On the Hardness of Signaling

There has been a recent surge of interest in the role of information in strategic interactions. Much of this work seeks to understand how the realized equilibrium of a game is influenced by uncertainty in the environment and the information available to players in the game. Lurking beneath this literature is a fundamental, yet largely unexplored, algorithmic question: how should a "market maker" who is privy to additional information, and equipped with a specified objective, inform the players in the game? This is an informational analogue of the mechanism design question, and views the information structure of a game as a mathematical object to be designed, rather than an exogenous variable. We initiate a complexity-theoretic examination of the design of optimal information structures in general Bayesian games, a task often referred to as signaling. We focus on one of the simplest instantiations of the signaling question: Bayesian zero-sum games, and a principal who must choose an information structure maximizing the equilibrium payoff of one of the players. In this setting, we show that optimal signaling is computationally intractable, and in some cases hard to approximate, assuming that it is hard to recover a planted clique from an Erdos-Renyi random graph. This is despite the fact that equilibria in these games are computable in polynomial time, and therefore suggests that the hardness of optimal signaling is a distinct phenomenon from the hardness of equilibrium computation. Necessitated by the non-local nature of information structures, en-route to our results we prove an "amplification lemma" for the planted clique problem which may be of independent interest. Specifically, we show that even if we plant many cliques in an Erdos-Renyi random graph, so much so that most nodes in the graph are in some planted clique, recovering a constant fraction of the planted cliques is no easier than the traditional planted clique problem.