Abstract
We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs’ condition. The dynamics is an ordinary differential equation parametrized by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. This value is taken in the sense of the limit of any time discretization, as the mesh of the time partition tends to zero. We characterize it in terms of the unique viscosity solution in some dual sense of a Hamilton–Jacobi–Isaacs equation. Here we do not suppose the Isaacs’ condition, which is usually assumed in differential games.
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Acknowledgments
We thank the anonymous referees for the careful reading of the manuscript and the useful remarks and comments. The work is partially supported by the Commission of the European Communities under the project SADCO, FP7-PEOPLE-2010-ITN, No. 264735, the French National Research Agency ANR-10-BLAN 0112 and Natural Science Foundation of Jiangsu Province and China (No. BK20140299; No. 14KJB110022; No. 11401414) and the collaborative innovation center for quantitative calculation and control of financial risk.
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Buckdahn, R., Quincampoix, M., Rainer, C. et al. Differential games with asymmetric information and without Isaacs’ condition. Int J Game Theory 45, 795–816 (2016). https://doi.org/10.1007/s00182-015-0482-x
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DOI: https://doi.org/10.1007/s00182-015-0482-x
Keywords
- Zero-sum differential game
- Asymmetric information
- Isaacs’ condition
- Viscosity solution
- Subdynamic programming principle
- Dual game