In its simplest form, the prisoner's dilemma involves two players, A and B. Imagine both are accused of a crime and are being interrogated in separate rooms. If both maintain their innocence, they are released for lack of evidence. If A claims innocence but B says A is guilty, then A goes to jail and B gets a reward. The symmetric result follows if B claims innocence and A says B is guilty. If each says the other is guilty, both go to jail. The outcome for each player is called the player's payoff. Maintaining the innocence of both is called cooperating, saying the other person is guilty is called defecting.
Imagine the scenario just described is one round of a game, that we will now iterate. Under fairly general conditions, the most robust strategy is tit-for-tat. That is, start by cooperating and then on all successive rounds do what your opponent did in the previous round. Genetic algorithms have been used to evolve strategies in this game, but that is not a direction we pursue here.
Instead, we consider iterated play on a square grid of players, introduced by Nowak and May. In Nowalk's and May's version, when two cooperators play, both receive a payoff of 1; when two defectors play, both receive 0; when a defector and a cooperator play, the cooperator receives 0 and the defector receives b. The dynamics of the interactions depends on the value of b. In each iteration, every player plays all its eight immediate neighbors (and itself). The sum of the payoffs from these nine games is the payoff of that player. After each iteration, each player looks at the nine players in its immediate neighborhood, and adopts the strategy of whichever has the highest payoff.
For example, in a grid of 51 by 51 players with an initial distribution of 80 percent cooperators, ten generations gives the pictures below, for b = 1.72, b = 1.77, and b= 1.82. The color coding is this: blue and green cells are cooperators, blue were cooperators in the previous generation, green were defectors in the previous generation; red and yellow cells are defectors, red were defectors in the previous generation, yellow were cooperators in the previous generation.
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Nowak and May uncover many interesting dynamical behavors, and extend this study to stochastic and continuous-time rules in Nowak, Bonhoeffer, and May. We mention one more example. Starting from a field of all cooperators with a single defector, with b = 1.85 Nowak and May observe a dynamical fractal. Note the pattern growing on the corners replicates earlier stages of the entire pattern.
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Some have used the iterated spatial prisoner's dilemma as a model for international realtions. We shall not address that issue, but reiterate that, depending on the value of the payoff, iteration can generate intricate fractal-like shapes. This should not necessarily be such a surprise, because the iterated prisoner's dilemma is a form of cellular automaton, and we know some cellular automata can generate fractals. Still, this may signal an undelying relation between political science and fractals.
We mention that similar models have been used to model the evolution of cities, and to explain the self-similar organization of the distribution of services in large cities.