# Game Theory

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## Presentation Transcript

Game Theory

### Game Theory :

Game Theory Basic concepts The payoff matrix Nash equilibrium Dominant strategies Dominated strategies Maximin strategies Mixed strategies Game theory and oligopoly Noncooperative games: the prisoner’s dilemma Cooperative games: enforcing a cartel Repeated games: dealing with cheaters Sequential games: the advantage of being first

### Introduction :

Introduction In 1950s – By John von Neumann and Oskar Morgenstern Application - political, courtship, economic isses It could be used to analyze the bargaining process between 2 parties : Wage rate negotiation: unions and firms Peace talks: beteen 2 countris Courship: men and women

### 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

### 2 types of games :

2 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 :

The Payoff Matrix Strategy: It is a course of action taken by 1 of the participants in a game 2 types: 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

### Payoff matrix: advertising game :

Payoff matrix: advertising game

### NASH EQUILIBRIUM :

NASH EQUILIBRIUM 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.

### i.e.- each firm has the knowledge about other firm’s strategy.- their actions are dependant on eachother. (like in oligopoly)for e.g. :

i.e.- each firm has the knowledge about other firm’s strategy.- their actions are dependant on eachother. (like in oligopoly)for e.g.

### Payoff Matrix (crores) :

Payoff Matrix (crores)

### Here, no price changes is an equilibrium, because neither firm can benefit by increasing its price if other firm does not. :

Here, no price changes is an equilibrium, because neither firm can benefit by increasing its price if other firm does not.

### 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 srategy.

### Dominant Strategy (crores) :

Dominant Strategy (crores)

### Slide 14:

Dominated Strategy

### Slide 15:

Maximin Strategies

### Slide 16:

Firm 2 Firm 1 No New Product New Product No New Product New Product 4 , 3 , 6 , 2 , 3 2 6 6 4 , 6 , Nash Equilibrium

### Slide 17:

Firm 2 Firm 1 No New Product New Product No New Product New Product 4 , 3 , 6 , 2 , 3 2 6 6 4 , 6 , Minimum 3 3 2 2 Minimum Minimum Minimum 3 2 2 3 2 3 2 3 2

### Slide 18:

Firm 2 Firm 1 No New Product New Product No New Product New Product 4 , 3 , 6 , 2 , 3 2 6 6 4 , 6 , Minimum 3 3 2 2 Minimum Minimum Minimum 3 2 2 3 2 3 2 3 2 Find Out The Maximum 3 3 No New Product New Product No New Product No New Product

### Slide 19:

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

### Slide 20:

Just Remember that its 2 Step Process 1.Find Minimum(Least) Profit 2.Select maximum Out of Minimum Profit

Mixed Strategy

### Slide 22:

Why We Should Study This?

Tennis Match S R

Striking S R

Serving S R

### Slide 26:

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

Game Table

### Slide 28:

Game Table For Striker: Best response to defend left is to Strike right Best response to defend right is to Strike left For receiver: Just the opposite

### Slide 29:

Receiver - 70-30 Striker - 60-40

### Slide 30:

Expects Throws Receiver Striker Probablity Left Left .70 * .60 = .42 Left Right .30 * .40 = .12 Right Left .70 * .60 = .42 Right Right .30 * .40 = .12

### Slide 31:

Probablity % OF Payoff Matrix Chance Of Success .42 75% .315 .12 25% .03 .42 25% .105 .12 75% .09 .6654 Chance Of Success = 66.54%

### Slide 32:

Receiver - 50-50 Striker - 70-30

### Slide 33:

Expects Throws Receiver Striker Probablity Left Left .50 * .70 = .35 Left Right .50 * .30 = .15 Right Left .50 * .70 = .35 Right Right .50 * .30 = .15

### Slide 34:

Probablity % OF Payoff Matrix Chance Of Success .35 75% .2625 .15 25% .0375 .35 25% .0875 .15 75% .01125 .50 Chance Of Success = 50%

### Slide 35:

Receiver - 50-50 Striker - 60-40

### Slide 36:

Expects Throws Receiver Striker Probablity Left Left .50 * .60 = .30 Left Right .50 * .40 = .20 Right Left .50 * .60 = .30 Right Right .50 * .40 = .20

### Slide 37:

Probablity % OF Payoff Matrix Chance Of Success .30 75% .225 .20 25% .05 .30 25% .075 .20 75% .15 .50 Chance Of Success = 50%

### Slide 38:

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.

