Some Properties of the Nash Equilibrium in $2 \times 2$ Zero-Sum Games

TL;DR

This paper reviews properties of Nash equilibria in 2x2 zero-sum games, providing formulas and characterizations based on payoff matrix entries, offering a synthetic and original presentation of known results.

Contribution

It offers a concise, original synthesis of known properties of Nash equilibria in 2x2 zero-sum games with explicit formulas and characterizations.

Findings

Cardinality of NE set based on payoff matrix entries

Closed-form expressions for NE strategies and game value

Original synthetic presentation of known results

Abstract

In this report, some properties of the set of Nash equilibria (NEs) of $2 \times 2$ zero-sum games are reviewed. In particular, the cardinality of the set of NEs is given in terms of the entries of the payoff matrix. Moreover, closed-form expressions for the NE strategies and the payoff at the NE (the value of the game) are provided in terms of the entries of the payoff matrix. The results presented in this report are not necessarily new knowledge, as they follow from the definition of the NE after some tedious calculations. Nevertheless this synthetic presentation is original in the literature.