Rational exaggeration and counter-exaggeration in information aggregation games

We study an information aggregation game in which each of a finite collection of “senders” receives a private signal and submits a report to the center, who then makes a decision based on the average of these reports. The integration of three features distinguishes our framework from the related literature: players’ reports are aggregated by a mechanistic averaging rule, their strategy sets are intervals rather than binary choices and they are ex ante heterogeneous. In this setting, players engage in a “tug-of-war,” as they exaggerate and counter-exaggerate in order to manipulate the center’s decision. While incentives to exaggerate have been studied extensively, the phenomenon of counter-exaggeration is less well-understood. Our main results are as follows. First, the cycle of counter-exaggeration can be broken only by the imposition of exogenous bounds on the space of admissible sender reports. Second, in the unique pure-strategy equilibrium, all but at most one player is constrained with positive probability by one of the report bounds. Our third and fourth results hold for a class of “anchored” games. We show that if the report space is strictly contained in the signal space, then welfare is increasing in the size of the report space; but if the containment relation is reversed, welfare is independent of the size of the space. Finally the equilibrium performance of our heterogeneous players can be unambiguously ranked: a player’s equilibrium payoff is inversely related to the probability that her exaggeration will be thwarted by the report bounds.