### Slide 39:

Suppose p is the probablity 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. Receiver’s Best Response

### Slide 40:

Receiver’s Best Response p Left Right ½

### Slide 41:

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 Best Response

### Slide 42:

Server’s Best Response Left Right ½ q

### Slide 43:

Putting Things Together ½ q p S’s best response 1/2 R’s best response

### Slide 44:

Equilibrium ½ q p S’s best response 1/2 R’s best response Mutual best responses

### Noncooperative Games :

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’s Dilemma :

Prisoner’s Dilemma Prisoner 2 Prisoner 1 Prisoner 1 Maximum Prisoner 2 Maximum 15 5 15 5

(1)

(2)‏

### Slide 49:

There are two prisoners whose aim is to minimize the years of imprisonment. They have committed a crime jointly. Each prisoner is interviewed separately and there are not any contacts whatsoever between them. They decide individually to confess or deny the crime taking into account possible decisions of the other prisoner (strategic game).

### Slide 50:

Each prisoner chooses his dominant strategy, that is the behavior giving the best result regardless of the decision of the other prisoner. Two prisoner's are interrogated separately, they have no idea whether the other prisoner will confess or not. Hence this is an example of non cooperative game.

### Slide 51:

Firm 2 Firm 1 Low-Level Advertising Low-Level Advertising High-Level Advertising 30 , 30 10 , 40 40 , 10 20 , 20 High-Level Advertising

### cooperative Games :

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

Repeated Games

### Slide 54:

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.

### Slide 55:

Firm 2 Firm 1 Low-Level Advertising Low-Level Advertising High-Level Advertising 30 , 30 10 , 40 40 , 10 20 , 20 High-Level Advertising

### Slide 56:

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 What About Cheating at last round?

### Slide 57:

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

### Slide 59:

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. Sequential Games and Credibility

### Slide 60:

When the players are A monopolist and a new firm, The new firm faces a number of decisions beginning with whether to enter the industry.

### Entry in Vacuum cleaner market :

Entry in Vacuum cleaner market Market currently has one incumbent, I-cleaners (I) Potential entrant, E-cleaners (E), deciding to enter or not If E enters, I has 2 choices: Accommodate: accept a lower market share Price war: punish E Need to construct game tree

### Slide 62:

Joint moves & resulting payoffs for E E-cleaners I-cleaners Enter Out Accommodate Price War \$100,000 = Profit - Entry Cost =\$150,000 - \$50,000 − \$25,000 = \$25,000 - \$50,000 \$0

### What should E do? :

What should E do? E needs to forecast I’s response How does it do this? Put itself in I’s shoes Work out I’s payoffs

### Same solution procedure as before: work backwards :

Same solution procedure as before: work backwards E-cleaners I-cleaners Enter Out Accommodate Price War E : \$100,000 I : \$150,000 E :-\$25,000 I : \$25,000 E : \$0 I : \$300,000 I accommodates: no return to vindictive behavior – threat of price war not credible

### If I accommodates, E should enter :

If I accommodates, E should enter E-cleaners I-cleaners Enter Keep Out Accommodate Fight Price War E: \$100,000 I: \$150,000 E:-\$25,000 I: \$25,000 E: \$0 I: \$300,000

### Slide 66:

Benifits to the one Who acts first

### Slide 67:

Firm 2 Firm 1 No New Product New Product No New Product New Product -5 , , -7, -5 -7 2 10 2 , 10 Minimum -5 -7 Minimum Minimum Minimum -7 -5

### Slide 68:

Firm 2 Firm 1 No New Product New Product No New Product New Product -5 , , -7, -5 -7 2 10 2 , 10