Nash Equilibrium in Iterated Multiplayer Games Under Asynchronous Best-Response Dynamics

As an important branch of evolutionary game theory, iterated games describe the situations that interacting agents play repeatedly based on previous outcomes by using the conditional strategies. A new class of zero-determinant (ZD) strategies, which can control a linear relation between the expected payoffs of a single agent and the coplayers, has dramatically changed the viewpoint on iterated games. Here, in this article, we focus on the decision-making behaviors in iterated multiplayer gaming (IMG) systems with the underlying scenarios of two competing ZDs. The results show that, under the asynchronous best-response dynamics, IMG systems starting from any initial state will converge to Nash equilibrium (NE) in finite time. Particularly, the convergence occurs not only in finite time, but can be limited by the number of strategy switches, which is no more than the total amount of agents in the population. Further studies on calculating the NE points reveal that, there is a threshold for the ZD slope, above which agents with higher baseline payoff dominate, while below which agents of lower baseline payoff prevail. The results of system convergence and NE states highlight the fixation of long-run decision-making behaviors in IMG. Finally, an example of the iterated public goods games is provided for the application of the proposed IMG model.