Involution game with migration and spatial heterogeneity of social resources

Abstract

Involution, a phenomenon of excessive competition with diminishing returns, has become a pressing socio-economic concern in contemporary China, prompting both academic inquiry and policy interventions. This paper proposes an evolutionary game model of involution that incorporates agent migration and spatial heterogeneity in resource distribution. The model captures realistic features such as effort-based resource allocation, local interactions on a lattice, and mobility driven by payoff comparisons. We explore how varying conditions of migration and resource allocation influence the dynamics of involution. The key findings from our simulations are as follows: when total resources are held constant, similar resource levels across different regions tend to suppress involution; conversely, an increase in total resources exacerbates it. In addition, the probability of migration does not significantly affect the final evolutionary outcome. We further identify threshold effects in the effort ratio and utility multiplier, revealing critical conditions under which involution emerges or subsides. To further elucidate these simulation results, we conduct a theoretical analysis using mean-field theory, which provides analytical expressions for the equilibria and stability conditions. The theoretical predictions are in excellent qualitative agreement with simulation outcomes. Finally, we discuss real-world counterparts of the model, including competition among food delivery riders and between stores offering similar services.

Figures (7)

• Figure 1: Proportion of competitive agents ($F_D$) as a function of migration rate $\mu$ and selection intensity $k$ under asymmetric resource distribution ($M_1 = 1$, $M_2 = 2$). (a) $F_D$ vs. $\mu$ for different $k$; (b) $F_D$ vs. $k$ for different $\mu$; (c) contour plot over the whole parameter space. Averaged over 50 independent runs on a $100\times100$ lattice with 2000 agents. For runs not reaching an absorbing state within 3000 steps, the last 300 steps are averaged.
• Figure 2: Proportion of agents located in region $M_1$, $P_{M_1}$, as a function of the resource ratio $M_1/M_2$ for different migration rates $\mu$. Simulations with total resources $M_1+M_2 = 3.0$, 2000 agents on a $100\times100$ lattice, averaged over 50 independent runs. The dashed line at $P_{M_1}=0.5$ indicates uniform distribution.
• Figure 3: Proportion of competitive agents $F_D$ as a function of the resource ratio $M_1/M_2$ and migration rate $\mu$ under different total resource endowments. Each panel corresponds to a fixed total resource $M_1+M_2$ (2, 3, and 5, from left to right), with selection intensity fixed at $k=1$. Color intensity represents the fraction of agents adopting the high-effort strategy D. Results are averaged over 50 independent runs on a $100\times100$ lattice with 2000 agents.
• Figure 4: Effect of utility multiplier $\beta$ on involution and spatial distribution for three total resource levels. Each pair of panels corresponds to a fixed total resource: (a,b) $M_1+M_2=2.0$, (c,d) $M_1+M_2=3.0$, (e,f) $M_1+M_2=5.0$. Simulations with $k=1.0$, $\mu=1.0$, 2000 agents, averaged over 50 independent runs.
• Figure 5: Effect of the effort ratio $e_D/e_C$ on involution. Simulations with $M_1+M_2 = 3.0$, $\mu = 1.0$, $k = 1.0$, and 2000 agents, averaged over 50 independent runs.

Theorems & Definitions (2)

• Proposition 1: Migration equilibrium
• Proposition 2: Strategy equilibria