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Feedback Stackelberg-Nash equilibria in difference games with quasi-hierarchical interactions and inequality constraints

Partha Sarathi Mohapatra, Puduru Viswanadha Reddy, Georges Zaccour

TL;DR

The paper tackles two-player finite-horizon difference games with coupled inequality constraints under a quasi-hierarchical structure where each player acts through a sequential variable and a simultaneous variable. It defines the feedback Stackelberg-Nash (FSN) equilibrium and provides a recursive dynamic-programming framework to compute FSN under cost separability, linking FSN to a parametric feedback Stackelberg solution of an unconstrained game. In the linear-quadratic setting, the FSN solution is shown to reduce to a large-scale linear complementarity problem, enabling efficient computation as demonstrated on a dynamic duopoly example. The results offer a rigorous, implementable approach for hierarchical interaction problems with constraints, with potential applications in supply chains, competition, and multi-agent control, and point to MPCC challenges for broader cross-term or mixed-constraint extensions.

Abstract

In this paper, we study a class of two-player deterministic finite-horizon difference games with coupled inequality constraints, where each player has two types of decision variables: one involving sequential interactions and the other simultaneous interactions. We refer to this class of games as quasi-hierarchical dynamic games and define a solution concept called the feedback Stackelberg-Nash (FSN) equilibrium. Under separability assumption on cost functions, we provide a recursive formulation of the FSN solution using dynamic programming. We show that the FSN solution can be derived from the parametric feedback Stackelberg solution of an associated unconstrained game involving only sequential interactions, with a specific choice of the parameters that satisfy certain implicit complementarity conditions. For the linear-quadratic case, we show that an FSN solution is obtained by reformulating these complementarity conditions as a single large-scale linear complementarity problem. Finally, we illustrate our results using a dynamic duopoly game with production constraints.

Feedback Stackelberg-Nash equilibria in difference games with quasi-hierarchical interactions and inequality constraints

TL;DR

The paper tackles two-player finite-horizon difference games with coupled inequality constraints under a quasi-hierarchical structure where each player acts through a sequential variable and a simultaneous variable. It defines the feedback Stackelberg-Nash (FSN) equilibrium and provides a recursive dynamic-programming framework to compute FSN under cost separability, linking FSN to a parametric feedback Stackelberg solution of an unconstrained game. In the linear-quadratic setting, the FSN solution is shown to reduce to a large-scale linear complementarity problem, enabling efficient computation as demonstrated on a dynamic duopoly example. The results offer a rigorous, implementable approach for hierarchical interaction problems with constraints, with potential applications in supply chains, competition, and multi-agent control, and point to MPCC challenges for broader cross-term or mixed-constraint extensions.

Abstract

In this paper, we study a class of two-player deterministic finite-horizon difference games with coupled inequality constraints, where each player has two types of decision variables: one involving sequential interactions and the other simultaneous interactions. We refer to this class of games as quasi-hierarchical dynamic games and define a solution concept called the feedback Stackelberg-Nash (FSN) equilibrium. Under separability assumption on cost functions, we provide a recursive formulation of the FSN solution using dynamic programming. We show that the FSN solution can be derived from the parametric feedback Stackelberg solution of an associated unconstrained game involving only sequential interactions, with a specific choice of the parameters that satisfy certain implicit complementarity conditions. For the linear-quadratic case, we show that an FSN solution is obtained by reformulating these complementarity conditions as a single large-scale linear complementarity problem. Finally, we illustrate our results using a dynamic duopoly game with production constraints.

Paper Structure

This paper contains 16 sections, 7 theorems, 77 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related literature
  3. Contributions
  4. Preliminaries
  5. Dynamic game with inequality constraints
  6. Quasi-hierarchical interaction and information structure
  7. Feedback Stackelberg-Nash solution
  8. Recursive formulation of feedback Stackelberg-Nash solution
  9. Simultaneous interaction
  10. Sequential interaction
  11. Sufficient conditions for FSN solution
  12. Parametric feedback Stackelberg solution and its relation to FSN solution
  13. Linear-quadratic case
  14. FSN solution as a linear complementarity problem
  15. Numerical Illustration
  16. ...and 1 more sections

Key Result

Lemma IV.1

Consider the CNZDG described by eq:state-eq:objective. Let Assumptions ass:A1, ass:seperability and ass:convex hold. Then, for every $(k,x_{k})\in \mathsf{K}\times \mathsf{X}_{k}$, $\texttt{SOL}(\mathrm{pCP}_{k}(x_{k}))$ is unique (a singleton), and the FSN control actions $\mathsf{v}_{k}^{\star }$

Figures (3)

  • Figure 1: Three step decision process involving sequential and simultaneous interactions between the time instants $k$ and $k+1$.
  • Figure 2: Illustration of set $\mathsf{Q}$ defined in \ref{['eq:Qset']} as the intersection of the continuous map \ref{['eq:Xvectoreq']} (gray dashed curve) and piecewise single valued map \ref{['eq:pCompliment']} (gray normal curve).
  • Figure 3: Evolution of production capacities and production quantities (panel (a)), stock of knowledge (panel (b)), production capacities (panel (c)), quantity produced (panel (d)), investments in R$\&$D (panel (e)) and investments in production capacity (panel (f)).

Theorems & Definitions (33)

  • Remark II.1
  • Definition II.1: Quasi-hierarchical interaction
  • Definition II.2: Constrained feedback information structure
  • Definition III.1: Feedback Stackelberg-Nash solution
  • Remark III.1
  • Remark III.2
  • Remark III.3
  • Example IV.1
  • Remark IV.1
  • Lemma IV.1
  • ...and 23 more