  Suppose we observe an infinite series of coin flips $X_1,X_2,\ldots$, and
wish to sequentially test the null that these binary random variables are
exchangeable. Nonnegative supermartingales (NSMs) are a workhorse of sequential
inference, but we prove that they are powerless for this problem. First,
utilizing a geometric concept called fork-convexity (a sequential analog of
convexity), we show that any process that is an NSM under a set of
distributions, is also necessarily an NSM under their "fork-convex hull".
Second, we demonstrate that the fork-convex hull of the exchangeable null
consists of all possible laws over binary sequences; this implies that any NSM
under exchangeability is necessarily nonincreasing, hence always yields a
powerless test for any alternative. Since testing arbitrary deviations from
exchangeability is information theoretically impossible, we focus on Markovian
alternatives. We combine ideas from universal inference and the method of
mixtures to derive a "safe e-process", which is a nonnegative process with
expectation at most one under the null at any stopping time, and is upper
bounded by a martingale, but is not itself an NSM. This in turn yields a level
$\alpha$ sequential test that is consistent; regret bounds from universal
coding also demonstrate rate-optimal power. We present ways to extend these
results to any finite alphabet and to Markovian alternatives of any order using
a "double mixture" approach. We provide an array of simulations, and give
general approaches based on betting for unstructured or ill-specified
alternatives. Finally, inspired by Shafer, Vovk, and Ville, we provide
game-theoretic interpretations of our e-processes and pathwise results.
