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# Multilinear Formulations for Computing a Nash Equilibrium of Multi-Player Games

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## Acknowledgements

This research was initiated as part of the MSc dissertation of the first author in the School of Mathematics at the University of Edinburgh.

## Cite As

Miriam Fischer and Akshay Gupte. Multilinear Formulations for Computing a Nash Equilibrium of Multi-Player Games. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SEA.2023.12

## Abstract

We present multilinear and mixed-integer multilinear programs to find a Nash equilibrium in multi-player noncooperative games. We compare the formulations to common algorithms in Gambit, and conclude that a multilinear feasibility program finds a Nash equilibrium faster than any of the methods we compare it to, including the quantal response equilibrium method, which is recommended for large games. Hence, the multilinear feasibility program is an alternative method to find a Nash equilibrium in multi-player games, and outperforms many common algorithms. The mixed-integer formulations are generalisations of known mixed-integer programs for two-player games, however unlike two-player games, these mixed-integer programs do not give better performance than existing algorithms.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Exact and approximate computation of equilibria
• Theory of computation → Nonconvex optimization
##### Keywords
• Noncooperative n-person games
• Nash equilibrium
• Multilinear functions
• Nonconvex problems
• Mixed-integer optimization

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