The Computable but Not Learnable Information-Value-Free Equilibria and Regulation of Algorithmic Collusion

A correlated equilibrium is information-value-free if every player has an action that yields the same payoff as their equilibrium strategy when all other players follow their respective equilibrium strategies. An equilibrium is learnable if the empirical history of play generated by certain learning algorithms using only full feedback on each player's own payoff converges to it. Our main result is that although an information-value-free equilibrium can be computed efficiently offline, it is not learnable by a broad class of learning algorithms. This separation stands in sharp contrast to canonical equilibrium concepts, where offline computation and online learning typically have comparable difficulty. In information-economics terms, our results imply that learning correlated equilibria in games cannot avoid generating valuable information, which has implications for current debates on the regulation of algorithmic collusion: Valuable and implicit information exchange is unavoidable under rationalizable learning, and traditional antitrust regulation against such exchange is therefore incompatible with rationalizable learning. Our results also imply an informational impossibility result for time-average convergence to a Nash equilibrium by a broad class of learning algorithms.