Dynamically Unstable ESS in Matrix Games Under Time Constraints

Matrix games under time constraints represent a natural extension of conventional matrix games. They take into account the additional factor that, apart from the payoff, a pairwise interaction results in a delay for the contestants before they can engage in subsequent interactions. Each matrix game can be associated with a continuous dynamical system, known as the replicator equation, which describes the evolution of phenotype frequencies within the population. One of the fundamental theorems of evolutionary matrix games asserts that the state corresponding to an evolutionarily stable strategy is an asymptotically stable rest point of the replicator equation (Taylor and Jonker in Math Biosci 40:145–156, https://doi.org/10.1016/0025-5564(78)90077-9, 1978; Hofbauer et al. in J Theor Biol 81:609–612, https://doi.org/10.1016/0022-5193(79)90058-4, 1979; Zeeman in Global theory of dynamical systems. Lecture notes in mathematics. Springer, New York, vol 819. https://doi.org/10.1007/BFb0087009, 1980). Garay et al. (J Math Biol 76:1951–1973, https://doi.org/10.1086/681638, 2018) and Varga et al. (J Math Biol 80:743–774, https://doi.org/10.1007/s00285-019-01440-6, 2020) generalized the statement to two-strategy and, in some particular cases, three- or more strategy matrix games under time constraints. However, the general applicability of this implication has remained an open question. In this paper, we present examples that demonstrate the negative answer. Moreover, we illustrate, through the rock-paper-scissors game, that even slight disparities in waiting times can lead to the destabilization of the equilibrium corresponding to an ESS. Additionally, we establish that a stable limit cycle can emerge around the unstable equilibrium in a supercritical Hopf bifurcation.