Mixed extensions of generic finite games embedded into products of real projective spaces

TL;DR

This paper studies the properties of generic finite games and their mixed extensions by embedding the strategy space into a product of real projective spaces, proving smoothness and transversality of relevant hypersurfaces for most games.

Contribution

It introduces a novel embedding of mixed strategies into real projective spaces and proves generic properties like smoothness and transversality of hypersurfaces in this setting.

Findings

Generic games have smooth and transversal hypersurfaces in the embedded space.

Most games (outside a measure-zero set) exhibit these geometric properties.

The approach uses Sard's theorem and Khovanskii's argument.

Abstract

Finite games in normal form and their mixed extensions are a corner stone of noncooperative game theory. Often generic finite games and their mixed extensions are considered. But the properties which one expects in generic games and the existence of games with these properties are often treated only in passing. The paper considers strong properties and proves that generic games have these properties. The space of mixed strategy combinations is embedded in a natural way into a product of real projective spaces. All relevant hypersurfaces extend to this bigger space. The paper shows that for all games in the complement of a semialgebraic subset of codimension at least one all relevant hypersurfaces in the bigger space are smooth and maximally transversal. The proof uses the theorem of Sard and follows an argument of Khovanskii.