# 2-Player Nash and Nonsymmetric Bargaining Games: Algorithms and Structural Properties

## TL;DR

This paper demonstrates that solutions to 2-player Nash and nonsymmetric bargaining games with linear constraints can be efficiently computed in polynomial time, especially when the game is succinct, using linear programming methods.

## Contribution

It establishes polynomial-time algorithms for solving certain 2-player bargaining games with linear constraints, including cases with small coefficients and non-succinct linear games.

## Key findings

- Solutions are rational and computable in polynomial time for games with linear constraints.
- Succinct games with small coefficients admit strongly polynomial algorithms.
- A non-succinct linear game also has a strongly polynomial solution.

## Abstract

The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2-player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the coefficients in its convex program are ``small'', then its solution can be found in strongly polynomial time. We also give a non-succinct linear game whose solution can be found in strongly polynomial time.

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Source: https://tomesphere.com/paper/0908.1181