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When Nash Meets Stackelberg

Margarida Carvalho, Gabriele Dragotto, Felipe Feijoo, Andrea Lodi, Sriram Sankaranarayanan

TL;DR

The paper introduces NASPs, a class of complete-information simultaneous games where each player solves a Stackelberg game with linear objective and linear constraints, while followers solve convex quadratic programs under optimistic responses. It proves $oldsymbol{ ext{$oldsymbol{ extSigma^p_2}$}}$-hardness for existence of a Nash equilibrium and presents two exact algorithms—an enumeration over complementarity polyhedra and an inner-approximation scheme—to compute $MNE$s or certify nonexistence, leveraging a convex reformulation in which a PNE corresponds to an $MNE$ of the original game. The methods exploit Balas’ extended formulations and LCP/NP-like solvers to achieve finite termination, with extensions to compute $PNE$s and adaptively refine inner approximations. Empirical results in international energy markets demonstrate that NASPs yield informative managerial insights, including counterintuitive effects of carbon-tax revenue strategies and robust reductions in emissions when markets trade energy; a Chilean-Argentinian case study highlights practical policy implications and the value of hierarchical regulatory modeling. Overall, the work provides a rigorous benchmark at the intersection of bilevel optimization, EPECs, and equilibrium computation with concrete applications in energy regulation and policy analysis.

Abstract

This article introduces a class of $Nash$ games among $Stackelberg$ players ($NASPs$), namely, a class of simultaneous non-cooperative games where the players solve sequential Stackelberg games. Specifically, each player solves a Stackelberg game where a leader optimizes a (parametrized) linear objective function subject to linear constraints while its followers solve convex quadratic problems subject to the standard optimistic assumption. Although we prove that deciding if a $NASP$ instance admits a Nash equilibrium is generally a $Σ^2_p$-hard decision problem, we devise two exact and computationally-efficient algorithms to compute and select Nash equilibria or certify that no equilibrium exists. We apply $NASPs$ to model the hierarchical interactions of international energy markets where climate-change aware regulators oversee the operations of profit-driven energy producers. By combining real-world data with our models, we find that Nash equilibria provide informative, and often counterintuitive, managerial insights for market regulators.

When Nash Meets Stackelberg

TL;DR

The paper introduces NASPs, a class of complete-information simultaneous games where each player solves a Stackelberg game with linear objective and linear constraints, while followers solve convex quadratic programs under optimistic responses. It proves oldsymbol{ extSigma^p_2}-hardness for existence of a Nash equilibrium and presents two exact algorithms—an enumeration over complementarity polyhedra and an inner-approximation scheme—to compute s or certify nonexistence, leveraging a convex reformulation in which a PNE corresponds to an of the original game. The methods exploit Balas’ extended formulations and LCP/NP-like solvers to achieve finite termination, with extensions to compute s and adaptively refine inner approximations. Empirical results in international energy markets demonstrate that NASPs yield informative managerial insights, including counterintuitive effects of carbon-tax revenue strategies and robust reductions in emissions when markets trade energy; a Chilean-Argentinian case study highlights practical policy implications and the value of hierarchical regulatory modeling. Overall, the work provides a rigorous benchmark at the intersection of bilevel optimization, EPECs, and equilibrium computation with concrete applications in energy regulation and policy analysis.

Abstract

This article introduces a class of games among players (), namely, a class of simultaneous non-cooperative games where the players solve sequential Stackelberg games. Specifically, each player solves a Stackelberg game where a leader optimizes a (parametrized) linear objective function subject to linear constraints while its followers solve convex quadratic problems subject to the standard optimistic assumption. Although we prove that deciding if a instance admits a Nash equilibrium is generally a -hard decision problem, we devise two exact and computationally-efficient algorithms to compute and select Nash equilibria or certify that no equilibrium exists. We apply to model the hierarchical interactions of international energy markets where climate-change aware regulators oversee the operations of profit-driven energy producers. By combining real-world data with our models, we find that Nash equilibria provide informative, and often counterintuitive, managerial insights for market regulators.

Paper Structure

This paper contains 39 sections, 7 theorems, 20 equations, 2 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Game.
  3. Applications.
  4. Literature Review
  5. Algorithms.
  6. Complexity and Equilibria.
  7. Definitions and Background
  8. Simultaneous Games
  9. Stackelberg Games
  10. NASPs
  11. Hardness of Finding a Nash Equilibrium
  12. The Enumeration Algorithm
  13. The Enumeration Algorithm.
  14. Proof of Statement (i.).
  15. Proof of Statement (ii.).
  16. ...and 24 more sections

Key Result

Theorem 1

It is $\Sigma^p_2$-hard to decide if a trivial NASP has a PNE.

Figures (2)

  • Figure 1: A schematic representation of a NASP with $n$ players with $2$ followers each. The vertical arrows represent sequential “Stackelberg” interactions, while the horizontal ones are simultaneous “Nash” interactions.
  • Figure 2: A representation of \ref{['Alg:FullEnumeration']} for two players (blue and red).

Theorems & Definitions (35)

  • Remark 1
  • Example 1
  • Definition 1: Simultaneous “Nash” Game
  • Definition 2: Nash equilibrium
  • Definition 3: Stackelberg game
  • Definition 4: NASP
  • Remark 2
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • ...and 25 more