Introduction to Game Theory · Introduction to Game Theory Introduction Normal Form Games De–nition 2.1: A normal form game: 1 N players whose names are listed in the set I f1,2,

Introduction to Game Theory


March 7, 2018


Introduction

Basic Concepts in Noncooperative Game Theory

Actions (welfare or profits)

Help us to analyze industries with few firms

What are the firms’actions?

Two types of games:

1 Normal Form Game2 Extensive Form game

Two types of actions: Pure and Mixed

Information: Perfect and Imperfect


Introduction

Examples of Noncooperative Game Theory


Introduction

Normal Form Games

Definition 2.1: A normal form game:

1 N players whose names are listed in the set I ≡ {1, 2, ...,N}2 Each player i , i ∈ I , has an action set Ai , whereAi = {ai1, ai2, ..., aiki }

3 List of actions chosen by each player:a ≡ (a1, a2, ..., ai , ..., aN )

4 Each player has a payoff function πi ∈ R


Introduction

Normal Form Game

"Peace-War Game" (Prisoners’Dilemma)

Country 1

Country 2War Peace

War 1, 1 3, 0Peace 0, 3 2, 2

Let us apply Definition 2.1.........


Introduction

Equilibrium Concepts

We would like to obtain one outcome (unique eq.)

Outcome of the game: a ≡ (a1, a2, ..., ai , ..., aN )a−i ≡ (a1, ..., a−i , ai+1, ..., aN )

let’s talk about a−i

Hence, an outcome a can be expresses as a ≡ (ai , a−i )


Introduction

Equilibrium in dominant actions

Definition: A particular action ai ∈ Ai is said to be adominant action for player i if no matter what all otherplayers are playing ai always maximizes i’s payoff

πi (ai , a−i )

for every ai ∈ Ai .Example: War-Peace gameWhat is a Dominant Strategy for player 1?

Country 1

Country 2War Peace

War 1, 1 3, 0Peace 0, 3 2, 2

Note that an outcome is always composed by a DominantStrategy


Introduction

Payoff matrix (Normal Form Game)

Firm A

Firm BLow Prices High Prices

Low Prices 5, 5 9, 1High Prices 1, 9 7, 7

Low prices yield a higher payoff than high prices both when afirm’s rival chooses low prices and when it selects high prices

Low prices is strictly dominant strategy for both firmsHigh prices is referred to as a strictly dominated strategy


Introduction

A strictly dominated strategy can be deleted from the set ofstrategies a rational player would use.

This helps to reduce the number of strategies to consider asoptimal for each player.

In the above payoff matrix, both firms will select “low prices”in the unique equilibrium of the game.

However, games do not always have a strictly dominatedstrategy.


Introduction

Battle of the Sexes (coordination games)

Jacob

RachelOpera Football

Opera 2, 1 0, 0Football 0, 0 1, 2


Introduction


No Dominant Strategies!

Hence, there does not exist an equilibrium in dominant actions


Introduction

Nash Equilibrium (NE)

Definition: An outcome a = (a1, a2, ..., ai , ..., aN ) is said to bea NE if no player would find it beneficial to deviate providedthat all other players do not deviate from their strategiesplayed at the Nash outcome

πi (ai , a−i ) ≥ πi (ai , a−i )

for every ai ∈ Ai .An equilibrium in dominant action is a NE but a NE ; eq.D.A


Introduction

Nonexistence of a Nash Equilibrium

After 30 years of marriage................. ;)

Jacob


Opera 2¯, 0 0,2

¯Football 0,1¯

1¯, 0


Introduction

Best-Response functions to solve for NE

Definition: in a two-player game, the BRF of player i is thefunction R i (aj ), that for every given action aj of player jassigns an action ai = R i (aj ) that maximizes player i’s payoffπi (ai , aj )

Example: Battle of the sexes


Introduction


Jacob


Opera 2, 1 0, 0Football 0, 0 1, 2

Example Battle of the sexes

RJ (aR ) ={

Opera if aR = OperaFootball if aR = Football

RR (aJ ) ={

Opera if aJ = OperaFootball if aJ = Football

Then if a is a NE , then ai = R i (a−i ) for every player i .

Documents

Introduction to Game Theory · Introduction to Game Theory Introduction Normal Form Games De–nition 2.1: A normal form game: 1 N players whose names are listed in the set I f1,2,