Network reciprocity created in prisoner's dilemma games by coupling two mechanisms

We found that a nontrivial enhancement of network reciprocity for 2 × 2 prisoner's dilemma games can be achieved by coupling two mechanisms. The first mechanism presumes a larger strategy update neighborhood than the conventional first neighborhood on the underlying network. The second is the strategy-shifting rule. At the initial time step, the averaged cooperation extent is assumed to be 0.5. In the case of strategy shifting, an agent adopts a continuous strategy definition during the initial period of a simulation episode (when the global cooperation fraction decreases from its initial value of 0.5; that is, the enduring period). The agent then switches to a discrete strategy definition in the time period afterwards (when the global cooperation fraction begins to increase again; that is, the expanding period). We explored why this particular enhancement comes about, and to summarize, the continuous strategy during the initial period relaxes the conditions for the survival of relatively cooperative clusters, and the large strategy adaptation neighborhood allows those cooperative clusters to expand easily.

Figure 1

Schematic view of the evolution of cooperation in the spatial prisoner's dilemma game, with the concepts of END and EXP demonstrated. Enduring (END) period: Initial cooperators are rapidly plundered by defectors, allowing only a few cooperators to survive through the formation of compact $C$ clusters. Expanding (EXP) period: $C$ clusters start to expand, as a cooperator on a cluster's border can attract a neighboring defector into the cluster. We presume $P_c^{\mathrm{initial}}=0.5$ for this visualization.

Figure 2

Averaged cooperation fractions over all PD regions $0\le D_g\le 1$ and $0\le D_r\le 1$, for the following four cases: (1) the conventional model (standard IN and LN and a discrete strategy system), (2) a model assuming standard IN, large LN, and a discrete strategy system, (3) a model assuming standard IN and LN and shifting from a continuous to a discrete strategy system, and (4) our proposed model assuming standard IN, large LN, and shifting from a continuous to a discrete strategy system.

Figure 3

Ensemble-averaged cooperation fractions over 100 realizations of PD for $0\le D_g\le 1$ and $0\le D_r\le 1$: (a) the conventional model (standard IN and LN and a discrete strategy system), (b) a model assuming standard IN, large LN, and a discrete strategy system, (c) a model assuming standard IN and LN and shifting from a continuous to a discrete strategy system, and (d) our proposed model assuming standard IN and large LN and shifting from a continuous to a discrete strategy system. Open circles in (a) and (b) represent $(D_g,D_r)=(0,0.8)$, for which detailed information is shown in Fig. 5. The open square and triangle in (d) respectively represent $(D_g,D_r)=(0.7,0.8)$ and $(0,1)$, shown in more detail in Figs. 6 and 7.

Figure 4

Schematic configuration in which a perfect $C$ cluster is surrounded by defectors: (a) a perfect $C$ cluster surrounded by defectors, (b) a concave configuration of a perfect $C$ cluster and two additional cooperators surrounded by defectors.

Figure 5

Snapshots and time series of one of the representative paths of $(D_g,D_r)=(0,0.8)$ for the model that assumes standard IN, standard LN, and a discrete strategy system (i.e., the conventional case), as shown in Fig. 3, and for the model that assumes standard IN, large LN, and a discrete strategy system shown in Fig. 3. (a) and (b) show snapshots of the conventional case and the latter case, respectively. (c) shows those time series in terms of the global cooperation fraction.

Figure 6

Snapshots and time series of one of the representative paths of $(D_g,D_r)=(0.7,0.8)$ by the proposed model assuming standard IN, large LN, and a shift from the continuous to the discrete strategy system, as shown in Fig. 3.

Figure 7

Snapshots and time series of one of the representative paths of $(D_g,D_r)=(0,1)$ for the proposed model, assuming standard IN, large LN, and a shift from the continuous to the discrete strategy system, as shown in Fig. 3.