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A game theory analysis of decentralized epidemic management with opinion dynamics

Olivier Lindamulage De Silva, Samson Lasaulce, Irinel-Constantin Morarescu, Vineeth S. Varma

TL;DR

The paper addresses decentralized epidemic management in a network of regions by coupling a networked SIR model with opinion dynamics and casting the problem as a static generalized Nash equilibrium (GNE) game. It introduces an auxiliary convex game to establish existence/uniqueness and to compute the centralized optimum, enabling precise assessment of decentralization via the Price of Anarchy (PoA). Numerical experiments on a COVID-19–type scenario quantify how PoA grows with epidemic connectivity and how joint epidemic-and-opinion control can curb efficiency losses, while relying solely on opinion or information campaigns yields much larger costs. The framework offers a tractable, data-absent method to quantify decentralization costs and suggests extensions to dynamic games, additional constraints, and mean-field or data-driven approaches for scalability and realism.

Abstract

In this paper, we introduce a static game that allows one to numerically assess the loss of efficiency induced by decentralized control or management of a global epidemic. Each player represents a region which is assumed to choose its control to implement a tradeoff between socio-economic aspects and health aspects; the control comprises both epidemic control physical measures and influence actions on the region opinion. The Generalized Nash equilibrium $(\mathrm{GNE})$ analysis of the proposed game model is conducted. The direct analysis of this game of practical interest is non-trivial but it turns out that one can construct an auxiliary game which allows one: to prove existence and uniqueness; to compute the GNE and the optimal centralized solution (sum-cost) of the game. These results allow us to assess numerically the loss (measured in terms of Price of Anarchy ($\mathrm{PoA}$)) induced by decentralization with or without taking into account the opinion dynamics.

A game theory analysis of decentralized epidemic management with opinion dynamics

TL;DR

The paper addresses decentralized epidemic management in a network of regions by coupling a networked SIR model with opinion dynamics and casting the problem as a static generalized Nash equilibrium (GNE) game. It introduces an auxiliary convex game to establish existence/uniqueness and to compute the centralized optimum, enabling precise assessment of decentralization via the Price of Anarchy (PoA). Numerical experiments on a COVID-19–type scenario quantify how PoA grows with epidemic connectivity and how joint epidemic-and-opinion control can curb efficiency losses, while relying solely on opinion or information campaigns yields much larger costs. The framework offers a tractable, data-absent method to quantify decentralization costs and suggests extensions to dynamic games, additional constraints, and mean-field or data-driven approaches for scalability and realism.

Abstract

In this paper, we introduce a static game that allows one to numerically assess the loss of efficiency induced by decentralized control or management of a global epidemic. Each player represents a region which is assumed to choose its control to implement a tradeoff between socio-economic aspects and health aspects; the control comprises both epidemic control physical measures and influence actions on the region opinion. The Generalized Nash equilibrium analysis of the proposed game model is conducted. The direct analysis of this game of practical interest is non-trivial but it turns out that one can construct an auxiliary game which allows one: to prove existence and uniqueness; to compute the GNE and the optimal centralized solution (sum-cost) of the game. These results allow us to assess numerically the loss (measured in terms of Price of Anarchy ()) induced by decentralization with or without taking into account the opinion dynamics.
Paper Structure (14 sections, 3 theorems, 55 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 3 theorems, 55 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Dynamical System Model
  4. Generalized Strategic Form Game Model
  5. Generalized Nash Equilibrium Analysis
  6. Existence and uniqueness analysis
  7. Efficiency measures
  8. GNE determination algorithm
  9. Numerical performance analysis
  10. Summary and conclusions
  11. Auxiliary Game
  12. Proof of the Proposition \ref{['Prop:Existence_Uniqueness']}
  13. Proof of the Proposition \ref{['Prop:Efficiency']}
  14. Proof of Proposition \ref{['Prop:Determination']}

Key Result

Proposition 1

If $b_k^{\mathrm{global}}>0, \forall k$, the game $\mathcal{G}$ possesses a unique $\mathrm{GNE}$; which is denoted by $(u^\star, v^\star)$.$\Box$

Figures (4)

  • Figure 1: Bottom curve: When each region controls the epidemic both through physical measures ($u$) and opinion ($v$), the maximum value reached for the PoA is $2$, which is already significant. Middle curve: When only physical measures are controlled and the opinion is left to evolve freely, the PoA can be as large as $3.6$, showing the loss of non-controlling the opinion. Top curve: when each region only controls its opinion, very large values for the PoA can be reached ($>10$), showing the irrelevance for decentralized management when based only on opinion control.
  • Figure 3: Evolution of the fractions of infected and opinion levels for the different regions. The effect of influence campaigns on the fractions of infected appears very clearly.
  • Figure 4: The figure provides the control action intensity for the different regions. The corresponding values have to be put in correlation with the local situation of the epidemic, which is in part related to the values of the natural reproduction numbers.
  • Figure : (a) Plot of ${ \sum_{n=0}^N\frac{\rho\left(\boldsymbol{{D}}_{\gamma}^{-1} \left(\boldsymbol{{B}^0}-\boldsymbol{U}+\mathrm{Diag}(\theta(n))\widehat{\boldsymbol{B}}\right)\right)}{N+1}}$ vs $\alpha$.

Theorems & Definitions (5)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Definition 2