# Nash Equilibria

## Overview

This tutorial shows how to find stable equilibria in asymmetric games. It assumes that you have already completed the Stable Strategies tutorial for symmetric games and have a basic understanding of asymmetric games, from starting either the Conflict II or Parental Care tutorial. If you work through all the example problems in detail, this tutorial should take about 30 minutes.

## Introduction

In a symmetric game, you can calculate the evolutionarily stable strategy (ESS). In an asymmetric game, since there are two roles with different strategy sets, stability consists of a pair of strategies, one for each role. A stable state in an asymmetric game is called a Nash equilibrium, and its calculation is a bit different from that for an ESS. Mathematically, the ESS is actually a special case of the Nash equlibrium. Since the two concepts are related, let’s start with a recap of the ESS.

An evolutionarily stable strategy is a strategy that cannot be invaded by another strategy.

That is, if the entire population plays the ESS strategy, a mutation that made some members play another strategy would be eliminated. The idea behind the Nash equilibrium is the similar, although it is stated in terms of players switching strategies, rather than invasion of a population by a rare mutant1. For stability in evolutionary games, we want to find strict Nash equilibria, defined as follows:

A pair of strategies is a strict Nash equilibrium if neither player can unilaterally switch to another strategy without reducing its payoff.

Here is a simple example (remember, the payoff to the left of the comma is for the strategy to the left against the strategy above; the payoff to the right of the comma is for the strategy above against the strategy to the left):

Male
ABC
Female X2 , 31 , 21 , 1
Y1 , 12 , 13 , 2
Z1 , 22 , 22 , 1

Male A Female X (the upper-left cell) is a strict Nash equilibrium. Looking at the male’s payoffs (to the right of the comma) in the X row, we see that his payoff would drop from 3 to 2 or 1 if he switched from A to B or C. Looking at the female’s payoffs (to the left of the comma) in the A column, we see that her payoff would drop from 2 to 1 if she switched from X to Y or Z. Neither player can unilaterally switch to another strategy without reducing its payoff.

To find all the Nash equilibria in a game, you must test each pair of strategies in this way. Are there any other strict Nash equilibria in the above game?

Male C Female Y is also a strict Nash equilibrium. If the male switched from C to A or B, his payoff would drop from 2 to 1, while if the female switched from Y to X or Z, her payoff would drop from 3 to 2 or 1.It may also look like Male B Female Z is another strict Nash equilibrium, but the male could switch from B to A or the female could switch from Z to Y without suffering reduced payoff.

1. Game theory was developed by mathematicians interested in economics, and some of its terminology reflects that history. The Nash equilibrium was described by John Nash, known in popular culture as the subject of the Hollywood movie A Beautiful Mind. In 1990, Nash won a Nobel prize in economics for this work, done when he was a 21-year old graduate student [link to paper]. These ideas were further developed in the context of evolution by John Maynard Smith and others [link to paper].
2. s−t/2 > s/2 becomes s/2−t/2 > 0, or s > t. It is easier to visualize the effects of changing a variable when the inequalities are simplfied like this. Inequalities are algebraically simplified in the same manner as equalities, with one exception. If you multiply both sides of an inequality by −1, the direction of the inequality changes (e.g. −s > r−t becomes s < t−r).
3. This is known as Selten’s Theorem. [link to Selten, R. (1980). A note on evolutionarily stable strategies in asymmetric animal contests. J. Theoret. Biol. 84, 93–101.]
4. A (non-strict) Nash equilibrium is defined as a pair of strategies in which neither player could improve its payoff by switching to another strategy. The distinction between this and a strict equilibrium is subtle. A strict equilibrium demands strict inequality (>), while a non-strict equilibrium allows equality (≥), between payoffs to different strategies.