Dilemma solving by the coevolution of networks and strategy in a $2\times 2$ game

A $2\times 2$ game model implemented by a coevolution mechanism of both networks and strategy, inspired by the work of Zimmermann and Eguiluz [Phys. Rev. E72, 056118 (2005)] is established. Network adaptation is the manner in which an existing link between two agents is destroyed and how a new one is established to replace it. The strategy is defined as whether an agent offers cooperation $\left(C\right)$ or defection $\left(D\right)$. Both the networks and strategy are synchronously renovated in a simulation time step. A series of numerical experiments, considering various $2\times 2$ game structures, reveals that the proposed coevolution mechanism can solve dilemmas in several game classes. The effect of solving a dilemma means mutual-cooperation reciprocity ($R$ reciprocity), which is brought about by emerging several cooperative hub agents who have plenty of links. This effect can be primarily observed in game classes of the prisoner’s dilemma and stag hunt. The coevolution mechanism, however, seems counterproductive for game classes of leader and hero, where the alternating reciprocity (ST reciprocity) is meaningful.

Figure 1

Scene of the $2\times 2$ game world. According to Tanimoto and Sagara [23], any $2\times 2$ games can be parametrized by two game structural parameters $r$ and $\theta$. Any game classes including both dilemma games such as PD, chicken, SH and so on, and even trivial game, can be drawn schematically. Avatamsaka is a special game that is on the border of donor-recipient game (one of the special PDs) and trivial game.

Figure 2

(Color online) Result of the numerical experiments (a) the cooperation fraction among $N$ in SEN, (b) the payoff difference between SEN and the analytical solution, (c) the payoff difference between SEN and the fixed random network, (d) the payoff difference between SEN and the fixed scale-free network, (e) the maximum degree in the SEN network, (f) the payoff difference between the evolutionary network based on severing method No. 1 and connecting method No. 2 and the analytical solution, (g) the payoff difference between the evolutionary network based on severing method No. 2 and connecting method No. 1 and the analytical solution, and (h) the payoff difference between the evolutionary network based on severing method No. 2 and connecting method No. 2, and the analytical solution. Each payoff indicates a payoff per single game. Degree distribution of the closed plot in (e) that locates on $\theta =3\pi ∕4$ and $r=1.8$ is shown in Fig. 3. Marks $+$ and $-$ indicate positive and negative differences, respectively.

Figure 3

Bold line indicates the degree distribution of the closed plot in Fig. 2e ($\theta =3\pi ∕4$ and $r=1.8$ in Fig. 1), where the game structure is antileader (see Fig. 1) with an SH-type dilemma. The thin and dotted lines are those of the fixed random network (maximum degree is 19) and the fixed, scale-free network (maximum degree is 74).

Figure 4

Degree distributions of $C$ agents, $D$ agents, and entire agents in case of SEN. The game structure is same as Fig. 3 ($\theta =3\pi ∕4$ and $r=1.8$). The gray bold line (“altogether”) is consistent with “SEN” in Fig. 3. The agent having the maximum degree 57 is a $C$ agent. The maximum degree of $D$ agents is 12.

Figure 5

Payoff (per an agent) distribution of $C$ agents, $D$ agents, and entire agents in case of SEN. The game structure is same as Fig. 3 ($\theta =3\pi ∕4$ and $r=1.8$).

Figure 6

Payoff distributions for two agents relations of both $C$-$C$ link and $C$-$D$ link in the case of SEN. The game structure is same as Fig. 3 ($\theta =3\pi ∕4$ and $r=1.8$). (A) is for $C$-$C$ link distribution. $X$ axis indicates a $C$ agent’s payoff. $Y$ axis indicates another $C$ agent’s payoff who is connected with the first $C$ agent. Hence this subgraph is asymmetric to the $45°$ line. (B) is for $C$-$D$ link distribution. $X$ axis indicates a $C$ agent’s payoff. $Y$ axis indicates a $D$ agent’s payoff who has a link with the first $C$ agent.