Identifying the Deviator
Noga Alon, Benjamin Gunby, Xiaoyu He, Eran Shmaya, Eilon Solan
TL;DR
This work studies identifying a single deviator among multiple players when a prescribed strategy profile would yield a target set with high probability but a deviation causes rejection. It frames the problem as testability with a blame function $f$ and shows that for any goal $(\sigma^*,D)$ with ${\bf P}_{\sigma^*}(D)>1-\varepsilon$, a blame rule exists that limits misidentification to at most $2\sqrt{(|I|-1)\varepsilon}$; the bound is tight up to constants and 0-testability holds when $D$ is reached with probability 1. The authors connect to Nash equilibria and statistical decision theory, and employ a minimax-based (non-constructive) proof along with finite-horizon approximations to establish existence; they also provide concrete examples (e.g., a one-dimensional random walk, and adjacent-ones processes) where explicit blame functions can be constructed. The results have implications for dynamic games and tail-measurable payoffs by ensuring that unilateral deviations can be credibly identified, which supports equilibrium construction and enforcement mechanisms in stochastic settings. Future work includes obtaining explicit blame rules with tighter quantitative guarantees, extending to higher-dimensional processes, and refining the bounds via group-testing-style iterations.
Abstract
A group of players are supposed to follow a prescribed profile of strategies. If they follow this profile, they will reach a given target. We show that if the target is not reached because some player deviates, then an outside observer can identify the deviator. We also construct identification methods in two nontrivial cases.