Semi-algebraic sets and equilibria of binary games

TL;DR

This paper demonstrates that any nonempty, compact, semi-algebraic set within the unit cube can be represented as the projection of the equilibrium set of a finite binary-action game, with constructive proofs.

Contribution

It provides a constructive and elementary method to represent semi-algebraic sets as equilibrium projections of finite binary games, extending understanding of game equilibrium structures.

Findings

Any semi-algebraic set in [0,1]^n is a projection of a finite binary game equilibrium set.

Results apply to sets of equilibrium payoffs as well.

Proofs are constructive and elementary.

Abstract

Any nonempty, compact, semi-algebraic set in [0, 1] n is the projection of the set of mixed equilibria of a finite game with 2 actions per player on its first n coordinates. A similar result follows for sets of equilibrium payoffs. The proofs are constructive and elementary.