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 10year 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 fiveyear 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 suboptimal. 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 socalled "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:

B1

B2

A1

A serves six months and B
serves six months

A serves ten years and B
goes free

A2

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 coreligionists,
or they might be sworn comrades in a struggle against tyranny. [They might even be fellow protesters
arrested in an antiapartheid 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:

B1

B2

A1

second, second

fourth, first

A2

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 suboptimal 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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