THE PRISONER'S DILEMMA PART TWO

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:

	B1	B2
A1	A serves one day and B serves one day	A serves 40 years and a day  and B goes free
A2	A goes free and B serves 40 years and a day	A serves 40 years and B serves 40 years

            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, regardless of how long or short that is.  It is therefore still the case that O11 is preferred by both players to O22, and it is still the case that IF each player's preference order is determined solely 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 solely 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.

            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."