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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2403.07467 |
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| _version_ | 1866908840030633984 |
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| author | Novikov, Ivan |
| author_facet | Novikov, Ivan |
| contents | We study $λ$-discounted zero-sum games as the discount factor $λ$ approaches $0$ (that is, the players are more and more patient), in the context of games with stage duration. In stochastic games with stage duration $h$, players act at times $0, h, 2h$, and so on. The payoff and leaving probabilities are proportional to $h$. When $h$ tends to $0$, such discrete-time games approximate games played in continuous time. The asymptotic behavior of the values (when both $λ$ and $h$ tend to $0$) has already been studied for stochastic games with full state observation and for state-blind games. We consider the same question for the case of stochastic games with deterministic public signals on the state. We construct a stochastic game with public signals, with no asymptotic value (as the discount factor $λ$ goes to $0$) if the stage duration is $1$, but with an asymptotic value when the stage duration $h$ and the discount factor $λ$ both tend to $0$. Informally, this means that the asymptotic value in discrete time does not exist, whereas it does exist in continuous time. This situation cannot occur in stochastic games with full state observation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic Value in Zero-Sum Stochastic Games with Vanishing Stage Duration and Public Signals Novikov, Ivan Optimization and Control We study $λ$-discounted zero-sum games as the discount factor $λ$ approaches $0$ (that is, the players are more and more patient), in the context of games with stage duration. In stochastic games with stage duration $h$, players act at times $0, h, 2h$, and so on. The payoff and leaving probabilities are proportional to $h$. When $h$ tends to $0$, such discrete-time games approximate games played in continuous time. The asymptotic behavior of the values (when both $λ$ and $h$ tend to $0$) has already been studied for stochastic games with full state observation and for state-blind games. We consider the same question for the case of stochastic games with deterministic public signals on the state. We construct a stochastic game with public signals, with no asymptotic value (as the discount factor $λ$ goes to $0$) if the stage duration is $1$, but with an asymptotic value when the stage duration $h$ and the discount factor $λ$ both tend to $0$. Informally, this means that the asymptotic value in discrete time does not exist, whereas it does exist in continuous time. This situation cannot occur in stochastic games with full state observation. |
| title | Asymptotic Value in Zero-Sum Stochastic Games with Vanishing Stage Duration and Public Signals |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2403.07467 |