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Workshop

A vector bundle approach to Nash equilibria

  • Luca Sodomaco (KTH Royal Institute of Technology, Sweden)
Felix-Klein-Hörsaal Universität Leipzig (Leipzig)

Abstract

Using vector bundle techniques, we study the algebraic scheme of totally mixed Nash equilibria of an n-person game. This approach reveals intriguing similarities and differences among totally mixed Nash equilibria in Game Theory, eigenvectors, and singular vector tuples in Spectral Theory of higher-order tensors.

First, a generic n-person game of format d=(d_1,...,d_n) does not admit totally mixed Nash equilibria if and only if the corresponding Segre variety of rank-one tensors of format d is dual defective.

Consequently, we define the Nash resultant variety of games with dual defective format whose scheme of totally mixed Nash equilibria is nonempty. In particular, we show its irreducibility and compute its codimension and degree.

Secondly, we study the algebraic variety of n-person games of format d whose scheme of totally mixed Nash equilibria is either non-reduced or has a positive dimensional component. We call it the Nash discriminant. In this talk, we describe entirely the Nash discriminant of a 3-person game of format (2,2,2), and we determine an irreducible decomposition of its singular locus. Finally, with the help of symbolic computations in Macaulay2, we show that the Nash discriminant of a 3-person game of format (2,2,3) is not irreducible.

conference
29.07.24 02.08.24

MEGA 2024

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Felix-Klein-Hörsaal

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Christian Lehn

Ruhr-Universität Bochum

Irem Portakal

Max Planck Institute for Mathematics in the Sciences

Rainer Sinn

Universität Leipzig

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Simon Telen

Max Planck Institute for Mathematics in the Sciences

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