An evolutionary game with reputation-based imitation-mutation dynamics

TL;DR

The study extends evolutionary game theory by introducing imitation-mutation dynamics tied to reputation, allowing irrational updates through a mutation rate $m$ and fitness-modulated imitation with $F_i(t)=R_i(t)P_i(t)$ and $R_i(t)=R_i(t-1)\pm\alpha$. Across lattices, small-world, and scale-free networks, the IM-MUTA framework generally increases cooperation in the Prisoner's Dilemma and reduces cooperation in the Snow Drift Game, with the effects of $\alpha$ and $m$ showing context-dependent trade-offs tied to temptation levels. These findings highlight the nuanced role of reputation and irrational updates in social dilemmas and point to future improvements like heterogeneous mutation rates to enhance realism. The results have implications for understanding cooperative dynamics in complex networks where reputation and exploratory decision-making coexist.

Abstract

Reputation plays a crucial role in social interactions by affecting the fitness of individuals during an evolutionary process. Previous works have extensively studied the result of imitation dynamics without focusing on potential irrational choices in strategy updates. We now fill this gap and explore the consequence of such kind of randomness, or one may interpret it as an autonomous thinking. In particular, we study how this extended dynamics alters the evolution of cooperation when individual reputation is directly linked to collected payoff, hence providing a general fitness function. For a broadly valid conclusion, our spatial populations cover different types of interaction topologies, including lattices, small-world and scale-free graphs. By means of intensive simulations we can detect substantial increase in cooperation level that shows a reasonable stability in the presence of a notable strategy mutation.

Figures (8)

• Figure 1: Schematic diagram of the imitation-mutation process and reputation mechanism. At $t-1$ time, player 'a' is a cooperator while others, player 'b', 'c', and 'd' are defectors. In the next step player 'a' imitates the strategy of player 'b' with probability $1-m$, while player 'd' chooses random mutation with probability $m$. Meanwhile, the reputation index is updated for both players as indicated.
• Figure 2: The evolution of cooperation in PDG for different dynamics and topology. Curves show the time evolution of $f_C$ starting from a random initial state for different $r_1$ values as indicated in the legend. The top row illustrates the evolution of the traditional model where players always follow imitation (IM) during strategy updates. Panel (a) to (c) represent different interaction graphs, as shown in the labels. As a comparison, the bottom low depicts those cases where the extended imitation-mutation (IM-MUTA) strategy update is applied. Other parameters are $\alpha=0.05$ and $m=0.2$. Note that we used a semi-log plot to stress the time dependence faithfully. The time evolution of $f_C$, "first down, later up", demonstrates how network reciprocity works.
• Figure 3: Stationary fraction of cooperators on $r_1 - \alpha$ parameter plane. The results are obtained for WS interaction graph by using $m=0.2$. The impact of parameter $\alpha$ is ambiguous, while for low $r_1$ values, it is better to use large $\alpha$ steps during the reputation update, for larger $r_1$ values smaller $\alpha$ provide a larger cooperation level. The largest cooperation level can be reached in the low $r_1$ - large $\alpha$ corner of the parameter plane. Note that cooperation level cannot reach 1 even for large $r_1$ due to mutation mechnaism.
• Figure 4: Time evolution of cooperation on WS graph obtained for different $\alpha$ values. Panel (a) to (d) respectively shows the cases for $\alpha=0.05$, $0.2$, $0.3$, and $0.4$. In all cases, $m=0.2$ was fixed. The applied $r_1$ values are indicated in the legend for each panel. The comparison of trajectories suggests that the impact of network reciprocity is practically diminished if $\alpha$ exceeds $0.3$.
• Figure 5: The average fitness of strategies at different temptation values in PDG on WS graph. With IM-MUTA, Panel (a) depicts how fitness values change for different $\alpha$ for low temptation at $r_1=0.1$. Panel (b) shows the same quantities for $r_1=0.7$ which represents high temptation. The fitted lines indicate clearly that at low temptation cooperators benefit more if we increase the reputation step $\alpha$. At large temptation, however, cooperators suffer more from the usage of larger $\alpha$.