Slide 1: 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
Slide 21: Mixed Strategy
Slide 22: Why We Should Study This?
Slide 23: Tennis Match S R
Slide 24: Striking S R
Slide 25: 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
Slide 27: 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) : (1)
(2) : (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.
Slide 53: 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
Slide 58: 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
Slide 69: Your Queries