Myopic Best Response as a Double-Edged Mechanism in Networked Social Dilemmas with Individual Solutions

Abstract

Myopic best-response dynamics (MBRD) capture agents' bounded rationality and can generate evolutionary outcomes that differ from those produced by widely examined imitation dynamics. In this study, we apply MBRD to a three-strategy social dilemma -- the snowdrift game with an individual solution -- in which not only defection but also an individual solution that guarantees a safe, constant payoff can undermine cooperation. Monte Carlo simulations show that, on a square lattice, the evolutionary dynamics result in distinct equilibria, including the dominance of the individual solution, the coexistence of cooperators and defectors, or all-strategy coexistence. By combining simulations with a simple heuristic that approximates the transition condition between the dominance of the individual solution and the all-strategy coexistence, the analysis reveals a dual role of neighborhood size. Specifically, smaller neighborhoods can promote cooperation even when the individual solution is relatively inexpensive; however, achieving cooperation under these conditions requires greater benefits from cooperation. Notably, this hindrance to cooperation contrasts with evolutionary outcomes observed under imitation dynamics. Analysis of local strategy configurations explains the transition between the all-strategy coexistence and the coexistence of cooperators and defectors while showing that this transition is absent in a one-dimensional lattice. These observations indicate that the persistent availability of individual solutions constitutes an additional inhibiting factor of cooperation in populations of boundedly rational agents.

Figures (8)

• Figure 1: Emergence of the $I$-, $CDI$-, and $CD$-equilibria under MBRD on a square lattice. As the cost of the individual solution $c_I$ increases, the $CDI$-equilibrium emerges, and this transition becomes more pronounced at higher selection intensities $\beta$. For larger values of the benefit parameter $b$, the same increase in $c_I$ instead leads to the $CD$-equilibrium. Simulation parameters: $L = 100$ and $\mu = 10^{-6}$.
• Figure 2: Spatial coexistence of cooperation, defection, and individual solution on a square lattice. Defection yields the highest payoff when agents are surrounded by cooperators, whereas the individual solution yields the highest payoff when agents are surrounded by defectors. Simulation parameters: $L = 30, b = 2.6, c_I = 1.8, \beta = 10$, and $\mu = 10^{-6}$.
• Figure 3: Distinction between the $I$- and $CDI$-equilibria based on two threshold conditions. The heatmaps show that the $CDI$-equilibrium emerges when $c_I > 7/4$ and $b > 7/3$, with the black lines representing these two thresholds. For smaller values of $b$, cooperation and defection emerge only after the value of $c_I$ approaches that of unilateral cooperation (i.e., $c_I = 2$). Parameters: $L = 100, \beta = 10$, and $\mu = 10^{-6}$.
• Figure 4: Emergence of the $CDI$-equilibrium at values of $c_I$ smaller than those predicted by the heuristic when $b$ is large. Specifically, for $b > 3$, cooperation---and consequently the $CDI$-equilibrium---emerges at $c_I$ values below 7/4, which is the critical $c_I$ suggested by the heuristic. Simulation parameters: $L = 100, \beta = 10$, and $\mu = 10^{-6}$.
• Figure 5: Dual effects of limited interactions with a small number of neighbors on the emergence of cooperation (and the $CDI$-equilibrium). When the neighborhood size ($k$) is small, cooperation can emerge at relatively low values of the cost parameter $c_I$, but this emergence requires a large benefit $b$. The three lines indicate the critical values of $c_I$ that distinguish the $I$- and $CDI$-equilibria for $k = 8$, $4$, and $2$, ordered from the top panel to the bottom. The heuristic predictions are largely consistent with the simulation results, except in the case of a well-mixed population. This figure presents the results obtained under deterministic MBRD, implemented by assuming $\beta \to \infty$. Simulation parameters: $N = 10^4$ and $\mu = 10^{-6}$.