<p>Matrix games under time constraints generalize classical matrix games by incorporating the need for players to wait after interactions before engaging in new ones. As a result, the population divides into active and inactive individuals, where only active individuals are capable of engaging in interactions. Consequently, differences in the fitness of strategies are determined solely by the payoffs of active individuals. Similarly to classical matrix games, the concept of evolutionarily stable strategy (ESS) can also be defined in this model as a strategy that, if adopted by the majority of the population, has a higher fitness than any mutant phenotype (Garay et al. in J Theor Biol 415:1–12, 2017. <a href="https://doi.org/10.1016/j.jtbi.2016.11.029">https://doi.org/10.1016/j.jtbi.2016.11.029</a>) . We recently introduced a generalized replicator dynamics that takes time constraints into account (Varga in J Math Biol 90:6, 2024. <a href="https://doi.org/10.1007/s00285-024-02170-0">https://doi.org/10.1007/s00285-024-02170-0</a>). Using this, we proved that if a strategy is an ESS under time constraints, then the associated fixed point of the dynamics is asymptotically stable. However, evolutionary stability is not necessary for asymptotic stability. In other words, asymptotic stability does not provide a full characterization of ESS, even under the standard replicator dynamics in matrix games (Taylor and Jonker in Math Biosci 40(1):145–156, 1978. <a href="https://doi.org/10.1016/0025-5564(78)90077-9">https://doi.org/10.1016/0025-5564(78)90077-9</a>). To address this, Cressman proposed the concept of strong stability: a strategy <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13235_2025_637_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">p</mi> </math></EquationSource> </InlineEquation> is strongly stable if it is a convex combination of some strategies, the average strategy of the population converges to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13235_2025_637_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">p</mi> </math></EquationSource> </InlineEquation> under the replicator dynamics with respect to those strategies (Cressman in J Theor Biol 145:319–330, 1990. <a href="https://doi.org/10.1016/S0022-5193(05)80112-2">https://doi.org/10.1016/S0022-5193(05)80112-2</a>). This criterion already provides a necessary and sufficient condition for a strategy to be an ESS. Here, we extend this approach to matrix games under time constraints, showing that a strategy is evolutionarily stable if and only if it is strongly stable under the generalized replicator dynamics.</p>

Strong Stability for Matrix Games Under Time Constraints

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发表日期：

2025年4月16日

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Tamás Varga,
József Garay

摘要

Matrix games under time constraints generalize classical matrix games by incorporating the need for players to wait after interactions before engaging in new ones. As a result, the population divides into active and inactive individuals, where only active individuals are capable of engaging in interactions. Consequently, differences in the fitness of strategies are determined solely by the payoffs of active individuals. Similarly to classical matrix games, the concept of evolutionarily stable strategy (ESS) can also be defined in this model as a strategy that, if adopted by the majority of the population, has a higher fitness than any mutant phenotype (Garay et al. in J Theor Biol 415:1–12, 2017. https://doi.org/10.1016/j.jtbi.2016.11.029) . We recently introduced a generalized replicator dynamics that takes time constraints into account (Varga in J Math Biol 90:6, 2024. https://doi.org/10.1007/s00285-024-02170-0). Using this, we proved that if a strategy is an ESS under time constraints, then the associated fixed point of the dynamics is asymptotically stable. However, evolutionary stability is not necessary for asymptotic stability. In other words, asymptotic stability does not provide a full characterization of ESS, even under the standard replicator dynamics in matrix games (Taylor and Jonker in Math Biosci 40(1):145–156, 1978. https://doi.org/10.1016/0025-5564(78)90077-9). To address this, Cressman proposed the concept of strong stability: a strategy $\textbf{p}$ $p$ is strongly stable if it is a convex combination of some strategies, the average strategy of the population converges to $\textbf{p}$ $p$ under the replicator dynamics with respect to those strategies (Cressman in J Theor Biol 145:319–330, 1990. https://doi.org/10.1016/S0022-5193(05)80112-2). This criterion already provides a necessary and sufficient condition for a strategy to be an ESS. Here, we extend this approach to matrix games under time constraints, showing that a strategy is evolutionarily stable if and only if it is strongly stable under the generalized replicator dynamics.