The Iterated Prisoner's Dilemma

 

The opponent

Cooperate

Defect

You

Cooperate

(R, R)

(T, S)

Defect

(S, T)

(P, P)

  

If two players play prisoners' dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called Iterated Prisoners' Dilemma (IPD).

 

Two conditions must be hold for the above general matrix of PD. Firstly, the order of the payoffs is important. The best a player can do is T (temptation to defect). The worst a player can do is to get the sucker payoff, S. If the two players cooperate then the reward for that mutual cooperation, R, should be better than the punishment for mutual defection, P. Therefore, the following must hold.


T > R > P > S


Secondly, players should not be allowed to get out of the dilemma by taking it in turns to exploit each other. Or, to be a little more pedantic, the players should not play the game so that they end up with half the time being exploited and the other half of the time exploiting their opponent. In other words, an even chance of being exploited or doing the exploiting is not as good an outcome as both players mutually cooperating. Therefore, the reward for mutual cooperation should be greater than the average of the payoff for the temptation and the sucker. That is, the following must hold.


R > (S + T) / 2

 

IPD is fundamental to certain theories of human cooperation and trust. On the assumption that the game can model transactions between two people requiring trust, cooperative behaviour in populations may be modeled by a multi-player, iterated, version of the game. It has, consequently, fascinated many scholars over the years. In 1975, Grofman and Pool estimated the count of scholarly articles devoted to it at over 2,000. The iterated prisoners' dilemma has also been referred to as the "Peace-War game".

 

·         An IPD is finite if it repeats exactly n rounds. Mutual defection is the only Nash equilibrium in this situation according to classic game theory. The proof is inductive by means of the so called backward induction. Both players might as well defect in the last round, since they will not have a chance to punish the opponent. Thus, they might as well defect in the second-to-last round since both will defect in the last round no matter what is done. The same induction applies till the first round, and therefore both will defect throughout the game.

 

·         An IPD is infinite if its iteration n→∞. Under this circumstance, mutual cooperation is equilibrium as well. A series of Folk Theorems in game theory contains theoretical explanation of it.

 

·         An IPD is indefinite if its iteration is stochastic. A discount factor w is the probability that the next round will be played. For example, the expect length of an IPD with discount factor w=0.98 is 50. Mutual cooperation is equilibrium of indefinite IPD according to Folk Theorems.