game theory 3

From Last Time

• How to solve/ make predictions for the game?

• What is Iterated Elimination of Dominated Strategies?

• What is a Nash Equilibrium?

Nash Equilibrium

Nash Equilibrium:• A set of strategies, one for each player, such

that each player’s strategy is a best response to others’ strategies

Everybody is playing a best response• No incentive to unilaterally change my strategy

BEST RESPONSE

Solving the game

• How to solve the game when there are no dominated strategies?

Coordination Games

Opera Movie

Opera 2,1 0,0

Movie 0,0 1,2

Matching Pennies

L R

L 1,-1 -1, 1

R -1, 1 1,-1

Best Response

• Form beliefs about what others will do

• If you believe opponent will play Opera: Best Response is to play Opera

• Similarly, if you believe opponent will play Movie: Best Response is to play Movies

Coordination Games

Opera Movie

Opera 2,1 0,0

Movie 0,0 1,2

Best Response

• Best Response: Action that gives highest payoff given a belief about others play

• Best Response changes with different beliefs about opponents play.

• There may be more than one Best Response for a belief

L C

R

U 8, 3 0, 4 4,4

M 8,5 1,5 5,3

D 3,7 0,1 2,0

Forming Beliefs

• Forming one’s belief is the important part of strategy

• Success depends upon belief formation

Best Response Functions

Consider best response function of i:

Bi(a-i) = {ai is element of Ai: u I (ai,a-i ) ≥ ui(ai',a-i ) for all ai’ that is element of Ai}

• Set-valued, each member of Bi(a-i) is a best responseto a-i

Best Response Functions

a* is a Nash equilibrium if and only if ai* is element of Bi(a-i*) for every i

Best Response Function Examples

Best Response Function Examples

Best Response Function Examples

Player 2B1 B2 B3

Player 1 A1 10,10 14,12 14,15

A2 12,14 20,20 28,15

A3 15,14 15,28 25,25

•What is Player 1’s best response to Player 2’sstrategy of B1, B2 or B3?•What is Player 2’s best response to Player 1’sstrategy of A1, A2 or A3?

Best Response in 2-player game

Using best response function to find Nash equilibrium in a 2-player game

( s1,s2) is a Nash equilibrium if and only if

• player 1’s strategy s1 is her best response to player 2’s strategy s2 • player 2’s strategy s2 is her best response to player 1’s strategy s1

Battle of Sexes

•Ball is Player 1’s best response to Player 2’s strategy Ball•Ball is Player 2’s best response to Player 1’s strategy Ball•Hence, (Ball, Ball) is a Nash equilibrium•Theatre is Player 1’s best response to Player 2’s strategy Theatre•Theatre is Player 2’s best response to Player 1’s strategy Theatre•Hence, (Theatre, Theatre) is a Nash equilibrium

Player2

Ball Theatre

Player1 Ball 2,1 0,0

Theatre 0,0 1,2

Matching Pennies

•Head is Player 1’s best response to Player 2’s strategy Tail•Tail is Player 2’s best response to Player 1’s strategy Tail•Tail is Player 1’s best response to Player 2’s strategy Head•Head is Player 2’s best response to Player 1’s strategy Head•Hence, NO Nash equilibrium

Player2

Head Tail

Player1 Head -1,1 1,-1

Tail 1,-1 -1,1

ITERATED ELIMINATION OF STRICTLY DOMINATED STRATEGY

Iterated Elimination of Strictly Dominated Strategies

X Y Z

A 3, 3 0, 5 0,4

B 0,0 3,1 1,2

Common Knowledge

• First Level: You and Opponent know the matrix• Second Level: -You know that Opponent knows the matrix-Opponent knows that you know the matrix• Third Level- You know that opponent knows that you know the

matrix- Opponent knows that you know that opponent knows

the matrix

Iterated Elimination of Strictly Dominated Strategies

X Y Z

A 3, 3 0, 5 0,4

B 0,0 3,1 1,2

Iterated Elimination of Strictly Dominated Strategies

• If a strategy is strictly dominated for some player, eliminate it

• Repeat, eliminating any strictly dominated strategies in reduced game

Dominance example

Dominance exampleB is dominated for Player 1 After eliminating B,R is dominated for Player 2

IEDS Example

IEDS Example

IEDS Example

IEDS Example

L C R

T 2,3 2,2 5,0

Y 3,2 5,3 3,1

Z 4,3 1,1 2,2

B 1,2 0,1 4,4

L C R

T 2,3 2,2 5,0

Y 3,2 5,3 3,1

Z 4,3 1,1 2,2

B 1,2 0,1 4,4

L C R

T 2,3 2,2 5,0

Y 3,2 5,3 3,1

Z 4,3 1,1 2,2

B 1,2 0,1 4,4

L C R

T 2,3 2,2 5,0

Y 3,2 5,3 3,1

Z 4,3 1,1 2,2

B 1,2 0,1 4,4

L C R

T 2,3 2,2 5,0

Y 3,2 5,3 3,1

Z 4,3 1,1 2,2

B 1,2 0,1 4,4

Another IEDS Example

Order of Elimination Question: Does the order of elimination matter?

Answer: Although it is not obvious, the end result of iterated strict dominance is always the same regardless of the sequence of eliminations.

Payoff Matrix for Bottled Water Game

Firm Coca Cola

Raise Decrease

Raise

Decrease

Firm Pepsi Co

+1, +1 -1, +2

0, 0+2, -1

ITERATED ELIMINATION OF WEAKLY DOMINATED STRATEGY

Weakly Dominated Strategies

ai weakly dominates a’i if for all strategy profiles a-i of the other players

ui(ai, a-i) ≥ui(a’i, a-i) and there is at least one a-i'

such that ui(ai, a-i') >ui(a’i, a-i')

Example A B C

I -1, 3 0, 3 3, 7

II -1, 4 2, 7 6, 5

Weakly Dominated Strategies

• A weakly dominated strategy can be chosen in a Nash equilibrium

Firm Coca Cola

Raise Decrease

Raise

Decrease

Firm Pepsi Co

+1, +1 0, 0

0, 00,0

Weakly Dominated Strategies

• A weakly dominated strategy can be chosen in a Nash equilibrium

• Order of eliminating weakly dominated strategies matters

Example A B C

I -1, 3 0, 3 3, 7

II -1, 4 2, 7 6, 5

Example A B C

I -1, 3 0, 3 3, 7

II -1, 4 2, 7 6, 5

• Eliminate I, • Eliminate A, I

A B

I 0,0 2,5

II 5,5 100,5

III 5,5 0,0

• I is weakly dominated by II• B is weakly dominated by A• (II,A) AND (III,A) are two possible outcomes

A B

I 0,0 2,5

II 5,5 100,5

III 5,5 0,0

• III is weakly dominated by II• A is weakly dominated by B• (II,B) is a possible outcome

A B

I 0,0 2,5

II 5,5 100,5

III 5,5 0,0

Order of Elimination

• If you eliminate a strategy when there is some other strategy that yields payoffs that are higher or equal no matter what the other players do, you are doing iterated weak dominance

• In this case you will not always get the same answer regardless of the sequence of eliminations.

•This is a serious problem, and this is the reason why iterated strict dominance is mostly used.