Abstract
We study repeated games with frequent actions and frequent imperfect public signals, where the signals are the aggregate of many discrete events, such as sales or tasks. The high-frequency limit of the equilibrium set depends on both the probability law governing the discrete events and on how many events are aggregated into a single signal. When the underlying events have a binomial distribution, the limit equilibria correspond to the equilibria of the associated continuous-time game with diffusion signals, but other event processes that aggregate to a diffusion limit can have a different set of limit equilibria. Thus the continuous-time game need not be a good approximation of the high-frequency limit whenever the underlying events have three or more possible values.
Highlights
We study the limits of equilibria of repeated games with imperfect public information as the frequency of observations and actions grows to infinity
We feel that this is of relevance for interpreting results about continuous time games where players observe the state of a diffusion process, as in Faingold and Sannikov (2007), Fudenberg and Levine (2007a), Sannikov (2007a), and Sannikov and Skrypcaz (2008), because in most settings of interest the diffusion assumption is an approximation for a sum of discrete events
Many different arrays converge to a given diffusion process, and the limit equilibria of these arrays is not in general determined by the parameters of the limiting diffusions, but binomial arrays are an exception to this result
Summary
We study the limits of equilibria of repeated games with imperfect public information as the frequency of observations and actions grows to infinity. We focus on cases where the public signal is an aggregate of several or many discrete events, such as sales, price changes, or components of quality, and in particular on the case where the distribution of this aggregate converges to a diffusion process We feel that this is of relevance for interpreting results about continuous time games where players observe the state of a diffusion process, as in Faingold and Sannikov (2007), Fudenberg and Levine (2007a), Sannikov (2007a), and Sannikov and Skrypcaz (2008), because in most settings of interest the diffusion assumption is an approximation for a sum of discrete events.. Our work is similar to that of Sannikov and Skrypacz (2008) They consider a game with two long-run players observing the infinite-dimensional sample path of a continuous-time Levy process at discrete intervals and provide an upper bound on the set of pure-strategy equilibrium payoffs as the time interval shrinks to zero. Whether or not the continuous time results are relevant is an empirical issue and is not a necessary consequence of the periods being short
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