Delayed-response strategies in repeated games with observation lags

Abstract

We extend the folk theorem of repeated games to two settings in which players' information about others' play arrives with stochastic lags. In our first model, signals are almost-perfect if and when they do arrive, that is, each player either observes an almost-perfect signal of period-t play with some lag or else never sees a signal of periodt play. In the second model, the information structure corresponds to a lagged form of imperfect public monitoring, and players are allowed to communicate via cheap-talk messages at the end of each period. In each case, we construct equilibria in "delayedresponse strategies," which ensure that players wait long enough to respond to signals that with high probability all relevant signals are received before players respond. To do so, we extend past work on private monitoring to obtain folk theorems despite the small residual amount of private information.

Some notes and comments on the efficient use of information in repeated games with Poisson signals

“…Figure 4 shows the punishment probability and the signals used to provide incentives for varying : Figure 2 shows the payo¤s for varying : The payo¤s increase for K = 0; They present a u-shape for K = 1; :::; 13; and are convex decreasing for K = 14; :::; 19: The largest payo¤ is achieved at 9 = 1 or 10 = 0; i.e., the coordinate ( ; v) = (8:43; 1:99) : This observation is consistent with Abreu et al [1], i.e., time lags favor the provision of incentives and payo¤s. Fudenberg et al [3] exploit this argument to show a non-limit folk theorem.…”

Combining local and global evolutionary trajectories of brain-behavior equilibrium through game theory

“…In the classical example, called the prisoner's dilemma, the years in jail (payoff) of two prisoners (players) depends on their choice (action) to confess their crime or not ( Figure 3A). Among the many variants of game theory, games can involve more than two players (n-players game; Nash, 1950;Czumaj et al, 2017), can be sequential or simultaneous depending on the temporal delay between choices (Fudenberg et al, 2014), can entail stochastic elements if the strategies or the payoff follow probabilistic laws (Shapley, 1953;Solan and Vieille, 2015), and can be cooperative or non-cooperative depending on whether or not players are allowed to form temporarily stable coalitions (Nash, 1951).…”

Approachability with delayed information

“…Levy [12] analysed zero-sum stochastic games when one or both players observe past actions of their opponent with a time-dependent delay. Recently Fudenberg et al [4] examined equilibria in repeated games when players' signals about the actions of others arrive with a certain lag. However, the problem of approachability with delayed imperfect monitoring was not yet considered.…”

“…We use (X) to denote the set of probability distributions on the set X 4. We consider the L 2 -norm when using the · notation.…”