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On Signed Network Coordination Games

Martina Vanelli, Laura Arditti, Giacomo Como, Fabio Fagnani

TL;DR

This work introduces signed network coordination (SNC) games on directed signed graphs with binary actions, capturing both coordination and anti-coordination via weights $W_{ij}$ and external fields $h_i$. It proves existence of pure Nash equilibria that exhibit consensus or polarization on a cohesive subset $\mathcal{R}$ under structural assumptions, and analyzes their stability under asynchronous best-response dynamics. The authors develop a unified approach using graph cohesiveness and supermodularity, with gauge transformations to handle structural balance by reducing to unsigned coordinating subgraphs. The results provide robustness guarantees for equilibrium convergence in heterogeneous networks and connect classic network coordination theory with signed-graph concepts, offering practical criteria for existence and stability of equilibria in complex social/economic networks.

Abstract

We study binary-action pairwise-separable network games that encompass both coordinating and anti-coordinating behaviors. Our model is grounded in an underlying directed signed graph, where each link is associated with a weight that describes the strenght and nature of the interaction. The utility for each agent is an aggregation of pairwise terms determined by the weights of the signed graph in addition to an individual bias term. We consider a scenario that assumes the presence of a prominent cohesive subset of players, who are either connected exclusively by positive weights, or forms a structurally balanced subset that can be bipartitioned into two adversarial subcommunities with positive intra-community and negative inter-community edges. Given the properties of the game restricted to the remaining players, our results guarantee the existence of Nash equilibria characterized by a consensus or, respectively, a polarization within the first group, as well as their stability under best response transitions. Our results can be interpreted as robustness results, building on the supermodular properties of coordination games and on a novel use of the concept of graph cohesiveness.

On Signed Network Coordination Games

TL;DR

This work introduces signed network coordination (SNC) games on directed signed graphs with binary actions, capturing both coordination and anti-coordination via weights and external fields . It proves existence of pure Nash equilibria that exhibit consensus or polarization on a cohesive subset under structural assumptions, and analyzes their stability under asynchronous best-response dynamics. The authors develop a unified approach using graph cohesiveness and supermodularity, with gauge transformations to handle structural balance by reducing to unsigned coordinating subgraphs. The results provide robustness guarantees for equilibrium convergence in heterogeneous networks and connect classic network coordination theory with signed-graph concepts, offering practical criteria for existence and stability of equilibria in complex social/economic networks.

Abstract

We study binary-action pairwise-separable network games that encompass both coordinating and anti-coordinating behaviors. Our model is grounded in an underlying directed signed graph, where each link is associated with a weight that describes the strenght and nature of the interaction. The utility for each agent is an aggregation of pairwise terms determined by the weights of the signed graph in addition to an individual bias term. We consider a scenario that assumes the presence of a prominent cohesive subset of players, who are either connected exclusively by positive weights, or forms a structurally balanced subset that can be bipartitioned into two adversarial subcommunities with positive intra-community and negative inter-community edges. Given the properties of the game restricted to the remaining players, our results guarantee the existence of Nash equilibria characterized by a consensus or, respectively, a polarization within the first group, as well as their stability under best response transitions. Our results can be interpreted as robustness results, building on the supermodular properties of coordination games and on a novel use of the concept of graph cohesiveness.
Paper Structure (14 sections, 14 theorems, 64 equations, 8 figures)

This paper contains 14 sections, 14 theorems, 64 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Signed Network Coordination Games
  4. Best Responses and Nash Equilibria
  5. Preliminary results
  6. SNC games on undirected networks
  7. SNC games on unsigned networks
  8. Signed network games on structurally balanced networks
  9. Main results
  10. Existence of Nash equilibria
  11. Stability of Nash equilibria
  12. Concluding remarks
  13. Proof of Lemma \ref{['lemma:NashGauge']}
  14. Proof of Lemma \ref{['lemma:balanced']}

Key Result

Proposition 1

Consider a SNC game with binary actions on a network $\mathcal{G}=(\mathcal{V},\mathcal{E},W)$ with external field $h$. Then, Moreover, if $\mathcal{G}$ is undirected, then:

Figures (8)

  • Figure 1: A network with 13 nodes. Weights are represented by the values on the links. Nodes in the set $\mathcal{R}= \{1, \dots, 8\}$ are colored in gray, while nodes in the complement set $\mathcal{S}= \{9, \dots, 13\}$ are white.
  • Figure 2: Two signed graphs representing: (a) the discoordination game and (b) a directed anti-coordination game.
  • Figure 3: A signed graph with $8$ nodes and a coordinating set made of $7$ players (in gray).
  • Figure 4: Graph studied in Example \ref{['ex:sb']}. The subset $\mathcal{R}=\{1, \dots, 4\}$ (in gray) is such that $\mathcal{G}_{\mathcal{R}}$ is structurally balanced.
  • Figure 5: Graph considered in Example \ref{['ex:dec1']}.
  • ...and 3 more figures

Theorems & Definitions (39)

  • Example 1
  • Definition 1
  • Remark 1
  • Remark 2
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Proposition 1
  • proof
  • ...and 29 more