Evolutionary Game Theory - Prisoners Dilemma

Prisoners Dilemma

One of the great difficulties of Darwinian Theory, and one recognised by Darwin himself was the problem of altruism. If the basis for selection is at the individual level, altruism makes no sense at all. But universal selection at the group level (for the good of the species, not the individual) fails to pass the test of the mathematics of game theory and is certainly not found to be the general case in nature. Yet in many social animals altruistic behaviour can be found. The solution to this paradox can be found in the application of Evolutionary Game Theory to the “Prisoners Dilemma” game - a game which tests the payoffs of cooperating or in defecting from cooperation. It is certainly the most studied game in all of Game Theory.

As with all games in Evolutionary Game Theory the analysis of Prisoners Dilemma is as a repetitive game. This repetitive nature affords competitors the possibility of retaliating for “bad behaviour” (defection) in previous rounds of the game. There is a multitude of strategies which have been tested by the mathematics of EGT and in computer simulations of contests and the conclusion is that the best competitive strategies are general cooperation with a reserved retaliatory response if necessary. The most famous and certainly one of the most successful of these strategies is Tit for Tat which carries out this approach by executing the simplest of algorithms.

The pay-off for any single round of the game is defined by the pay-off matrix for a single round game (shown in bar chart 1 below). In multi-round games the different choices - Co-operate or Defect - can be made in any any particular round, resulting in a certain round payoff. It is, however, the possible accumulated pay-offs over the multiple rounds that count in shaping the overall pay-offs for differing multi-round strategies such as Tit-for-Tat.

Example 1: This is the straightforward single round Prisoners Dilemma Game (the game is executed in only one round) The classic Prisoners Dilemma game payoffs gives you a maximum payoff if you defect and your "partner" co-operates (this choice for you is known as TEMPTATION). If however you co-operate and your partner defects you get the worst possible result ( the SUCKERS PAYOFF ). In these payoff conditions the best choice (a Nash Equilibrium in classic game theory) is to defect.

Example 2: This is Prisoners Dilemma played repeatedly The strategy employed is Tit-for-Tat which alters behaviours based on the action taken by a partner in the previous round - ie. reward co-operation and punish defection. The affect of this strategy in ACCUMULATED payoff over many rounds is to produce a higher payoff for both players co-operation and a lower payoff for defection. This removes the Temptation to defect. The suckers payoff also becomes less, although "invasion" by a pure defection strategy is not entirely eliminated (this can be overcome by clustering of cooperative strategies - see spatial games below)