Involution-Cooperation-Lying Flat Game on a Network-Structured Population in the Group Competition

Currently, on the Chinese internet, the term "involution" refers to intense but meaningless competition for limited resources, also known as "the inflation of effort." Involution is a prevalent phenomenon in various social competition scenarios and can lead to significant social contradictions. On the other hand, "lying flat" refers to a conscious choice by individuals to withdraw from intense competition and pursue a self-centered and hollow existence. The propagation of lying flat culture may lead to nihilism or irrationalism. However, existing literature on involution and lying flat has two main limitations. First, previous studies have separately examined involution and lying flat as two opposite yet closely related outcomes of intense social competition. This approach fails to provide a comprehensive and in-depth understanding of involution and lying flat in the context of increasingly fierce social competition. Second, existing literature predominantly adopts value-based payoff allocation methods, disregarding the prevalence of rank-based payoff allocation methods. Consequently, the other prevalent rank-based allocation (RA) method remains understudied. To address these limitations, we propose an evolutionary game model on a square lattice to abstract the fierce social competition between involution, cooperation, and lying flat. Our experimental results demonstrate that various factors, including social resources, the cost of involution, the relative utility of the involution strategy, and the variance of agents' competition abilities, play distinct yet significant roles in social competition. Furthermore, these factors exhibit interaction effects on the levels of cooperation, involution, and lying flat. The rank threshold ( $T$ ) positively affects the level of cooperation, exhibits a U-shaped effect on the level of lying flat, and an inverted U-shaped effect on the level of involution. The RA rule tends to result in a higher level of involution, whereas the value-based rule tends to result in a higher level of lying flat. These findings remain robust across different network sizes. Our study provides insights into strategies for suppressing the levels of involution and lying flat in social competitions.