Reputation assimilation mechanism for sustaining cooperation

TL;DR

The paper addresses how reputation feedback that aggregates individual and neighborhood information can sustain cooperation in spatial public goods settings. It introduces an assimilated reputation mechanism that affects both strategy imitation and group payoff via a reputation-dependent multiplier, with an assimilation parameter and a perturbation term. Through extensive Monte Carlo simulations on lattices and small-world networks, the study shows that high-reputation clusters form and stabilize cooperative behavior even under strong social dilemmas due to positive feedback between reputation and cooperation. The results highlight the potential for reputation-based mechanisms to promote cooperation in structured populations and suggest extensions to more dynamic networks and noisy reputation dynamics.

Abstract

Keeping a high reputation, by contributing to common efforts, plays a key role in explaining the evolution of collective cooperation among unrelated agents in a complex society. Nevertheless, it is not necessarily an individual feature, but may also reflect the general state of a local community. Consequently, a person with a high reputation becomes attractive not just because we can expect cooperative acts with higher probability, but also because such a person is involved in a more efficient group venture. These observations highlight the cumulative and socially transmissible nature of reputation. Interestingly, these aspects were completely ignored by previous works. To reveal the possible consequences, we introduce a spatial public goods game in which we use an assimilated reputation simultaneously characterizing the individual and its neighbors' reputation. Furthermore, a reputation-dependent synergy factor is used to capture the high (or low) quality of a specific group. Through extensive numerical simulations, we investigate how cooperation and extended reputation co-evolve, thereby revealing the dynamic influence of the assimilated reputation mechanism on the emergence and persistence of cooperation. By fostering social learning from high-reputation individuals and granting payoff advantages to high-reputation groups via an adaptive multiplier, the assimilated reputation mechanism promotes cooperation, ultimately to the systemic level.

Figures (6)

• Figure 1: Strategy, reputation update, and payoff adjustment process. Blue nodes represent cooperators, while red nodes represent defectors. The shading indicates different game states: green shading corresponds to the game state at time step $t-1$, and blue shading represents the updated game state. The gray numbers next to the nodes indicate each node's reputation. The model consists of three processes: (a) Reputation update: The central node updates its reputation based on its current behavior. (b) Payoff adjustment: After the central node's reputation is updated, the multiplication factor of the PGG group is adjusted according to the updated reputation of the central node. (c) Strategy update: The central node $i$ selects a neighbor $j$ with probability $P_j$, and adopts the strategy of node $j$ with probability $W_{s_i \rightarrow s_j}$.
• Figure 2: Dependence of $f_c$ and $\bar{R}$ on the $r_0$ under different $\lambda$ and $\beta$ values. Other parameters are set as $\alpha=0.5$ and $\delta=5$. In Figs. \ref{['fig:lamuda']}(\ref{['fig:lamuda_0_fc']}) and \ref{['fig:lamuda']}(\ref{['fig:lamuda_0_R']}), $\lambda=0$, while in Figs. \ref{['fig:lamuda']}(\ref{['fig:lamuda_2_fc']}) and \ref{['fig:lamuda']}(\ref{['fig:lamuda_2_R']}), $\lambda=2$. Figs. \ref{['fig:lamuda']}(\ref{['fig:lamuda_0_fc']}) and \ref{['fig:lamuda']}(\ref{['fig:lamuda_2_fc']}) show the fraction of cooperators $f_c$ as a function of $r_0$ under different payoff enhancement parameters $\beta$, whereas Figs. \ref{['fig:lamuda']}(\ref{['fig:lamuda_0_R']}) and \ref{['fig:lamuda']}(\ref{['fig:lamuda_2_R']}) show the corresponding average reputation $\bar{R}$. Different colored lines represent different $\beta$ values as shown in the legend.
• Figure 3: Spatial evolution of strategy, reputation, and payoff distributions. The top row shows the strategy distribution at four representative stages. Here, red (green) represents defector (cooperator) players. The middle and bottom rows show the related reputation and payoff values. These snapshots demonstrate a strong interdependence between cooperation, reputation, and payoff values. The parameters are $r_0=2$, $\lambda=2$, $\beta=2$, $\alpha=0.5$, and $\delta=5$.
• Figure 4: Evolutionary trajectories of the cooperation fraction $f_c$ and average reputation $\bar{R}$ under different levels of reputation perturbation factor $\delta$. Different colored curves correspond to different values of $\delta$, as indicated in the legend. Other parameters are fixed as $r_0=2$, $\lambda=2$, $\beta=2$, and $\alpha=0.5$. Panel (a) illustrates the well-known "first down, later up" dynamics of cooperation level, which is a trademark of enhanced network reciprocity perc_pre08bszolnoki_epjb09. Panel (b) demonstrates that the average reputation evolves with the same dynamics, which supports our findings shown in Fig. \ref{['fig:snapshots']}.
• Figure 5: Color coded values of stationary cooperation level $f_c$ and average reputation $\bar{R}$ under different parameter combinations. Figs. \ref{['fig:heatmaps_combined']}(\ref{['subfig:lamuda_deta_cooperation']}) and \ref{['fig:heatmaps_combined']}(\ref{['subfig:lamuda_deta_reputation']}) show the joint effects of $\lambda$ and $\delta$, Figs. \ref{['fig:heatmaps_combined']}(\ref{['subfig:alpha_delta_fc']}) and \ref{['fig:heatmaps_combined']}(\ref{['subfig:alpha_delta_R']}) show the joint effects of $\alpha$ and $\delta$, and Figs. \ref{['fig:heatmaps_combined']}(\ref{['subfig:alpha_beta_fc']}) and \ref{['fig:heatmaps_combined']}(\ref{['subfig:alpha_beta_R']}) show the joint effects of $\beta$ and $\alpha$. Except for the parameters being varied, all other parameters are fixed as $r_0=2$, $\lambda=2$, $\beta=2$, $\alpha=0.5$, and $\delta=5$.