On the hardness of designing public signals

We use computational complexity as a lens to study the design of information structures in games of incomplete information. We focus on one of the simplest instantiations of the information structure design problem: Bayesian zero-sum games, and a principal who must design a public signal maximizing the equilibrium payoff of one of the players. In this setting, we show that optimal information structure design is computationally intractable, and in some cases hard to approximate, assuming that it is hard to recover a planted clique from an Erdős–Rényi random graph. Our result suggests that there is no “simple” characterization of optimal public-channel information structures in multi-player settings.