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The Big Match under translation-invariant charges

  • Abraham Neyman
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Abstract

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 ε>0, a strategy that is ε-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 1/2, every strategy of Player 1 is worthless: for every strategy of Player 1 and every δ>0, Player 2 has a pure reply, and there is a translation-invariant charge under which Player 1's payoff is less than δ. 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.

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Cite

BibTeXRIS
BibTeX
@misc{neyman2026match,
  title = {{The Big Match under translation-invariant charges}},
  author = {Abraham Neyman},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19646},
  eprint = {2610.19646},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19646},
}
RIS
TY  - UNPB
TI  - The Big Match under translation-invariant charges
AU  - Abraham Neyman
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19646
UR  - https://arxiv.org/abs/2610.19646
AB  - 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 ε>0, a strategy that is ε-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 1/2, every strategy of Player 1 is worthless: for every strategy of Player 1 and every δ>0, Player 2 has a pure reply, and there is a translation-invariant charge under which Player 1's payoff is less than δ. 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.
ER  -