Cooperation among individuals has been key to sustaining societies. However, natural selection favors defection over cooperation. Cooperation can be favored when the mobility of individuals allows cooperators to form a cluster (or group). Mobility patterns of animals sometimes follow a Lévy flight. A Lévy flight is a kind of random walk but it is composed of many small movements with a few big movements. Here, we developed an agent-based model in a square lattice where agents perform Lévy flights depending on the fraction of neighboring defectors. We focus on how the sensitivity to defectors when performing Lévy flights promotes the evolution of cooperation. Results of evolutionary simulations showed that cooperation was most promoted when the sensitivity to defectors was moderate. As the population density became larger, higher sensitivity was more beneficial for cooperation to evolve.

In 2012, Press and Dyson discovered a strategy set, called Zero-determinant (ZD) strategies, which enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy in the repeated prisoner's dilemma (RPD) game. In the RPD game, a discount factor and observation errors are both important because they often happen in society. Here, we examined strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first revealed that the payoffs of two players can be represented by the form of determinants even with these two factors. Then, we searched for all possible strategies that enforce linear payoff relationships and found that both ZD strategies and unconditional strategies are the only strategy sets which satisfy the condition.

The repeated prisoner’s dilemma (RPD) game has revealed how cooperation and competition arise among competitive players in long-run relationships. In the RPD game with no errors, zero-determinant (ZD) strategies allow a player to unilaterally set a linear relationship between the player’s own payoff and the opponent’s payoff regardless of the strategy of the opponent. On the other hand, unconditional strategies such as ALLD and ALLC also unilaterally set a linear relationship. However, little is known about the existence of such strategies in the RPD game with errors. Here, we analytically search for the strategies that enforce a linear payoff relationship under observation error in the RPD game. As a result, we found that, even in the case with observation errors, the only strategy sets that enforce a linear payoff relationship are either ZD strategies or unconditional strategies.

Direct reciprocity is one of the mechanisms for sustaining mutual cooperation in repeated social dilemma games. Zero-determinant (ZD) strategies allow a player to unilaterally set a linear relationship between the player’s own payoff and the co-player’s payoff regardless of the strategy of the co-player. The original ZD strategies were derived for infinitely repeated games. Here, we analytically search for ZD strategies in finitely repeated prisoner’s dilemma games. Our results can be summarized as follows. First, we present the forms of ZD in finitely repeated games, which are directly extended from the known results for infinitely repeated games. Second, for the three most notable ZD strategies, the equalizers, extortioners, and generous strategies, we derive the threshold discount factor value above which the ZD strategies exist. Finally, we show that the only strategy sets that enforce a linear payoff relationship are either the ZD strategies or unconditional strategies.

Cooperation is ubiquitous in every level of living organisms. It is known that spatial (network) structure is a viable mechanism for cooperation to evolve. Until recently, it has been difficult to predict whether cooperation can evolve at a network (population) level. To address this problem, Pinheiro et al. proposed a numerical metric, called Average Gradient of Selection (AGoS) in 2012. AGoS can characterize and forecast the evolutionary fate of cooperation at a population level. However, stochastic mutation of strategies was not considered in the analysis of AGoS. Here we analyzed the evolution of cooperation using AGoS where mutation may occur to strategies of individuals in networks. Our analyses revealed that mutation always has a negative effect on the evolution of cooperation regardless of the fraction of cooperators and network structures. Moreover, we found that mutation affects the fitness of cooperation differently on different social network structures.