Learning Against Nature: Minimax Regret and the Price of Robustness

Abstract: We study how a decision-maker (DM) learns from data of unknown quality to form robust, ''general-purpose'' posterior beliefs. We develop a framework for robust learning and belief formation under a minimax-regret criterion, cast as a zero-sum game: the DM chooses posterior beliefs to minimize ex-ante regret, while an adversarial Nature selects the data-generating process (DGP). We show that, in large samples of $n$ signal draws, Nature optimally induces ambiguity by choosing a process whose precision converges to the uninformative signals at the rate $1/\sqrt{n}$. As a result, learning against the adversarial DGP is nontrivial as well as incomplete: the DM's ex-ante regret remains strictly positive even with an infinite amount of data. However, when the true DGP is fixed and informative (even if only slightly), our DM with a robust updating rule eventually learns the state with enough data. Still, learning occurs at a sub-exponential rate -- quantifying the asymptotic price of robustness -- and it exhibits ''under-inference'' bias. Our framework provides a decision-theoretic dual to the local alternatives method in asymptotic statistics, deriving the characteristic $1/\sqrt{n}$-scaling endogenously from the signal ambiguity.