[Paper Review] Global Equilibria of Multi-leader Multi-follower Games

Ankur A. Kulkarni, Uday V. Shanbhag|arXiv (Cornell University)|Jun 13, 2012

Economic theories and models32 references4 citations

TL;DR

This paper proposes a modified multi-leader multi-follower game formulation with shared constraints, showing that global minimizers of a potential function over these constraints are equilibria of the modified game. It establishes existence via fixed point theory and links local optimality to Nash equilibria, enabling recovery of original game equilibria through multiplier sets.

ABSTRACT

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria for it has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over the shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.

Motivation & Objective

• To address the challenge of nonconvexity and lack of tractable existence conditions in multi-leader multi-follower games with hierarchical equilibrium constraints.
• To develop a modified game formulation where leaders are cognizant of all equilibrium constraints, ensuring equilibria of the original game are contained within the new solution set.
• To establish sufficient conditions for existence of global equilibria using potential functions and fixed point theory.
• To connect local optimality in the reformulated problem to local Nash equilibria, B-stationarity, and strong stationarity in the original game.
• To demonstrate how equilibria of the original game can be recovered from the modified shared-constraint formulation via multiplier sets.

Proposed method

• Introduce a modified multi-leader multi-follower game where all leaders are aware of the shared equilibrium constraints of all other leaders, forming a shared-constraint structure.
• Show that if leader objectives admit a potential function, global minimizers of this function over the shared constraints are equilibria of the modified game.
• Establish existence of equilibria using fixed point theory without requiring the potential function condition.
• Prove that local minima, B-stationary, and strong-stationary points of the potential function minimization correspond to local Nash equilibria, Nash B-stationary, and Nash strong-stationary points in the original game.
• Analyze the relationship between variational equilibria of the shared-constraint game and equilibria of the original game through the lens of multiplier sets.
• Demonstrate recovery of original game equilibria by analyzing the structure of the multiplier sets associated with the shared constraints.

Experimental results

Research questions

• RQ1 Under what conditions can global equilibria be guaranteed in multi-leader multi-follower games with nonconvex agent problems?
• RQ2 How does a shared-constraint formulation improve the tractability and solution structure of hierarchical games with equilibrium constraints?
• RQ3 What is the relationship between the variational equilibria of the modified shared-constraint game and the equilibria of the original multi-leader multi-follower game?
• RQ4 In what ways do local optimality conditions in the reformulated problem correspond to meaningful equilibrium concepts in the original game?
• RQ5 Can equilibria of the original game be recovered from the modified formulation using multiplier set analysis?

Key findings

• Global minimizers of the potential function over the shared constraints are equilibria of the modified multi-leader multi-follower game.
• Existence of equilibria is established via fixed point theory without requiring the potential function condition.
• Local minima of the potential function minimization are local Nash equilibria of the original game.
• B-stationary and strong-stationary points of the potential function minimization correspond to Nash B-stationary and strong-stationary points in the original game.
• Variational equilibria of the shared-constraint game are linked to equilibria of the original game through the structure of the multiplier sets.
• Equilibria of the original game can be recovered from the modified formulation by analyzing the associated multiplier sets.