Networked Anti-Coordination Games Meet Graphical Dynamical Systems: Equilibria and Convergence

TL;DR

This work analyzes networked evolutionary anti-coordination games by mapping them to inverted-threshold graphical dynamical systems to study NE existence and convergence under four classes (SN/SE with synchronous/sequential updates). It establishes a sharp complexity divide: EQE/EQF are NP-hard (#P-hard to count) under SE, while SN yields polynomial-time NE finding; convergence to NE or 2-cycle is shown to be polynomial in many regimes, with SN and synchronous updates particularly tractable. The authors provide potential-based proofs for convergence, derive bounds such as O(m^2) for NE discovery in SN modes, and propose efficient algorithms for special graph classes. Experimental results on synthetic and real networks corroborate the theoretical results, illustrating faster convergence in SN and richer NE structure under SN than SE. The findings advance understanding of equilibrium computation and dynamic behavior in networked anti-coordination settings with practical implications for routing, competition, and resource allocation.

Abstract

Evolutionary anti-coordination games on networks capture real-world strategic situations such as traffic routing and market competition. In such games, agents maximize their utility by choosing actions that differ from their neighbors' actions. Two important problems concerning evolutionary games are the existence of a pure Nash equilibrium (NE) and the convergence time of the dynamics. In this work, we study these two problems for anti-coordination games under sequential and synchronous update schemes. For each update scheme, we examine two decision modes based on whether an agent considers its own previous action (self essential ) or not (self non-essential ) in choosing its next action. Using a relationship between games and dynamical systems, we show that for both update schemes, finding an NE can be done efficiently under the self non-essential mode but is computationally intractable under the self essential mode. To cope with this hardness, we identify special cases for which an NE can be obtained efficiently. For convergence time, we show that the best-response dynamics converges in a polynomial number of steps in the synchronous scheme for both modes; for the sequential scheme, the convergence time is polynomial only under the self non-essential mode. Through experiments, we empirically examine the convergence time and the equilibria for both synthetic and real-world networks.

Figures (5)

• Figure 1: Example dynamics of SDS and SyDS under the SN mode. Specifically, $C$ is an initial configuration, and $C'$ is its successor under either the synchronous or the sequential (with vertex update order $(v_1, v_2, v_3, v_4)$) mode. State-1 vertices are highlighted in blue.
• Figure 2: Impact of network density on the average number of steps for the SE and SN modes to converge. The underlying Gnp networks have $10,000$ vertices with average degrees varying from $5$ to $100$. The variances for the SE and the SN modes are shown in the beige and blue shaded regions, respectively.
• Figure 3: The distribution of the number of instances with at most 28 NE. The underlying Gnp networks are of size $20$ with an average degree of $4$.
• Figure 4: An example (SE, IT)-SyDS$\mathcal{S}$ for Lemma \ref{['lemma:gadget']}, where $G_{\mathcal{S}}{}$ is a complete bipartite graph with bipartitions $\{A, B\}$. The thresholds $\tau_1$ of all vertices are $3$.
• Figure 5: An example graph $G$ where blue vertices are terminals, green vertices are gates, and red vertices are tree vertices.

Theorems & Definitions (29)

• Definition 3.1: Equilibrium existence/finding
• Definition 3.2: Convergence
• Theorem 4.2
• Lemma 4.3
• Lemma 4.4
• Theorem 4.5
• Theorem 4.6
• Lemma 5.1
• Lemma 5.2
• Theorem 5.3