A vector bundle approach to Nash equilibria
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915731942146048 |
|---|---|
| author | Abo, Hirotachi Portakal, Irem Sodomaco, Luca |
| author_facet | Abo, Hirotachi Portakal, Irem Sodomaco, Luca |
| contents | We use vector bundles to study the locus of totally mixed Nash equilibria of an $n$-player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A vector bundle approach to Nash equilibria Abo, Hirotachi Portakal, Irem Sodomaco, Luca Computer Science and Game Theory Algebraic Geometry 14A10, 14C17, 14F06, 14P05, 91A06, 91A12, 91A80 We use vector bundles to study the locus of totally mixed Nash equilibria of an $n$-player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree. |
| title | A vector bundle approach to Nash equilibria |
| topic | Computer Science and Game Theory Algebraic Geometry 14A10, 14C17, 14F06, 14P05, 91A06, 91A12, 91A80 |
| url | https://arxiv.org/abs/2504.03456 |