The Emerging Revolution in Game Theory

The Emerging Revolution in Game Theory
technologyreview.com

Each prisoner dilemma round has four possible outcomes: CC, CD, DC and DD (C=Cooperate,D=Defect). We study probabilistic strategies which depend only on the previous round, so for instance, P(player 1 plays C at round i+1|CD was played at round i)=0.3. From the paper:

Such games can be described by a Markov process defined by the four probabilities that characterize each of the two player’s strategies [9] (because this is an infinitely repeated game, the probability to engage in the first move–which is unconditional–does not play a role here). Each Markov process has a stationary state given by the left eigenvector of the Markov matrix, which in this case describes the equilibrium of the process. The expected payoff is given by the dot product of the stationary state and the payoff vector of the strategy. But while the stationary state is the same for either player, the payoff vector–given by the score received for each of the four possible plays CC, CD, DC, and DD–is different for the two players for the asymmetric plays CD and DC. Because the expected payoff is a linear function of the payoffs, it is possible for one strategy to enforce the payoff of the opponent by a judiciously chosen set of probabilities that makes the linear combination of determinants vanish (hence the name ZD strategies). Note that this enforcement is asymmetric because of the asymmetry in the payoff vectors introduced earlier: while the ZD player can choose the opponent’s payoff to depend only on their own probabilities, the payoff to the ZD player depends on both the ZD player’s as well as the opponent’s probabilities. This is the mathematical surprise: the expected payoff is usually a very complicated function of six probabilities (and four payoff values, for the four possible plays). When playing against the ZD strategy, the payoff that the opponent reaps is defined by the payoffs and only two remaining probabilities that characterize the ZD strategies.

Such games can be described by a Markov process defined by the four probabilities that char- acterize each of the two player’s strategies [9] (because this is an infinitely repeated game, the probability to engage in the first move–which is unconditional–does not play a role here). Each Markov process has a stationary state given by the left eigenvector of the Markov matrix, which in this case describes the equilibrium of the process.

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If we assume that two opponents play a sufficiently large number of games, their payoff approaches the payoff of the Markov stationary state [1,9]. We can use this mean expected payoff as the payoff to be used in the payoff matrix E that will determine the ESS.

TLDR: keep varying your strategy until you win in the long term, using your mean expected payoff as a payoff matrix in each subsequent state.