The evolution of stochastic strategies in the prisoner's dilemma
In a bimatrix game the conflict cannot necessarily be solved by Nash's equilibrium, and how children know such games get more interesting if they are played repeatedly. In the paper \(2\times 2\) games with the actions C (cooperative) and D (defect) are investigated. (A special view is directed to the known games ``Prisoner's Dilemma'' and ``Chicken'' where the 4 matrix entries satisfy some relations of ordering and magnitude, but most of the theorems are generally valid.) For repeated games, so-called reactive strategies \((y,p,q)\in [0,1]^ 3\) are treated where y is the probability to play C at the first stage, while p and q are the conditional probabilities to play C after an opponent's C (resp. D) at the previous stage. The treatment namely of stochastic strategies is used for modelling of uncertainties and errors in biological populations or computers. The sequence of random actions forms a Markov chain, and the final probabilities of it are given. The corresponding iterated game is a Markov game with one state, and the payoff A for the w-discounted (or w-stopped) as well as for the average case is calculated. For the undiscounted case now, the best reply for an individual mutant is searched for. For the corresponding adaptive dynamic which expresses the evolution of a population over time, a system of partial differential equations is derived in which A is involved. Many graphics illustrate the solution. In further chapters, also the w-discounted case is treated. Not all of the rich material can here be described in detail. The results can mainly be applied to population biology. Also ethical notions (escalation of conflict, foregiveness) are touched. The thorough introduction, comments, epilogue and list of references, underline the authors' great competence in this field.
scientific article; zbMATH DE number 4191440
scientific article