game theory
DESCRIPTION
TRANSCRIPT
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Basic concepts
• The payoff matrix
• Nash equilibrium
• Dominant strategies
• Maximin strategies
• Mixed strategies
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Game Theory And Oligopoly
• Non cooperative games the prisoner dilemma
• Cooperative games dealing with cheaters
• Sequential games the ad of being first
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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.
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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
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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.
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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
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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
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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.
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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
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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
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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
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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
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•
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?
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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
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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
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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
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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
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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.
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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.
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Left
1/2
Right
P
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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
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Left
1/2
Right
q
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Putting Things Together
p
1/2
q
1/2
R’s best response
S’s best response
Mutually best response
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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
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Prisoner
Prisoner 2
Prisoner 1
Strategy
Confess
Remain Silent
Confess
5,5
0,20
Remain Silent
20,0
1,1
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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
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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
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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
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• 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
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Advantages
•Easy to understand •Never initiates cheating •Never rewards cheating cause it punish in some way •Its about forgiving because cooperation is quickly restored
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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
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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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