# Toward Solving 2-TBSG Efficiently

## TL;DR

This paper introduces strongly polynomial algorithms for solving 2-TBSG, a two-player game model, by developing novel simplex strategy iteration methods inspired by Markov decision processes, and transforms general instances into simplified forms.

## Contribution

It presents the first strongly polynomial algorithms for 2-TBSG and introduces a transformation method to reduce general cases to simpler two-action scenarios.

## Key findings

- Algorithms exhibit geometric convergence.
- Proved strongly polynomial complexity.
- Transformation reduces problem complexity.

## Abstract

2-TBSG is a two-player game model which aims to find Nash equilibriums and is widely utilized in reinforced learning and AI. Inspired by the fact that the simplex method for solving the deterministic discounted Markov decision processes (MDPs) is strongly polynomial independent of the discounted factor, we are trying to answer an open problem whether there is a similar algorithm for 2-TBSG. We develop a simplex strategy iteration where one player updates its strategy with a simplex step while the other player finds an optimal counterstrategy in turn, and a modified simplex strategy iteration. Both of them belong to a class of geometrically converging algorithms. We establish the strongly polynomial property of these algorithms by considering a strategy combined from the current strategy and the equilibrium strategy. Moreover, we present a method to transform general 2-TBSGs into special 2-TBSGs where each state has exactly two actions.