Topology of the Generalized Nash Equilibrium Problem

TL;DR

This paper extends the existence results of Nash equilibria in abstract economies by removing convexity assumptions, using algebraic topology tools, and provides new examples fitting these conditions.

Contribution

It generalizes Nash equilibrium existence theorems to non-convex abstract economies using topological methods, broadening applicability.

Findings

Generalized Nash equilibrium existence without convexity assumptions

Application of algebraic topology to equilibrium problems

New examples of abstract economies satisfying the main theorem

Abstract

The Generalized Nash Equilibrium Problem refers to the question of the existence of a Nash equilibrium in an abstract economy. This model is due to Kenneth J. Arrow and Gerard Debreu in their pioneering work from 1954. An abstract economy is an extension of John Nash's original concept of a non-cooperative game from the 1950's. Here players selfishly seek to maximize their profits, which may depend on the others' choices. The novelty of an abstract economy is that the players may now mutually constrain each-other in their decision-making. A Nash equilibrium is reached when no player alone can increase his profit by a unilateral change of strategy. Abstract economies have found widespread applications from welfare economy, economic analysis and policy-making to constrained optimization, partial differential equations and optimal allocation. We generalize Leigh Tesfatsion's Nash equilibrium existence result to abstract economies without resorting to commonly employed convexity assumptions, thereby also re-proving all known Nash equilibrium existence results as special cases. A key element of our proof is the translation of the Generalized Nash Equilibrium Problem into the question of the existence of a coincidence of two particularly defined maps, and the application of a coincidence result from algebraic topology due to Samuel Eilenberg and Deane Montgomery in 1946.Additionally, we provide examples of abstract economies which satisfy the assumptions of our main result, but due to lacking convexity assumptions, do not satisfy the assumptions of other classical results.