Evolutionary game theory and models of learning provide powerful frameworks to describe strategic decision-making in social interactions. In the simplest case, these models describe games among two identical players. However, many interactions in everyday life are more complex. They involve more than two players who may differ in their available actions and in their incentives to choose each action. Such interactions can be captured by asymmetric multiplayer games. Recently, introspection dynamics has been introduced to explore such asymmetric games. According to this dynamics, at each time step players compare their current strategy to an alternative strategy. If the alternative strategy results in a payoff advantage, it is more likely adopted. This model provides a simple way to compute the players’ long-run probability of adopting each of their strategies. In this paper, we extend some of the previous results of introspection dynamics for 2-player asymmetric games to games with arbitrarily many players. First, we derive a formula that allows us to numerically compute the stationary distribution of introspection dynamics for any multiplayer asymmetric game. Second, we obtain explicit expressions of the stationary distribution for two special cases. These cases are additive games (where the payoff difference that a player gains by unilaterally switching to a different action is independent of the actions of their co-players), and symmetric multiplayer games with two strategies. To illustrate our results, we revisit several classical games such as the public goods game.

The prisoner’s dilemma, the snowdrift game, and the stag hunt are two-player symmetric games that are often considered as prototypical examples of cooperative dilemmas across disciplines. However, surprisingly little consensus exists about the precise mathematical meaning of the words “cooperation” and “cooperative dilemma” for these and other binary-action symmetric games, in particular when considering interactions among more than two players. Here, we propose definitions of these terms and explore their evolutionary consequences on the equilibrium structure of cooperative dilemmas in relation to social optimality. We show that our definition of cooperative dilemma encompasses a large class of collective action games often discussed in the literature, including congestion games, games with participation synergies, and public goods games. One of our main results is that regardless of the number of players, all cooperative dilemmas—including multi-player generalizations of the prisoner’s dilemma, the snowdrift game, and the stag hunt—feature inefficient equilibria where cooperation is underprovided, but cannot have equilibria in which cooperation is overprovided. We also find simple conditions for full cooperation to be socially optimal in a cooperative dilemma. Our framework and results unify, simplify, and extend previous work on the structure and properties of cooperative dilemmas with binary actions and two or more players.

We provide results of a deterministic approximation for non-Markovian stochastic processes modeling finite populations of individuals who recurrently play symmetric finite games and imitate each other according to payoffs. We show that a system of delay differential equations can be obtained as the deterministic approximation of such a non-Markovian process. We also show that if the initial states of stochastic process and the corresponding deterministic model are close enough, then the trajectory of stochastic process stays close to that of the deterministic model up to any given finite time horizon with a probability exponentially approaching one as the population size increases. We use this result to obtain that the lower bound of the population size on the absorption time of the non-Markovian process is exponentially increasing. Additionally, we obtain the replicator equations with distributed and discrete delay terms as examples and analyze how the memory of individuals can affect the evolution of cooperation in a two-player symmetric Snow-drift game. We investigate the stability of the evolutionary stable state of the game when agents have the memory of past population states, and implications of these results are given for the stochastic model.

We present an evolutionary model which allows us to study the impact relationship-specific investment has on bargaining. Agents are matched to play an investment and bargaining game. During bargaining, agents have an outside option to form a new relationship, but in exercising this option loses their current investment. We find that the stochastically stable post-investment bargaining convention is dependent on the cost of investment. In particular, the larger the cost of investment, the lower is the share of gross surplus that is received. This stands in contrast with previous studies. In addition, we find that there is under-investment. We disentangle the forces which lead to these two results.