The Whole and the Parts – The Prisoner’s Dilemma

The justly famous Prisoner’s Dilemma describes a situation in which the best solution for each part is different from the best solution for the whole.  We’ll look at this situation as a first installment of a series of posts on situations where the whole is greater (or merely distinctly different) from the sum of the parts.

What is nice about the Prisoner’s Dilemma is that it can be stated simply and looked at from different angles.  It is a model, not necessarily to be taken literally as a true representation about prisoners.  It suggests a phenomenon that we can look for in real situations.  As in any model, certain aspects are pushed into the foreground at the expense of other details which are relegated to the background.

The Prisoner’s Dilemma deals with two hypothetical suspects of a crime, who have been arrested and who both happen to be guilty.  Yet, guilty or not, it may not be possible to prosecute them successfully without evidence in court, evidence obtained from the other suspect.   The prosecutor wants each suspect to  cooperate by providing evidence for the guilt of the other.  Of course, the suspects aren’t interested in providing that evidence.  So each suspect, independent of the other one, is offered a deal.  If one provides evidence against the other, the betrayer goes free and the silent partner will receive a full prison sentence.   If both betray each other, they will each end up with half of the full sentence.   If neither betrays the other, they will each get a sentence on a minor charge only.  Each of the suspects has to choose without knowing what the other one will choose, and the only choices are silence or betrayal.
A useful way to represent the possible combinations is through a table with the pay-offs:

prisoners-dilemma-pay-offs1Joey looks at the first table for his pay-offs, Jeremy looks at the second one for his.  The pay-offs indicate the length of their prison sentence, so a small number is better than a large number.  Note that though the tables are different, the situation is entirely symmetric.

Joey notes that regardless of what Jeremy chooses, he is better off if he betrays Jeremy.  For if Jeremy stays silent, Joey ends up with no prison term at all if he betrays Jeremy.  And if Jeremy rats on Joey, Joey ends up with a 4-year sentence rather than an 8-year sentence by turning evidence.  Yet the situation is symmetric, and Jeremy notices that in his table of pay-offs, choosing betrayal over silence guarantees him a better outcome regardless of what Joey does.

Left to their own devices, and applying the logic of the parts, each may decide that betrayal is the best course of action – even if knowing very well that the best course of action for the whole of them would be to both stay silent.  For each, choosing the path of silence is very dangerous because it may lead to the very worst outcome: that the other one rats on him.  The logic of the whole is different from the sum of the logic of the parts.  Neither of the parts appears to be able to afford to apply the logic of the whole – and yet, if only all of them applied the logic of the whole, the outcome (for every single one of the participants) would be far superior to the sum of the logic of the parts.

Again, the Prisoner’s Dilemma is just a model, and it’s useful to look to see where the model applies and where it doesn’t.  But even if you don’t yet see a situation where it applies, you may start to appreciate that it is possible for all of us to do what we think is best and yet end up with a situation that is far from the best.  A closely related model is that of the Tragedy of the Commons.  Lest you think the logic of the whole compared to the logic of the parts is inherently tragic, I will point you to two books that show examples of the opposite: The Evolution of Cooperation by Robert Axelrod, and Nonzero: The Logic of Human Destiny by Robert Wright.

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One Response to The Whole and the Parts – The Prisoner’s Dilemma

  1. Pingback: The Whole and the Parts – The Series « Learning and Unlearning Math

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