“Information Acquisition and Reputation Dynamics” by Qingmin LIU
We study reputation games where a long-lived player with a possible commitment type faces a sequence of short-lived players who must pay to observe the long-lived player’s past behavior. In this costly information model we show that equilibrium behavior is cyclical. The long-lived player builds her reputation up only to exploit it; then builds it up again, and so on. We call this behavior reputation renewal and show that for a wide class of reputation games as well as for a host of possible alternatives to the information cost structure, all equilibria display reputation renewal. We provide a method to construct the equilibria, and explicitly construct an essentially unique equilibrium for the case of linear cost functions. We also find that in the reputation renewal equilibria, the short-lived players always randomize how much information they purchase, but play pure strategies once the information is obtained.