A Beauty Contest with Flexible Information Acquisition

This paper studies a beauty-contest coordination game. A continuum of players get payoffs based on the squared distance of their action from an unobserved fundamental state of the world and the average action among all players. Each player receives a signal whose probability distribution conditional on the value of the fundamental is part of their strategy. This flexible information acquisition technology allows players to choose not only how precise but also what kind of information they want to get about the fundamental. Information is costly, in particular cost is linear in Shannon’s mutual information measure between the prior of the fundamental and the player’s chosen conditional distribution. When unit costs are high enough, there is a unique equilibrium where players do not obtain information. For lower information unit costs, players restrict their attention around the expected value of the fundamental while paying little attention to fundamental values away from it. As costs get lower, players follow the value of the fundamental more closely. A stronger coordination motive or a more concentrated distribution of the fundamental have the same effect as a higher information cost. When information costs exceed a certain threshold, players do not acquire any information and play the ex-ante expected value of the fundamental with probability 1. The case of a normally distributed fundamental is examined in more detail. Only in this case there exist equilibria whereby the average action of the population is an affine function of the realized value of the fundamental. In most parameter combinations, there exists a unique equilibrium within the classes of affine equilibria and equilibria without information acquisition. Interestingly, when the coordination motive is high and for relatively high information costs, there is multiplicity of equilibria within the classes considered.