggregates, we can safely use this assumption when discussing prices. Any objection to this?

2. You say that RTC presupposes that we know all possible outcomes and their probabilities. Surely this is too strong? Does RTC not just say that we use the information concerning outcomes that we are aware of efficiently? Not that I am saying this is true, but it is more defensible.

It's all pretty vague in my mind, but I will try to explain what I meant.

ReplyDeleteIn the case of one shot gambles, I understood your point to be that the rationale for basing one's choice on mathematical expectation (i.e., that in the long run one's gain will average out to what's mathematically expected) doesn't apply. So the job I took to be one of providing a modified rationale on which to base one's choice in one-off cases.

Then it occurred to me that the modified rationale could be something like what Bernoulli suggests for the St. Petersburg Paradox. The more income I already have at my disposal, the less reluctant I'd be to part with some of it, and thus the more willing to take risks. So my preference for risk in one-shot gamble, and how much I'm willing to risk, depends in part on how much money I've already got, and in part on some unknown quantity to be empirically determined about my taste for gambling.

Not sure if I am representing Bernoulli's proposal correctly, and also what I had in mind, but this is the best I can come up with.

"The appropriate response to that claim is not Rational Choice Theory. It is the guillotine."

ReplyDeleteWell put.