Optimization of institutional incentives for cooperation in structured populations

TL;DR

This work establishes an index function for quantifying the cumulative cost during the process of incentive implementation, and theoretically derive the optimal positive and negative incentive protocols for cooperation on regular networks, finding that both types of optimal incentive protocols are identical and time-invariant.

Abstract

The application of incentives, such as reward and punishment, is a frequently applied way for promoting cooperation among interacting individuals in structured populations. However, how to properly use the incentives is still a challenging problem for incentive-providing institutions. In particular, since the implementation of incentive is costly, to explore the optimal incentive protocol, which ensures the desired collective goal at a minimal cost, is worthy of study. In this work, we consider the positive and negative incentives respectively for a structured population of individuals whose conflicting interactions are characterized by a prisoner's dilemma game. We establish an index function for quantifying the cumulative cost during the process of incentive implementation, and theoretically derive the optimal positive and negative incentive protocols for cooperation on regular networks. We find that both types of optimal incentive protocols are identical and time-invariant. Moreover, we compare the optimal rewarding and punishing schemes concerning implementation cost and provide a rigorous basis for the usage of incentives in the game-theoretical framework. We further perform computer simulations to support our theoretical results and explore their robustness for different types of population structures, including regular, random, small-world, and scale-free networks.

Figures (9)

• Figure 1: Evolutionary prisoner's dilemma game on a graph with institutional reward or punishment. Panel a shows the pairwise interaction between two connected neighbors in a network. Panel b (c) shows how incentives are implemented for the two connected agents who played the game when positive (negative) incentives from the incentive-providing institution are applied. Panel d shows the illustration of the four strategy update rules, depicting how agents update their strategies after obtaining payoffs from the pairwise interactions with neighbors and the incentive-providing institution.
• Figure 1: Time evolution of the fraction of cooperators for three different protocols of incentives on four different networks under DB updating. The applied protocols are marked by the legend, where the optimal one is indicated by $\ast$. We have also plotted the cumulative cost values for each incentive protocol. The results of Monte Carlo simulations for reward (punishment) are shown on top (bottom) row. Parameters: $N=100$, $L=10$, $b=2$, $c=1$, $\delta=0.01$, $\omega=0.01$, and $p_{0}=0.5$. For proper comparison the average degree is set to 4 for all graphs.
• Figure 2: Optimization of institutional incentives for cooperation for different strategy update rules. Here, the minimal $\mu_R$ ($\mu_P$) means the minimal amount of positive (negative) incentive needed for the evolution of cooperation for different strategy update rules. $\mu_{R}^{*}$ ($\mu_{P}^{*}$) represents the optimal positive (negative) incentive protocol for different strategy update rules. $J_R^*$ ($J_P^*$) means the cumulative cost produced by the optimal rewarding (punishing) protocol $\mu_{R}^{*}$ ($\mu_{P}^{*}$) for the dynamical system to reach the expected terminal state $1-\delta$ from the initial state $p_{0}$. In addition, $N$ denotes the population size and $k$ the degree of the regular network. $b$ represents the benefit of cooperation and $c$ the cost of cooperation. The parameters $\beta_{\textrm{DB}}=\frac{\omega(k-2)(ck-b)}{k-1}$ under DB updating, $\beta_{\textrm{BD}}=\frac{\omega k(k-2)c}{k-1}$ under BD updating, $\beta_{\textrm{IM}}=\frac{\omega k^{2}(k-2)[c(k+2)-b]}{(k+1)^{2}(k-1)}$ under IM updating, and $\beta_{\textrm{PC}}=\frac{\omega k(k-2)c}{2(k-1)}$ under PC updating.
• Figure 2: Time evolution of the fraction of cooperators for three different protocols of incentives on four different networks under BD updating. The applied protocols are marked by the legend, where the optimal one is indicated by $\ast$. We have also plotted the cumulative cost values for each incentive protocol. The results of Monte Carlo simulations for reward (punishment) are shown on top (bottom) row. Other parameter values are the same as those in figure S1.
• Figure 3: Time evolution of the fraction of cooperators for positive and negative incentives under different strategy update rules. Each panels shows the results derived from numerical calculations based on the obtained dynamical equation at different levels of incentives for reward ($R$) or punishment ($P$). The optimal incentive level is marked by $\ast$. For comparison we have also marked the $J_{R}$ and $J_{P}$ amounts of cumulative cost to reach the desired terminal state for each incentive protocol. Parameters: $N=100$, $b=2$, $c=1$, $\delta=0.01$, $\omega=0.01$, $p_{0}=0.5$, and $k=4$.