# Game Theory: The Genius of Nash (Part 5)

## Nash Equilibrium, Best Response

# Introduction

We discussed strict dominance solution concept in great detail in the last blog. Its application is limited and only applicable to some section of games( Games with strict dominant strategy). Strict dominant strategy often fails to exist.

Let’s consider **Battle of sexes** game.

There is no dominant strategy.

In the last blog, we discussed the concept of belief. Player will behave optimally (best response ) to their beliefs. Chris may behave optimally and go to football given his belief that Alex is going to the football game. But their beliefs can be wrong.

In this blog, we will discuss one of the most central and best known solution concept in the game theory. This overcomes many shortcoming faced by other solution concepts, this is developed by **John Nash**.

# Nash Equilibrium Solution Concept

Let’s define Nash’s solution concept. Nash equilibrium is as a profile of strategies for which each player is choosing a best response to the strategies of all other players.

Each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium. Lets formally define Nash equilibrium:

**Definition**: The pure-strategy profile **s*= (s***₁**, s***₂**, . . . , s*n) ∈ S** is a Nash equilibrium if** s*ᵢ** is a best response to **s*₋ᵢ** , for all i ∈ N, that is,

**v**ᵢ**(s∗ᵢ , s∗₋ᵢ) ≥ v**ᵢ**(sᵢ, s∗₋ᵢ)** **for all sᵢ ∈ Sᵢ and all i ∈ N.**

Please note that s* is strategy profile, not strategy. strategy profile refers of set of actions taken by all the players in a strategic environment/game.

## Example 1

lets try to understand this definition by working out an example.

Consider this matrix representation. Now lets write down all possible strategy profiles. **S** = {(L,U), (C,U),(R,U),(L,M), (C,M),(R,M),(L,D), (C,D),(R,D)}.

Now lets evaluate payoff functions vis-a-vis best response.

if player 1 chooses U best response for player 2 is L: BR₂(U) = L**BR**₂**(U) = L**, BR₂(M) = C, BR₂(D) = R**BR**₁**(L) = U**, BR₁(C) = D, BR₁(R) = U

Now closely observe If player 2 chooses L, then player 1’s best response is {U}; at the same time, if player 1 chooses U, then player 2’s best response {L}. It clearly fits the definition above.

So this is the

s*: {L, U}Nash equilibrium.

## Example 2

let’s apply Nash’s solution concept to prisoners dilemma.

S = {(RS,BE), (BE,BE), (BE,RS), (RS, RS)}

Nash equilibrium s* is (BE,BE)

I encourage readers to solve this and find out how (BE,BE) is Nash Equilibrium.

## Assumptions of Nash Equilibrium

Here are the assumptions for a Nash equilibrium:

- Each player is playing a best response to his beliefs.
- The beliefs of the players about their opponents are correct.

We will not dig too deep into these assumptions as it can put us in mid of some philosophical discussion.

# Comparing with other Solution Concepts

## Example 1

Lets compare Nash solution concept with other solution concepts:

Here it easy to deduce that there is ** no strictly dominant strategy** for both players: thus strict dominance concept fails.

There is no strictly dominated strategy for any player, so

**is not applicable.**

*iterative elimination method*Lets check if a pure-strategy Nash equilibrium does exist.

BR₁(L) = D, **BR**₁**(C) = M,** BR₁(R) = M

BR₂(U) = L,** BR**₂**(M) = C**, BR₂(D) = L

we find that **(M, C)** is the** pure-strategy Nash equilibrium **— and it is unique.

Solution concept is finest if it predicts or prescribes an unique strategy. It is necessary to understand if Nash equilibrium always yields unique strategy.

## Example 2

Lets consider the battle of sexes game.

Let’s solve Nash equilibrium for this game.

S = {(O, F), (O, O), (F, F), (F,O )}

BRa(O) = O, BRa(F) = F

BRc(O) = O, BRa(F) = F

We can clearly observe that we may not have a unique Nash equilibrium, but it usually lead to more refined predictions than those of strict dominant solution concept and iterative elimination.

Nash equilibrium solution concept has been applied widely in economics, political science, legal studies, and even biology.

# More Nash Equilibrium Examples

Let’s discuss an example where we can apply Nash’s solution concept to real life problem.

## Example 1: Stag Hunt (Social Cooperation)

Two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, they must have the cooperation of their partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag. This has been taken to be a useful analogy for social cooperation, such as international agreements on climate change. The payoff matrix is as follows

BR₁(S) = S, BR₁(H) = H

BR₂(H) = H, BR₂(S) = S

Game has two pure-strategy equilibria: (S, S) and (H, H). However, the payoff from (S, S) **Pareto dominates** that from (H, H).

If a player anticipates that the other individual is not cooperative, then he would choose to hunt a hare. But if he believes that other individual will cooperate then we would choose stag. When both individuals choose stag i.e when both believe other individual will cooperate, as a whole both of them would be better off

Let’s close the discussion on Nash Equilibrium for discrete actions here. We will discuss more about Nash Equilibrium in the next blog in my blog series.