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Equilibrium and Selfish Behavior in Network Contagion

Yi Zhang, Sanjiv Kapoor

TL;DR

To address selfish policy adoption under contagion, the paper models a non-atomic game where infinitesimal players select among $n$ policies in an SIR framework. It proves that Nash equilibria exist in general, but finding them is PPAD-hard for arbitrary transmissions; it then identifies polynomial-time solvable cases with uniform interaction policies and networked populations, using convex programming to compute final sizes and fixed-point analyses for equilibria. It derives PoA bounds that scale as $e^{R_0}$ or $e^{\omega R_0}$ depending on the model, highlighting potential efficiency losses under selfish behavior. The results yield practical polynomial-time algorithms to compute endemic final sizes and equilibria in both separable and network contagion settings, informing policy design for decentralized containment.

Abstract

In this paper we consider non-atomic games in populations that are provided with a choice of preventive policies to act against a contagion spreading amongst interacting populations, be it biological organisms or connected computing devices. The spreading model of the contagion is the standard SIR model. Each participant of the population has a choice from amongst a set of precautionary policies with each policy presenting a payoff or utility, which we assume is the same within each group, the risk being the possibility of infection. The policy groups interact with each other. We also define a network model to model interactions between different population sets. The population sets reside at nodes of the network and follow policies available at that node. We define game-theoretic models and study the inefficiency of allowing for individual decision making, as opposed to centralized control. We study the computational aspects as well.

Equilibrium and Selfish Behavior in Network Contagion

TL;DR

To address selfish policy adoption under contagion, the paper models a non-atomic game where infinitesimal players select among policies in an SIR framework. It proves that Nash equilibria exist in general, but finding them is PPAD-hard for arbitrary transmissions; it then identifies polynomial-time solvable cases with uniform interaction policies and networked populations, using convex programming to compute final sizes and fixed-point analyses for equilibria. It derives PoA bounds that scale as or depending on the model, highlighting potential efficiency losses under selfish behavior. The results yield practical polynomial-time algorithms to compute endemic final sizes and equilibria in both separable and network contagion settings, informing policy design for decentralized containment.

Abstract

In this paper we consider non-atomic games in populations that are provided with a choice of preventive policies to act against a contagion spreading amongst interacting populations, be it biological organisms or connected computing devices. The spreading model of the contagion is the standard SIR model. Each participant of the population has a choice from amongst a set of precautionary policies with each policy presenting a payoff or utility, which we assume is the same within each group, the risk being the possibility of infection. The policy groups interact with each other. We also define a network model to model interactions between different population sets. The population sets reside at nodes of the network and follow policies available at that node. We define game-theoretic models and study the inefficiency of allowing for individual decision making, as opposed to centralized control. We study the computational aspects as well.

Paper Structure

This paper contains 19 sections, 25 theorems, 64 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Models and Results: The Contagion Game
  3. Models
  4. Hardness of Nash Equilibrium in Interacting Policy Sets
  5. Existence of Nash Equilibrium
  6. PPAD-hardness of the Contagion Game
  7. Uniform Interaction Policy Sets
  8. Polynomial Time Convex Programming Approach to Compute the Final Size
  9. Computing the Nash Equilibrium
  10. Price of Anarchy
  11. Policy Sets over an Interacting Network
  12. Polynomial Time Convex Programming Approach to Compute the Final Size in the Network Interaction Model
  13. Computing the Nash Equilibrium in the Uniform Network Interaction Model
  14. Price of Anarchy
  15. Separable Policy Sets
  16. ...and 4 more sections

Key Result

Theorem 1

Nash equilibrium always exists in every contagion game.

Figures (5)

  • Figure 1: Transition diagram of the interacting model ( Model A). The red arrows indicate interactions across different groups.
  • Figure 2: Network interaction model ( Model B). The red arrows indicate interactions across different groups and different nodes.
  • Figure 3: A demonstration of the convex program in a 2-group setting. The red point is the final size point $F^*$.
  • Figure 4: Axis of $X_0$. When $X_0$ lies within a specific range $R$, $G_v$ is constructed.
  • Figure 5: $g(\overline{S})$ and the lower bound $LB$ and upper bound $UB$ of $\overline{S}(\infty)$.

Theorems & Definitions (41)

  • Theorem 1
  • proof
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • ...and 31 more