# Evolutionarily Stable Strategies

## Overview

This tutorial shows how to find stable states in symmetric games. It assumes that you have a basic understanding of symmetric games from starting the Conflict I tutorial. If you work through all the example problems in detail, this tutorial should take about 30 minutes.

## Introduction

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

We can determine whether a strategy is evolutionarily stable by a simple thought experiment. Imagine that the strategy in question is used by the whole population. All members play that strategy, so all get the payoff of that strategy played against itself. Now ask what would happen if a small number started using an alternate strategy. Playing against the majority strategy, would this minority do better or worse than the majority? If they do better, they would increase in number over time, and the original strategy is not evolutionarily stable. If they do worse, then they cannot invade, so the original strategy is an ESS.

That is, if the entire population plays the ESS strategy, a mutation that made some members play another strategy would be eliminated. Here’s a trivial example:

X Y Game 1 2 2 1 1

If all population members are X, all get a payoff of 2. If Y mutants appear, they get a payoff of 1 against all of the Xs they meet. Thus Y does worse than X, so Y cannot invade a population of Xs. If we instead start with a population of all Ys, everyone gets a payoff of 1. If an X mutant appears, it rapidly takes over, since it gets payoffs of 2 against the majority Ys, while the Ys only get payoffs of 0 against each other.

You can see this in the simulation below. Starting with a population of all Y (proportion of X = 0, proportion of Y = 100), change the proportion of X to 1. In about 200 generations, the population switches to all X. Now reverse the situation, setting the proportion of X to 100 and Y to 1. Can Y invade X?

A population of Xs cannot be invaded by Y, while a population of Ys can be invaded by X. Thus X is the only evolutionarily stable strategy in Game 1.

What about Game 2?

X Y Game 2 2 1 1 2

Test this in the simulator as well. Starting with a population of all Y (proportion of X = 0, proportion of Y = 100), change the proportion of X to 1. Does the proportion of X increase over time? Now reverse the situation, setting the proportion of X to 100 and Y to 1. Does the proportion of Y increase?

Given the definition of an ESS as a strategy that cannot be invaded by another strategy, what is the ESS in this game?