The worst-case payoff in games with stochastic revision opportunities
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The worst-case payoff in games with stochastic revision opportunities Yevgeny Tsodikovich1 Accepted: 7 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study infinitely repeated games in which players are limited to subsets of their action space at each stage—a generalization of asynchronous games. This framework is broad enough to model many real-life repeated scenarios with restrictions, such as portfolio management, learning by doing and training. We present conditions under which rigidity in the choice of actions benefits all players in terms of worst-case equilibrium payoff and worst-case payoff. To provide structure, we exemplify our result in a model of a two-player repeated game, where we derive a formula for the worst-case payoff. Moreover, we show that in zero-sum games, lack of knowledge about the timing of the revision can compensate for inability to change the action. Keywords Asynchronous Games · Rational Minimax · Worst-Case Payoffs · Commitment · Exogenous Timing Mathematics Subject Classification C73
1 Introduction This paper considers repeated interactions in which players do not necessarily act simultaneously. This is often the case, as different agents have different decision-making schedules that seldom coincide, and their capacity to change their actions often differs. The asynchronous nature of the play can lead to cooperation, similar to the collusion occurring in the priceleadership mechanism (MacLeod 1985), the commitment involved in the central bank setting the interest rate (Libich and Stehlík 2010, 2011), and other economic examples presented in the seminal works of Maskin and Tirole (1988) and Lagunoff and Matsui (1997).
The author wishes to thank Ehud Lehrer, Eilon Solan, David Lagziel, Rann Smorodinsky, Dotan Persitz, Galit Golan-Ashkenazi, Ilan Nehama, and two anonymous referees of the Annals of Operations Research journal for their highly valuable comments. The author acknowledges the support of the Israel Science Foundation, Grants #963/15 and #2510/17 and the support of the French National Research Agency Grants ANR-17-EURE-0020.
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Yevgeny Tsodikovich [email protected] Aix Marseille Univ, CNRS, AMSE, Marseille, France
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Annals of Operations Research
In all of these examples, whenever a player cannot revise his action, the action from the previous stage is replayed. However, there are other ways in which actions can be binding; for example, when a revision opportunity is given at every stage, but the new action has to be “close” to the old one (in some metric). To examine such constraints, this paper introduces a generalization of repeated games, namely Sub-Actions Repeated (SAR) Games, in which at each stage the players must choose an action from a subset of their mixed action space, stochastically determined by the history of the game. This generalizes the (simultaneousmove) repeated game as well as different models of asynchronous games. SAR games take place naturally in many real-life situations. One example is the pri
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