Fixation of competing strategies when interacting agents differ in the time scale of strategy updating

A commonly used assumption in evolutionary game theory is that natural selection acts on individuals in the same time scale; e.g., players use the same frequency to update their strategies. Variation in learning rates within populations suggests that evolutionary game theory may not necessarily be restricted to uniform time scales associated with the game interaction and strategy adaption evolution. In this study, we remove this restricting assumption by dividing the population into fast and slow groups according to the players' strategy updating frequencies and investigate how different strategy compositions of one group influence the evolutionary outcome of the other's fixation probabilities of strategies within its own group. Analytical analysis and numerical calculations are performed to study the evolutionary dynamics of strategies in typical classes of two-player games (prisoner's dilemma game, snowdrift game, and stag-hunt game). The introduction of the heterogeneity in strategy-update time scales leads to substantial changes in the evolution dynamics of strategies. We provide an approximation formula for the fixation probability of mutant types in finite populations and study the outcome of strategy evolution under the weak selection. We find that although heterogeneity in time scales makes the collective evolutionary dynamics more complicated, the possible long-run evolutionary outcome can be effectively predicted under technical assumptions when knowing the population composition and payoff parameters.

Figure 1

Fixation probabilities for the game whose dominance strategy is $B$ with different selection intensity: (a) $\omega =0.25$, and (b) $\omega =0.05$. The payoff parameter values of the prisoner's dilemma game: $a=3$, $b=1$, $c=5$, $d=2$. Here, the horizontal axis means the fixatation probability of strategy $A$ in the fast group, while the vertical axis represents the initial number of $A$ players in the fast group. The following settings are the same in Figs. 2 and 3. Simulation results (symbols) coincide perfectly with the approximation results (solid lines). The approximation results are from Eq. (10). Each simulation result corresponds to the average frequency of fixation of $A$ players from 100 independent realizations. Here, the results show that the heterogeneity of time scales on updating has only limited effects on the fixation probabilities.

Figure 2

Evolutionary outcomes for the games where $A$ and $B$ will stably coexist. (a) $\omega =0.25$, and (b) $\omega =0.05$. The payoff parameter values of snowdrift game: $a=3$, $b=2$, $c=5$, $d=1$. Here, the horizontal axis means the fixatation probability of strategy $A$ in the fast group, while the vertical axis represents the initial number of $A$ players in the fast group. Results show that the heterogeneity of time scales on updating has significant effects on the fixation of probabilities of strategies in their groups. Specifically, more opposite strategies in the other group promote a strategy to get fixation in its own group.

Figure 3

Evolutionary outcomes for the coordination games. (a) $\omega =0.25$, and (b) $\omega =0.05$. The payoff parameter values of stag-hunt game: $a=5$, $b=1$, $c=3$, $d=2$. Here, the results show that diversity of time scales on updating has significant effects on the fixation probabilities. Different with the results shown in Fig. 2, more opposite strategies in the other group inhibit a strategy to get fixation in its own group.

Figure 4

Simulation results of games whose dominating strategy is $B$. Here, the horizontal axis means the fixation probability of strategy $A$ in the fast group, while the vertical axis represents the initial number of $A$ players in the fast group. $\omega =0.25$ and $\omega =0.05$, respectively. The initial numbers of $A$ players in the slow groups are 1 (green lines), 20 (red lines), and 39 (blue lines) for comparison. Moreover, though a group of values $s=1,2,5,10,50,100$ are used here, the nondistinctive differences between them make it unnecessary to distinguish them with different colors. The simulation results here show notable consistence with the theoretical results in Fig. 1, suggesting the validity of our theoretical analysis.

Figure 5

Simulation results of games where $A$ and $B$ will stably coexist. Here, the horizontal axis means the fixation probability of strategy $A$ in the fast group, while the vertical axis represents the initial number of $A$ players in the fast group. $\omega =0.25$ and $\omega =0.05$, respectively. The initial numbers of $A$ players in the slow groups are 1 (green lines), 20 (red lines), and 39 (blue lines) for comparison. Still, we do not distinguish $s=1,2,5,10,50,100$ with different colors. The simulation results here show much consistence with the theoretical results in Fig. 2, suggesting the validity of our theoretical analysis.

Figure 6

Simulation results of the coordination games. Here, the horizontal axis means the fixation probability of strategy $A$ in the fast group, while the vertical axis represents the initial number of $A$ players in the fast group. $\omega =0.25$ and $\omega =0.05$, respectively. The initial numbers of $A$ players in the slow groups are 1 (green lines), 20 (red lines), and 39 (blue lines) for comparison. The simulation results here show notable consistence with the theoretical results in Fig. 3, suggesting the validity of our theoretical analysis.