Markovian randomized equilibria for general Markovian Dynkin games in discrete time
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866918122229858304 |
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| author | Christensen, Sören Lindensjö, Kristoffer Neumann, Berenice Anne |
| author_facet | Christensen, Sören Lindensjö, Kristoffer Neumann, Berenice Anne |
| contents | We study a general formulation of the classical two-player Dynkin game in a discrete time Markovian setting. We identify an appropriate class of mixed strategies -- \textit{Markovian randomized stopping times} -- in which players stop at any given state with a state-dependent probability. One main result is an explicit characterization of Wald-Bellman-type for Nash equilibria based on this notion of randomization. In particular, we derive a novel characterization of randomized equilibria in zero-sum Dynkin games, which we use to (i) establish the existence and explicit construction of Markovian randomized equilibria, (ii) provide necessary and sufficient conditions for the non-existence of pure strategy equilibria, and (iii) construct an example that admits a unique randomized equilibrium but no pure one. We also provide existence and characterization results in the symmetric version of our game. Finally, we establish existence of a characterizable equilibrium in Markovian randomized stopping times for the general game formulation under the assumption that the state space is countable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_13413 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Markovian randomized equilibria for general Markovian Dynkin games in discrete time Christensen, Sören Lindensjö, Kristoffer Neumann, Berenice Anne Probability Optimization and Control 91A55, 60G40, 91A15 We study a general formulation of the classical two-player Dynkin game in a discrete time Markovian setting. We identify an appropriate class of mixed strategies -- \textit{Markovian randomized stopping times} -- in which players stop at any given state with a state-dependent probability. One main result is an explicit characterization of Wald-Bellman-type for Nash equilibria based on this notion of randomization. In particular, we derive a novel characterization of randomized equilibria in zero-sum Dynkin games, which we use to (i) establish the existence and explicit construction of Markovian randomized equilibria, (ii) provide necessary and sufficient conditions for the non-existence of pure strategy equilibria, and (iii) construct an example that admits a unique randomized equilibrium but no pure one. We also provide existence and characterization results in the symmetric version of our game. Finally, we establish existence of a characterizable equilibrium in Markovian randomized stopping times for the general game formulation under the assumption that the state space is countable. |
| title | Markovian randomized equilibria for general Markovian Dynkin games in discrete time |
| topic | Probability Optimization and Control 91A55, 60G40, 91A15 |
| url | https://arxiv.org/abs/2307.13413 |