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Dynamic interventions with limited knowledge in network games

Mehran Shakarami, Ashish Cherukuri, Nima Monshizadeh

TL;DR

This work formulates a regulator-driven intervention framework for quadratic network games, where selfish players update their actions via projected pseudo-gradient dynamics and a central regulator modifies marginal returns to achieve the social optimum. It analyzes four knowledge regimes—full game information, network structure, estimated optimum, and adaptive with limited information—and provides corresponding static, dynamic, and adaptive intervention protocols with rigorous convergence guarantees using monotone operator and variational-inequality tools. The social optimum $x_{\mathrm{opt}}$ is characterized as the unique solution to ${\rm SOL}({\mathcal{X}},H)$ with $H(x)=(I-a(P+P^\top))x-b$, and feasibility is tied to the set ${\mathcal{S}}$, with results demonstrated on a Cournot competition example. The contributions include explicit protocol designs, proofs of convergence under each information regime, and a demonstration that adaptation can achieve $x_{\mathrm{opt}}$ even with partial knowledge, highlighting practical pathways for coordinating decentralized networks toward social welfare.

Abstract

This paper studies the problem of intervention design for steering the actions of noncooperative players in quadratic network games to the social optimum. The players choose their actions with the aim of maximizing their individual payoff functions, while a central regulator uses interventions to modify their marginal returns and maximize the social welfare function. This work builds on the key observation that the solution to the steering problem depends on the knowledge of the regulator on the players' parameters and the underlying network. We, therefore, consider different scenarios based on limited knowledge and propose suitable static, dynamic and adaptive intervention protocols. We formally prove convergence to the social optimum under the proposed mechanisms. We demonstrate our theoretical findings on a case study of Cournot competition with differentiated goods.

Dynamic interventions with limited knowledge in network games

TL;DR

This work formulates a regulator-driven intervention framework for quadratic network games, where selfish players update their actions via projected pseudo-gradient dynamics and a central regulator modifies marginal returns to achieve the social optimum. It analyzes four knowledge regimes—full game information, network structure, estimated optimum, and adaptive with limited information—and provides corresponding static, dynamic, and adaptive intervention protocols with rigorous convergence guarantees using monotone operator and variational-inequality tools. The social optimum is characterized as the unique solution to with , and feasibility is tied to the set , with results demonstrated on a Cournot competition example. The contributions include explicit protocol designs, proofs of convergence under each information regime, and a demonstration that adaptation can achieve even with partial knowledge, highlighting practical pathways for coordinating decentralized networks toward social welfare.

Abstract

This paper studies the problem of intervention design for steering the actions of noncooperative players in quadratic network games to the social optimum. The players choose their actions with the aim of maximizing their individual payoff functions, while a central regulator uses interventions to modify their marginal returns and maximize the social welfare function. This work builds on the key observation that the solution to the steering problem depends on the knowledge of the regulator on the players' parameters and the underlying network. We, therefore, consider different scenarios based on limited knowledge and propose suitable static, dynamic and adaptive intervention protocols. We formally prove convergence to the social optimum under the proposed mechanisms. We demonstrate our theoretical findings on a case study of Cournot competition with differentiated goods.
Paper Structure (16 sections, 7 theorems, 86 equations, 6 figures)

This paper contains 16 sections, 7 theorems, 86 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Notations
  4. Convex analysis
  5. Problem formulation
  6. Players' strategy
  7. Regulator's objective
  8. Intervention protocols
  9. Static open-loop intervention
  10. Static feedback intervention
  11. Dynamic intervention with estimated social optimum
  12. Adaptive intervention with known standalone marginal returns
  13. Illustrative examples
  14. Open-loop, static feedback and dynamic interventions
  15. Adaptive intervention
  16. ...and 1 more sections

Key Result

Lemma IV.1

Let Assumption asm:u-x-set hold. Then the social welfare maximization problem social has a unique solution if

Figures (6)

  • Figure 1: The directed network illustrating asymmetrical product substitutability.
  • Figure 2: Actions of the players and their distance to social optimum under static open-loop intervention.
  • Figure 3: Actions of the players and their distance to social optimum under static feedback intervention.
  • Figure 4: Actions of the players and their distance to social optimum under dynamic intervention.
  • Figure 5: The undirected network illustrating symmetrical product substitutability.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Remark III.2
  • Lemma IV.1
  • proof
  • Remark IV.3
  • Proposition IV.5
  • proof
  • Proposition IV.6
  • proof
  • Corollary IV.7
  • proof
  • ...and 7 more