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On the Equilibrium of a Class of Leader-Follower Games with Decision-Dependent Chance Constraints

Jingxiang Wang, Zhaojian Wang, Bo Yang, Feng Liu, Xinping Guan

TL;DR

This paper addresses the existence of equilibrium in a single-leader-multiple-follower game where followers face decision-dependent chance constraints (DDCCs) driven by decision-dependent uncertainties ($DDUs$). It develops a moment-based, strategy-dependent ambiguity set $D_i(x)$ and uses Cantelli's inequality to reformulate DDCCs into a second-order cone form, enabling a tractable, distribution-robust analysis. The authors prove the existence of a Stackelberg–Nash equilibrium by constructing continuous, compact, convex set-valued maps and applying Kakutani's fixed-point theorem, complemented by a numerical example illustrating the impact of $DDUs$ and risk parameters on the equilibrium. The framework provides a rigorous, practically relevant approach to robust decision-making under DDUs in hierarchical games, with potential applications in energy systems, V2G control, and disaster management.

Abstract

In this paper, we study the existence of equilibrium in a single-leader-multiple-follower game with decision-dependent chance constraints (DDCCs), where decision-dependent uncertainties (DDUs) exist in the constraints of followers. DDUs refer to the uncertainties impacted by the leader's strategy, while the leader cannot capture their exact probability distributions. To address such problems, we first use decision-dependent ambiguity sets under moment information and Cantelli's inequality to transform DDCCs into second-order cone constraints. This simplifies the game model by eliminating the probability distributions. We further prove that there exists at least one equilibrium point for this game by applying Kakutani's fixed-point theorem. Finally, a numerical example is provided to show the impact of DDUs on the equilibrium of such game models.

On the Equilibrium of a Class of Leader-Follower Games with Decision-Dependent Chance Constraints

TL;DR

This paper addresses the existence of equilibrium in a single-leader-multiple-follower game where followers face decision-dependent chance constraints (DDCCs) driven by decision-dependent uncertainties (). It develops a moment-based, strategy-dependent ambiguity set and uses Cantelli's inequality to reformulate DDCCs into a second-order cone form, enabling a tractable, distribution-robust analysis. The authors prove the existence of a Stackelberg–Nash equilibrium by constructing continuous, compact, convex set-valued maps and applying Kakutani's fixed-point theorem, complemented by a numerical example illustrating the impact of and risk parameters on the equilibrium. The framework provides a rigorous, practically relevant approach to robust decision-making under DDUs in hierarchical games, with potential applications in energy systems, V2G control, and disaster management.

Abstract

In this paper, we study the existence of equilibrium in a single-leader-multiple-follower game with decision-dependent chance constraints (DDCCs), where decision-dependent uncertainties (DDUs) exist in the constraints of followers. DDUs refer to the uncertainties impacted by the leader's strategy, while the leader cannot capture their exact probability distributions. To address such problems, we first use decision-dependent ambiguity sets under moment information and Cantelli's inequality to transform DDCCs into second-order cone constraints. This simplifies the game model by eliminating the probability distributions. We further prove that there exists at least one equilibrium point for this game by applying Kakutani's fixed-point theorem. Finally, a numerical example is provided to show the impact of DDUs on the equilibrium of such game models.
Paper Structure (15 sections, 8 theorems, 47 equations, 2 figures, 1 table)

This paper contains 15 sections, 8 theorems, 47 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Preliminaries
  5. Game Model and DDCC Reformulation
  6. Game model
  7. Definition of the equilibrium
  8. Construction of the ambiguity set
  9. DDCC reformulation
  10. Existence of Equilibrium for Leader-Follower Games with DDCCs
  11. Construction of set-valued maps
  12. Existence of the game equilibrium
  13. Illustrative Examples
  14. Conclusion
  15. Proof of Theorem \ref{['theorem1']}

Key Result

Lemma 1

(Continuity theorem of marginal functions and marginal maps jaubin1984differential) For the marginal function $V(y)$ and the marginal map $M: Y \rightrightarrows X$, which are defined in Definition marginal, we have the following conclusion. (i ) Suppose that $f(x,y)$ is lower semicontinuous on $X \

Figures (2)

  • Figure 1: The framework of leader-follower game with DDCCs
  • Figure 2: Payoff of players with different $\rho$

Theorems & Definitions (17)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 1: DDCC
  • Lemma 4
  • proof
  • ...and 7 more