<p>The paper deals with non-cooperative games in which the payoff function of its players is influenced by exogenous randomness. The main goal is to provide a general concept of stability in those games because the standard notion of Nash equilibrium is no longer satisfactory. This is because the best response strategy must be the best response for each different payoff scenario and if the corresponding payoff matrices are too different this results in an empty set of best responses. Therefore, different solution concepts are required. One could find a deterministic equivalent to the game with a random payoff by considering a collection of risk measures and defining a new game with a payoff function adjusted by applying specific risk measures to each player’s payoff. This, however, causes several problems as with the added payoff non-linearity in mixed strategies the Nash’s Theorem no longer holds for most of such equivalents, and the existence of equilibria in those games must be proven on an ad hoc basis. With an increasing number of parameters such as player-specific risk measures, this problem becomes increasingly difficult. In our article, we propose to loosen the standard concept of the best response to an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-best response which requires the strategy to be the best response only with a certain high probability. Based on this idea we define the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> </math></EquationSource> </InlineEquation>-Nash equilibria and we prove that for every finite game with random payoff non-trivial <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> </math></EquationSource> </InlineEquation>-Nash equilibria exist. Moreover, we show that those equilibria characterize equilibria in a broad class of deterministic equivalent games. Finally, we extend the idea of a static game with a random payoff to a game with multiple stages and we show that every finite stochastic game may be represented as a sequential game with a random payoff. In the numerical study, this theory is applied to a management problem of competition of hospitals for vaccines during a pandemic.</p>

Generalizing Nash equilibria for games with random payoffs

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发表日期：

2025年2月5日

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Miloš Kopa,
Petr Lachout,
Lukáš Račko

摘要

The paper deals with non-cooperative games in which the payoff function of its players is influenced by exogenous randomness. The main goal is to provide a general concept of stability in those games because the standard notion of Nash equilibrium is no longer satisfactory. This is because the best response strategy must be the best response for each different payoff scenario and if the corresponding payoff matrices are too different this results in an empty set of best responses. Therefore, different solution concepts are required. One could find a deterministic equivalent to the game with a random payoff by considering a collection of risk measures and defining a new game with a payoff function adjusted by applying specific risk measures to each player’s payoff. This, however, causes several problems as with the added payoff non-linearity in mixed strategies the Nash’s Theorem no longer holds for most of such equivalents, and the existence of equilibria in those games must be proven on an ad hoc basis. With an increasing number of parameters such as player-specific risk measures, this problem becomes increasingly difficult. In our article, we propose to loosen the standard concept of the best response to an $\alpha $ $α$ -best response which requires the strategy to be the best response only with a certain high probability. Based on this idea we define the $\varvec{\alpha }$ $α$ -Nash equilibria and we prove that for every finite game with random payoff non-trivial $\varvec{\alpha }$ $α$ -Nash equilibria exist. Moreover, we show that those equilibria characterize equilibria in a broad class of deterministic equivalent games. Finally, we extend the idea of a static game with a random payoff to a game with multiple stages and we show that every finite stochastic game may be represented as a sequential game with a random payoff. In the numerical study, this theory is applied to a management problem of competition of hospitals for vaccines during a pandemic.