On learning to cooperate. (Q1867551)

The authors consider a version of a finitely-repeated prisoner's dilemma (PD) with large but fixed average length of players' lifetimes. At each time period \(r=1,\dots,n\) every member of the population plays a game \(\text{PD}(a,r)\) where it receives \(1\) if both players use the strategies \((CC)\) and \(-a^r, 1+a^r, 0\) in the case \((CD), (DC), (DD)\) respectively; \(a\in (0,1)\). The problem is analysed for the set of strategies of the form: the strategy \(k\) prescribes to choose \(C\) in stages \(1,\dots,k-1\) and \(D\) in \(k,\dots,n\), but if the opponent chooses \(D\) in stage \(i<k-1\), then change it for \(D\) in stages \(i+1,\dots,n\). The distribution of players' ages is \((\varepsilon, \varepsilon (1-\varepsilon),\dots)\). Every player of the age \(\tau\) has beliefs regarding the behavior of his opponent. H changes his beliefs in dependence on the result of his experience in the current game PD. The Bellman's equation is applied to determine the optimal behaviour. The main result is that if PD has sufficient many stages, then the arbitrarily large fraction of the population will cooperate in all time periods, even though the beliefs of almost all players are eventually within an arbitrary \(\eta\) of being correct at all information sets. A similar model was studied by \textit{D. Fudenberg} and \textit{D. K. Levine} [Econometrica 61, 547--573 (1993; Zbl 0792.90096)].