Cooperation can be a Nash equilibrium in the IPD if you have a finite but unknown number of iterations (e.g. geometrically distributed). Also, if the number of iterations is known but very large, cooperating becomes an ϵ-Nash equilibrium for small ϵ (if we normalize utility by its maximal value), so agents which are not superrational but a little noisy can still converge there (and, agents are sometimes noisy by design in order to facilitate exploration).
Thank you for pointing this out. Here’s a source for the first claim.
Finitely repeated games with an unknown or indeterminate number of time periods, on the other hand, are regarded as if they were an infinitely repeated game. It is not possible to apply backward induction to these games.
And here’s a source that at least provides a starting point for the second claim about ϵ-Nash equilibria.
Given a game and a real non-negative parameter ϵ, a strategy profile is said to be an ϵ-equilibrium if it is not possible for any player to gain more than ϵ in expected payoff by unilaterally deviating from his strategy. Every Nash Equilibrium is equivalent to an ϵ-equilibrium where ϵ = 0.
Another simple example is the finitely repeated prisoner’s dilemma for T periods, where the payoff is averaged over the T periods. The only Nash equilibrium of this game is to choose Defect in each period. Now consider the two strategies tit-for-tat and grim trigger. Although neither tit-for-tat nor grim trigger are Nash equilibria for the game, both of them are ϵ-equilibria for some positive ϵ. The acceptable values of ϵ depend on the payoffs of the constituent game and on the number T of periods.
Cooperation can be a Nash equilibrium in the IPD if you have a finite but unknown number of iterations (e.g. geometrically distributed). Also, if the number of iterations is known but very large, cooperating becomes an ϵ-Nash equilibrium for small ϵ (if we normalize utility by its maximal value), so agents which are not superrational but a little noisy can still converge there (and, agents are sometimes noisy by design in order to facilitate exploration).
Thank you for pointing this out. Here’s a source for the first claim.
And here’s a source that at least provides a starting point for the second claim about ϵ-Nash equilibria.