The Big Match under translation-invariant charges

We study two-player zero-sum stochastic games in which a stream of stage payoffs is evaluated by a translation-invariant charge on the set of stages. Every patient valuation, that is, every diffuse charge that obeys the time value of money principle, is translation-invariant, and by Mertens and Neyman each player has, for every $\epsilon >0$, a strategy that is $\epsilon$-optimal for all patient valuations simultaneously. We show that this robustness breaks down for translation-invariant charges. In the Big Match with $0$–$1$ payoffs, whose uniform value is $1\mathrm{/}2$, every strategy of Player 1 is worthless: for every strategy of Player 1 and every $\delta >0$, Player 2 has a pure reply, and there is a translation-invariant charge under which Player 1’s payoff is less than $\delta$. Consequently, the Big Match has no value with respect to all translation-invariant charges. This contrasts with finite Markov decision processes, where every stationary Blackwell optimal strategy is optimal for every translation-invariant charge. We pose questions on robust guarantees in general stochastic games and on optimality with respect to a fixed translation-invariant charge.