United We Fall: On the Nash Equilibria of Multiplex and Multilayer Network Games

TL;DR

This work develops multiplex and multilayer network games to model strategic decisions across interacting networks and action dimensions. It recasts Nash equilibrium analysis through linear best-response dynamics and the LCP/P-matrix framework, deriving spectral conditions (involving $\lambda_{\min}$ and $\lambda_{\max}$) that govern existence, uniqueness, and stability of equilibria. The authors provide both necessary and sufficient conditions, including special cases for symmetric AD matrices, one-way inter-layer links, and games of complements, and they demonstrate how inter-layer interactions can fragilize or preserve equilibrium uniqueness. Numerical experiments on synthetic and real-world data corroborate the theory, illustrating practical implications for designing networked systems to achieve stable, unique outcomes. Overall, the paper offers spectral guidance for interventions in interconnected networks, with potential applications in interdependent security, economics, and multi-modal information sharing.

Abstract

Network games provide a framework to study strategic decision making processes that are governed by structured interdependencies among agents. However, existing models do not account for environments in which agents simultaneously interact over multiple networks, or when agents operate over multiple action dimensions. In this paper, we propose new models of multiplex network games to capture the different modalities of interactions among strategic agents, and multilayer network games to capture their interactions over multiple action dimensions. We explore how the properties of the constituent networks of a multiplex/multilayer network can undermine or support the existence, uniqueness, and stability of the game's Nash equilibria. Notably, we highlight that both the largest and smallest eigenvalues of the constituent networks (reflecting their connectivity and two-sidedness, respectively) are instrumental in determining the uniqueness of the multiplex/multilayer network game's equilibrium. Together, our findings shed light on the reasons for the fragility of equilibria when agents interact over networks of networks, and point out potential interventions to alleviate them.

Figures (5)

• Figure 1: Illustration of multiplex and multilayer networks.
• Figure 2: By using the results in Proposition \ref{['prop:multiplex-pert-pos']} and Proposition \ref{['prop:lambda-min-fails']} in a directed setting we observe that Proposition \ref{['prop:multiplex-pert-pos']} is most effective at providing guarantees for the uniqueness of Nash equilibria in smaller multiplex networks (\ref{['fig:multiplex-prop-pos']} and \ref{['fig:prop-pos-var-s']}) and Proposition \ref{['prop:lambda-min-fails']} is most effective at refuting the guarantees for the uniqueness of Nash equilibria in larger multiplex networks (\ref{['fig:multiplex-prop-neg']} and \ref{['fig:prop-neg-var-s']}).
• Figure 3: Uniqueness of equilibria of multilayer network games (in pink) is increasingly undermined as the strength of the interlayer interactions increases (left to right).
• Figure 4: With (without) the green links, we will have a cycle (chain) of networks.
• Figure 5: Determinant of 600 multiplex sub-networks, extracted from the Copenhagen Networks Data with $N=30$ (left) and from the Friendfeed-Twitter Data with $N=50$ (right), as $\kappa_1$ (the influence of in-person interactions) increases.

Theorems & Definitions (27)

• Definition 1
• Theorem 1
• Corollary 1
• Example 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6