Betting on Bets: Anytime-Valid Tests for Stochastic Dominance

How can we monitor, in real time, whether one uncertain prospect has any upside over another? To answer this question, the authors develop a novel family of sequential, anytimevalid tests for stochastic dominance (SD), a classical and popular notion for comparing entire distribution functions. The problem is distinct from that of testing mean dominance, and it is particularly useful when comparing distributions with similar means or with ordinal outcomes. The authors first derive powerful, nonparametric e-processes that quantify evidence against the null hypothesis that one prospect is stochastically dominated by another. For first-order SD, these e-processes are based on mixtures of growth-rate optimal e-variables, yielding a test of power one that retains validity under continuous monitoring. The authors then generalize the approach to sequential testing for higher-order SD and other integral stochastic orders. Empirically, they find that the tests are competitive in power with classical, non-anytime-valid SD tests. Their real-world application examines a controversial phenomenon in baseball analytics, known as the “third-time-through-the-order (3TTO) penalty,” viewed as a monitoring problem. The authors close by sketching the complementary problem of testing whether a prospect has a definite upside, formalizing conditions under which they can derive a powerful anytime-valid test.