Game theory

Basic concepts

• The payoff matrix

• Nash equilibrium

• Dominant strategies

• Maximin strategies

• Mixed strategies

Game Theory And Oligopoly

• Non cooperative games the prisoner dilemma

• Cooperative games dealing with cheaters

• Sequential games the ad of being first

Introduction

Introduced by

• In 1950s , By John von Neumann and Oskar Morgenstern

• Application –

1. political, courtship, economic issues

2. It could be used to analyze the bargaining

• process between two parties : Wage rate

• negotiation: unions and firms Peace talks:

• between 2 countries.

Assumptions

• Assumptions Finite sets of possible action

• Awareness of availability competitors strategies too Intelligent and rational

• Maximize gain and minimize loss If a’s gain is b’s loss,its 0-sum game (amt of gain=amt of loss) Players act; select their stategies simultaneously

Types of games

Cooperative

when players can negotiate a binding

contract to play joint strategies.

Non-cooperative when game is not

cooperative it is said to be non-cooperative

game.

The Payoff Matrix

It is a course of action taken by 1 of the participants in a game

Pure strategies Selects the same strategy

Mixed strategies-don’t selects the same stategy Payoff: It is the result or outcome of the strategy

Example: 2 children engaged in coin-flipping 2 competing firms whose objective is to increse their profits by price changes

Consider the competition between two department stores, each of which must what kind of clothing to promote

Payoff matrix

0,0 4,2

2,2 2,4

Store 1

Promote girl’s cloth

Promote women’s cloth

Promote girl’s cloth

Promote women’s cloth

Store 2

Profit in millions

A Nash Equilibrium is defined as a set of strategies such that none of the participants in the game can improve their payoff (profits), given the strategies of the other participants.

NASH EQUILIBRIUM

Firm

Firm 2

Firm 1

Strategy

No Price Change

Price Change

No Price Change

10,10

100,-30

Price Change

-20,30

140,25

Dominant Strategy(in millions)

Dominant Strategies : Dominant Strategies The dominant strategy is the optimal choice for a player no matter what the opponent does. One firm will be in dominant position in terms of change in strategy.

Firm

Firm 2

Firm 1

Strategy

No New Product

New Product

No New Product

4,6

3,6

New Product

6,3

2,2

Max min Strategy

New Product

No New Product

No New Product

Firm 1

New Product

Strategy

Firm 2

Nas

h

Equ

ilib

riu

m

3

2 6

4 6

2

6

,

, ,

, 3

Maxmin Strategy

New Product

No New Product

No New Product

Firm 1

New Product

Strategy

Firm 2

3

2 6

4 6

2

6

,

, ,

, 3

Maxmin Strategy

Minimum

Minimum

3

2

2 6

Maximum

Maximum

6

6

6 3

Profit in millions

Nash Equilibrium & Maximin Point Isn't Same

Why So?

• Decision Criterion is not Profit-Maximisation

• Its for avoiding highly unfavourable outcome

• Its for avoidance of risks

•

Just Remember that its 2 Step Process

1.Find Minimum(Least) Profit

2.Select maximum Out of Minimum Profit

Mixed Strategy Why We Should Study This?

Game Table

• The Game of Tennis Striker chooses to serve either left or right Receiver defends either left or right

• Better chance to get a good return if you defend in the area the striker is serving to

1. For Striker:

• Best response to defend left is to Strike right

• Best response to defend right is to Strike left

2. For receiver: Just the opposite

Receiver - 70-30 Striker - 60-40

Expects Throws Receiver Striker Probability

Percent of Payoff Matrix

Chances of Success

Left Left 0.70 0.60 0.42 75% 0.315

Left Right 0.30 0.40 0.12 25% 0.030

Right Left 0.70 0.60 0.42 25% 0.105

Right Right 0.30 0.40 0.12 75% 0.090

Total Success 0.540

Receiver – 50-50 Striker - 70-30

Expects Throws Receiver Striker

Probablity

Percent of Payoff Matrix

Chances of Success

Left Left 0.50 0.70 0.35 75% 0.263

Left Right 0.50 0.30 0.15 25% 0.038

Right Left 0.50 0.70 0.35 25% 0.088

Right Right 0.50 0.30 0.15 75% 0.113

Total Success 0.500

Receiver – 50-50 Striker - 40-60

Expects Throws Receiver Striker Probablity

Percent of Payoff Matrix

Chances of Success

Left Left 0.50 0.60 0.3 75% 0.225

Left Right 0.50 0.40 0.2 25% 0.050

Right Left 0.50 0.60 0.3 25% 0.075

Right Right 0.50 0.40 0.2 75% 0.150

Total Success 0.500

Mixed Strategy Equilibrium

•A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses.

• In the tennis example, this occurred when any player chose a 50-50 mixture of left and right.

Receiver’s Best Response

Suppose p is the probability of Strikers Serving towards left Clearly

•If p = 1, then the receiver should defend to the left

•If p = 0, the receiver should defend to the right.

Left

1/2

Right

P

Best Response

Suppose that the receiver goes left with probability q. Clearly,

•if q = 1, the server should serve right

•If q = 0, the server should serve left Server’s

Left

1/2

Right

q

Putting Things Together

p

1/2

q

1/2

R’s best response

S’s best response

Mutually best response

Prisoner

Prisoner 2

Prisoner 1

Strategy

Don’t Confess

Confess

Don’t Confess

0,0

15,5

Confess

5,15

5,5

Noncooperative Games A game is considered non cooperative if it not possible to negotiate with other participants and enter into some form of binding agreement.

Example : Prisoner's Dilemma

Prisoner

Prisoner 2

Prisoner 1

Strategy

Confess

Remain Silent

Confess

5,5

0,20

Remain Silent

20,0

1,1

cooperative Games A game is considered cooperative if it possible to negotiate with other participants and enter into some form of binding agreement.

Firm

Firm 2

Firm 1

Strategy

No New

Product

New Product

No New

Product

30,30

10,40

40,10

20.20

New Product

Profit in millions

Repeated Games

•A repeated game is a game that the same players play more than once In repeated games • the sequential nature of the relationship allows for the adoption of strategies that are contingent on the actions chosen in previous plays of the game

Firm

Firm 2

Firm 1

Strategy

Low-level

Advertising

High-level

Advertising

Low-level

Advertising

30,30

10,40

High-level

Advertising

40,10

20.20

Profit in millions

• Any 1 Firm breaks the agreement

• Adopts High-Level Advertising

• Temporary Loss to other firm due to cheating

• In next period, Other firm will do the same (Tit-For-Tat)

• If One Firm Cuts price-Other firm will cuts price in next period.

• If One firm Raise Price-Other firm will do so in next period.

• Tit-For-Tat is Win-Win Situation

Advantages

•Easy to understand •Never initiates cheating •Never rewards cheating cause it punish in some way •Its about forgiving because cooperation is quickly restored

Sequential Games •One Player acts First & Then other responds. •Games where players choose actions in a particular sequence are sequential move games. Examples: Chess, Bargaining/Negotiations. •Must look ahead in order to know what action to choose now

Firm

Firm 2

Firm 1

Strategy

Low-level

Advertising

High-level

Advertising

Low-level

Advertising

2,2

-5,10

High-level

Advertising

10,-5

-7,7

Benifits to the one Who acts first

Profit in millions

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