Informationally Robust Comparative Statics in Incomplete Information Games

A continuum of agents play an incomplete information game with continuous action. An agent's best-response function is linear on his expectation of a common payoff shock and the average action taken by the other agents. The interaction parameter is the weight that an agent places on the actions of the other agents. We give informationally robust predictions of the impact that a marginal change in the interaction parameter has on the agents' equilibrium expected utility for any given equilibrium outcome. Our predictions are informationally robust in the sense that they hold across all information structures that are consistent with the observed equilibrium outcome. We express our informationally robust predictions solely in terms of the equilibrium outcome; this gives a “Bayes-correlated-equilibrium approach” to comparative statics (cf. Bergemann and Morris (2013); Bergemann and Morris (2016)). The results are applied to give robust predictions on the impact that a marginal change in the interaction parameter has on any other statistics of the equilibrium outcome (e.g., other measures of total welfare or the actions' volatility). We use our results to study a competitive market with dispersed information (cf. Guesnerie (1992); Vives (1993)); we give robust predictions of the impact that a marginal change in the elasticity of the demand has on the firms' profits. We also use our results to study economies that are inefficient when agents have dispersed information (cf. Morris and Shin (2002); Angeletos and Pavan (2007)); we give robust predictions of the impact that a marginal change in a tax rate has on the total welfare