A few days ago I began a survey of Martin Nowak's treatment of modern game theory in his book Evolutionary Dynamics. Today I'm going to hit the Prisoner's Dilemma. Roughly, this scenario is one where two individuals are isolated, and if they both keep their mouths shut (cooperate) they get off, but, if one rats the other out while the other keeps silent, the silent partner is screwed while the snitch gets off. If both of them rat the other out they get a prison sentence, but a lighter one than if they had kept silent while the other ratted them out. In other words: ratting the other person out is the "rational" option. So, the question is, why don't we see a war of all against all in the world around us? That is what Nowak addresses in a whole chapter, and his exploration and treatment is highly ambitious, and it is clear that his long term aim is to characterize human action. Nowak aims to finish the house that Robert Trivers began constructing with his theory of reciprocal altruism. Here's a payoff matrix for a Prisoner's Dillemma:^1
The rows to the left are "your" strategies, while the top row indicates the strategy you play against. The integers represent your payoff. So, If you cooperate and the other person cooperates payoff = 3 If you cooperate and the other person defects payoff = 0 If you defect and the other person cooperates payoff = 5 If you defect and the other person defects the payoff = 1 The best group strategy is for both individuals to cooperate, the total payoff is 6 between the two individuals, but, that's not a Nash Equilibrium, it's rational to switch to defect and gain a payoff of 5. On the other hand, if you defect, and other individual defects, the Nash Equilibrium is attained, you can't get a better payoff by switching to another strategy. The aggregate payoff, 2, is lower than if you cooperate, but if your opponent switches you'll get a payoff of 5! There's no sucker's penalty here. More formally, consider the payoff (in biology it would be fitness) for the cooperative strategies as fC, and for the defecting strategies as fD. If the proportion of cooperators is given by x, then the defectors are by definition 1 - x. More formally the proportions would be multipled by the payoffs resulting in: fC = 3 * x + 0 * x fD = 5 * (1 - x) + 1 * (1 - x) Simplifying: fC = 3 * x fD = 4 * x + 1 If x is a proportion between the interval 0 and 1, then by definition 4x + 1 > 3x, so any way you square it, the defecting strategy is the best bet. So why cooperation? Well, reality is a bit more complicated. Let's modify the scenario above, and instead of 1 round we'll play the game for multiple rounds, defined by m. There are two strategies: GRIM,^2 where the player cooperates initially and continues to cooperate unless the other player defects, and which point it will defect permanently. And ALLD, always defect, which is pretty straightforward. The payoff matrix will be defined like so:
CooperateDefect
