Equilibrium seeking of higher-order networks under facet cover constraints

Agent intelligence involves specific requirements for social attributes. Intelligent agents make their decisions based on the groups they are part of, tend to satisfy co-members, and enlarge their own benefits. A fundamental question is whether this form of subgroup decision-making accommodate each individual's preferences. In this paper, we examine the evolution of an anticoordination game on a higher-order network in the form of a simplicial complex in relation to the facet cover problem, which ensures that each subgroup yields a positive benefit. We introduce and apply the facet update rule to regulate nodes' group-based interactions. We identify the payoff parameter condition that a strict Nash equilibrium (SNE) satisfies a facet cover. The proposed facet update rule enables the activated facet to reach a facet equilibrium, and all nodes would converge to an SNE with no more than 2m strategy switches, where m is the number of nodes in the simplicial complex. Additionally, we analyze the convergence of the asynchronous update rule, which can be seen as a special case of the facet update rule. Our simulations and extensive examples reveal that the SNE achieved by the facet update rule, on average, covers fewer nodes compared to the asynchronous update rule.