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Monday, February 3, 2014


I am back, tired but happy to be able, once again, to type with two fingers rather than one. I have a good deal to say on a variety of subjects, but I promised to reprise my take-down of the dreaded Prisoner's Dilemma so here it is, in three parts -- first part today. When I said that I would do this, Marinus Ferreira posted a long comment essentially saying that he had made some devastating comments about my discussion when it first appeared and I had not satisfactorily answered him. I am going to post all three parts of the discussion as it originally appeared and then turn to his, and perhaps other, comments. Note that this is part of an entire short book, available on, and at several points I refer back to things I said in earlier parts of that book. You can find the entire text on box if you are interested.

So, here we go:

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:
"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 (defects) and the other remains silent (cooperates), 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?"
            The following matrix is taken to represent the situation.

B1  cooperate
B2  defect
A1  cooperate
6 months, 6 months
10 years, Go free
A2  defect [
Go free, 10 years
5 years, 5 years

            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 both 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, but the players fail to coordinate on this strategy pair 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.
            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.
            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.
            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 Outcome Matrix for the game looks like this:

A serves six months and B serves six months
A serves ten years and B goes free
A goes free and B serves ten years
A serves 5 years and B serves five years

            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.  The totality of both sentences, and anything else that results from the playing of those two strategies, is the outcome.  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.

            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.] 
            "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?"
            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 payoff matrix of the game can then be constructed, and here it is:

second, second
fourth, first
first, fourth
third, third
            [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.  :) ]
            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.

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