Discontinuous Wealth-Gradient Transition Driving Cooperation

Abstract

The universal prevalence of cooperation is puzzling, as defection typically yields higher payoffs than cooperation, motivating searches for hidden pathways to cooperation. Here we study a game-theoretic model on a lattice structured population in which interaction payoffs are scaled by the minimum of participants' accumulated wealth, reflecting real-world heterogeneity and incorporating the influence of past strategic choices. This wealth scaling allows frequent cooperators to surpass defectors in payoffs through their greater wealth even at high cooperation costs where defection would otherwise dominate. At the elevated critical cost-benefit ratio, the wealth gradient at the cooperator-defector boundary in one dimension exhibits a discontinuous transition. We show that slowing and effective stalling of the boundary trigger an explosive buildup of the wealth gradient, driving the dominance of cooperation below the critical ratio. Remarkably, this promotion of cooperation is stronger at higher temperatures, revealing a constructive role of fluctuations.

Figures (7)

• Figure 1: Evolution of strategy and wealth. (a) Evolution of the strategy configuration $\{s_i(t)|1\leq i\leq N=4000, 1\leq t\leq T=10^6\}$ in a single run of simulation under periodic boundary condition. Blue (red) color represents cooperation (defection). Initially $s_i={\rm C}$ for $i\in \left[\frac{N}{4}+1, \frac{3N}{4} \right]$ and $s_i={\rm D}$ otherwise. (b) Log-wealth profiles $\{w_i(t)=\log_{10}W_i \mid 1\le i\le N\}$ at times $t_0,t_1,t_2,t_3$, from a single run. $x(t_i)$ marks the boundary between the central $C$ cluster and the right $D$ cluster at time $t_i$. (c) Wealth distribution at different times. (d) Time-evolution of the density of cooperators $\rho(t)$. (e) Frequency of fixation to cooperation versus cost-benefit ratio $c$. All results are obtained at $\varepsilon=0.01$, $\beta=100$, $c=0.68$, and $N=4000$ unless stated otherwise and the results in (c-e) are averaged over 100 simulation runs.
• Figure 2: Wealth gradient and phase diagram in one dimension. (a) Excess wealth gradient $r_x(t)-1$ at the domain boundary for different cost-benefit ratios $c$. Dotted lines represent $r_{\rm th}-1={1\over 2(1-c)}-1$ and the dashed line has a slope $1/2$. (b) Saturated wealth gradient $r_{\rm f}$ versus $c$ from simulations (open) and analytic prediction $r_{\rm f} = r_{\rm th}={1\over 2(1-c)}$ for $c<c_{\rm crit}^{\rm (theo)}$ (dotted) and the solution of Eq. \ref{['eq:self']} for $c>c_{\rm crit}^{\rm (theo)}$ (solid) with $c_{\rm crit}^{\rm (theo)} \approx 0.695$. Inset: Plots of $y=r$ and $y=f(r,c)$ with $f(r,c)$ given by the r.h.s. of Eq. \ref{['eq:self']} for different values of $c$. (c) Boundary occupation time $B_{x}(t)$. Dotted line represent $B_{\rm th}$ and the dashed line has slope $1/2$. (d) Critical ratios $c_{\rm crit}$ versus inverse temperature $\beta$, dividing the cooperation regime $c<c_{\rm crit}(\beta)$ and the defection regime $c>c_{\rm crit}(\beta)$ for $\varepsilon=0.01$ and different system sizes $N$. Shown are $c_{\rm crit}$ (points) and $c_{\rm crit}^{\rm (occu)}$ (dashed) from simulations, $c_{\rm crit}^{\rm (theo)}$ (solid) from Eq. \ref{['eq:self']}, and $c_{\rm crit}^{\infty}$ in the $\beta\to\infty$ limit from Eq. \ref{['eqn:beta_inf']}. All results are averaged over $100$ simulation runs and $\beta=100,\varepsilon=0.01$ and $N=4000$ in (a-c).
• Figure 3: Evolution of cooperator density and phase diagram in two dimension. (a) Time-evolution of the cooperator density $\rho(t)$ for different cost-benefit ratios $c$ from simulations on the two-dimensional lattice of size $N=L\times L$ with $N=4096 (L=64)$, $\beta=100$, and $\varepsilon=0.01$. (b) Critical ratios $c_{\rm crit}$ versus inverse temperature $\beta$ for different system sizes with $\varepsilon=0.01$. The critical ratios without wealth-scaled interactions $c_{\rm crit}^{{\rm (nw)}}$ for $N=4096$ is also shown (square). The inset shows the difference between $c_{\rm crit}$ and $c_{\rm crit}^{\rm (nw)}$ as a function of $\beta$.
• Figure B.1: Temperature-dependent critical points and their asymptotic behaviors for $\varepsilon=0.01$. (a) The critical point $c_{\rm crit}^{\rm (theo)}$ as a function of inverse temperature $\beta$ obtained by solving numerically Eqs. \ref{['eq:rfrcbe']} and \ref{['eq:df']} (points) and its asymptotic behaviors for $\beta$ large and small given by Eq. \ref{['eq:ccrit_large']} (dashed) and Eq. \ref{['eq:ccrit_small']} (solid), respectively. Inset: Plots of $1-c_{\rm crit}^{\rm (theo)}$ versus $\beta$ from numerical solutions (points) and Eq. \ref{['eq:ccrit_small']} (line) (b) The wealth gradient $r_{\rm crit}^{\rm (theo)}$ in the limit $c\downarrow {c_{\rm crit}^{\rm (theo)}}$ versus $\beta$ from numerical solutions (points) and its asymptotic behaviors given by Eq. \ref{['eq:rcrit_large']} (dashed) and Eq. \ref{['eq:rcrit_small']} (solid).
• Figure C.1: Time-evolution of the ensemble-averaged velocity of the C-D boundary for $c=0.4$, $c=0.6$, $0.68$, and $0.8$.