Friday, July 2, 2010

The Prisoner's Dilemma Second Installment

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."

    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.

    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 Black Athena 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 can 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.

    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.

    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."

    The little story called The Prisoner's Dilemma ignores just about every fact about a real Law and Order 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.

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].

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.

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.

    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 will act and also enable us to prescribe how rational individuals should 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.

    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.

    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]:

    A: O21 > O12 > O22 > O11

    B: O21 > O12 > O11 > O22

Another pair of preference orders has a story about collective bargaining attached to it:

    A: O21 > O11 > O12 > O22

    B: O12 > O11 > O21 > O22

    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:

    A: O11 > O21 = O22 > O12

    B: O11 > O12 = O22 > O21

    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.]

    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.

    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 not 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.

    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.

    Conclusion: Be not beguiled by 2 x 2 matrices.


  1. While I totally agree with you about the unhelpfulness of game theory in these more extended cases, this does leave me with perhaps a stupid question. What is it good for at all? There do seem to be cases when we have a monotonically rising, continuous utility functions, strictly opposed preferences, etc (say, when actually playing a game!). Have you come across any other possible uses of the formalism? It just seems to be such a powerful tool for decision-making under uncertainty that it should not just be left for gamblers!

  2. Odd as this may sound, I think its greatest use is negative. That is to say, you construct a formal model, analyse it, and then compare it to the real world. By studying the gap between the model and the world, you can learn important things about how the world works. That is also one of the most valuable uses of economic models. The trouble starts when people imagine that the world actually works they way the model does. [This is the seductive power of microeconomics!]

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  4. Having stumbled upon this site and your much needed book five years after the fact, I hope that it is not too late to mention my attempt to formalize a lassiez-faire fantasy in which the optimal outcome for self-regarding agents coincides with the best possible cooperative outcome, in which a tragedy of the commons is averted by players for whom cooperation is irrational. The payoff functions for this are highly unlikely to be realized--preposterous, in my opinion. Originally I had called this, "The Invisible Hand of Ayn Rand," but decided it would be better described as cooperation from self interest in one shot.