A folk theorem for repeated games played on a network

Communication in repeated network games with imperfect monitoring

“…At the beginning of phase III, I add a stage in which player j draws i.i.d. sequences of pure actions according to Q N (k) (k) (respectively Q N ( ) ( ) when both players k and are minmaxed) 23 for the minmaxing players for T (δ) periods. Player j announces the sequences publicly to all players except k (respectively player ).…”

“…Moreover, in that case, public announcements are not required, and players can be only allowed to send private messages (coordination in cases (i) and (ii) above is not required). In addition, it is possible to restrict communication along the network: the necessary and sufficient conditions for a folk theorem to hold are known (but are different from Condition DN) if (i) only private communication is allowed (no public announcements) and (ii) I consider Nash equilibrium of the repeated game or uniform sequential equilibrium for the undiscounted case ( [23]). If communication is restricted along the network, finding conditions for a folk theorem to hold is an open problem for (i) public announcements and (ii) sequential equilibria of repeated games with discounting.…”

“…This paper departs from [23] Second, contrary to [23] where a Nash folk theorem is established, I impose sequential rationality. The main difficulty here is to provide incentives to follow the communication protocol in case of a deviation, and to follow the equilibrium strategy (in particular for the minmaxing players during the punishment phase after a deviation).…”

“…The following example is taken from [23]. It shows that a necessary condition for a folk theorem is that the payoff functions are sufficiently rich to enable players to detect deviations.…”

“…The proof builds on the arguments developed in [23] (Section 3 page 720), and is thus relegated to Appendix A. Intuitively, when Condition DN is violated, there exists a player i, two of his neighbors j and k, and two deviations of j and k such that: for any action profile (possibly mixed), both j's and k's deviations induce the same distribution over player i's payoffs, and the same distribution over the messages received by player i.…”