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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Asymptotics of values in dynamic games on large intervals

Author: D. V. Khlopin
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 31 (2019), nomer 1.
Journal: St. Petersburg Math. J. 31 (2020), 157-179
MSC (2010): Primary 91A05; Secondary 49N90
Published electronically: December 3, 2019
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Abstract: Dynamic antagonistic games are treated. The dependence of the game value on the payoff function is explored for games with one and the same dynamics, running cost, and possibilities of the players. Each payoff function is viewed as a running cost, averaged in terms of a certain probability distribution on the semiaxis. It is shown that if the dynamic programming principle is fulfilled, then a Tauberian-type theorem holds, specifically, the existence of a uniform limit of the values (as the scaling parameter tends to zero) for games with exponential and/or uniform distribution implies that the same limit exists (as the scaling parameter tends to zero) with arbitrary piecewise-continuous density. Also, a version of this theorem for games with discrete time is proved. The general results are applied to various game settings, both in the deterministic and in the stochastic case. Also, asymptotics are studied as the time horizon tends to infinity.

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Additional Information

D. V. Khlopin
Affiliation: N. N. Krasovskii Institute of Mathematics and Mechanics of (IMM UB RAS), ul. S. Kovalevskoi 16, Ekaterinburg 620999, Russia

Keywords: Dynamic programming principle, antagonistic dynamic games, differential games, Abel mean, Cesaro mean
Received by editor(s): September 25, 2017
Published electronically: December 3, 2019
Additional Notes: This investigation was supported by Russian Science Foundation, grant No. 17-11-01093
Article copyright: © Copyright 2019 American Mathematical Society