Repeated games with asynchronous monitoring of an imperfect signal

We consider a long-run player facing a sequence of short-run opponents who receive noisy signals of the long-run player's past actions. We modify the standard, synchronous-action, model by supposing that players observe an underlying public signal of the opponent's actions at random and privately known times. In one modification, the public signals are Poisson events and either the observations occur within a small epsilon time interval or the observations have exponential waiting times. In the second modification, the underlying signal is the position of a diffusion process. We show that in the Poisson cases the high-frequency limit is the same as in the Fudenberg and Levine (2007, 2009) study of limits of high-frequency public signals, but that the limits can differ when the signals correspond to a diffusion.