Game Theory: The Genius of Nash (Part 5)

Nash Equilibrium, Best Response

Introduction

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

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

v(s∗ᵢ , s∗₋ᵢ) ≥ v(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

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(U) = L, BR₂(M) = C, BR₂(D) = R
BR(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

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

  1. Each player is playing a best response to his beliefs.
  2. 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

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 iterative elimination method is not applicable.

Lets check if a pure-strategy Nash equilibrium does exist.
BR₁(L) = D, BR(C) = M, BR₁(R) = M
BR₂(U) = L, BR(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

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

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.

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Kowshik chilamkurthy

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RL | ML | ALGO TRADING | TRANSPORTATION | GAME THEORY

Nerd For Tech

NFT is an Educational Media House. Our mission is to bring the invaluable knowledge and experiences of experts from all over the world to the novice. To know more about us, visit https://www.nerdfortech.org/.