best response curves and nash equilibria_3pp

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Week 3:Best responses and Nash equilibria(complementary notes)

Nash Equilibrium

In any Nash equilibrium (NE) each player chooses a “best” response to the choices made by all of the other players.

A game may have more than one NE.

Nash Equilibrium

How can we locate every one of a game’s Nash equilibria?

If there is more than one NE, can we argue that one is more likely to occur than another?

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Best Responses

Think of a 2×2 game; i.e., a game with two players, A and B, each with two actions.

A can choose between actions aA1 and aA2.

Best Responses

B can choose between actions aB1 and aB2.

There are 4 possible action pairs;(aA1, a

B1), (a

A1, a

B2), (a

A2, a

B1), (a

A2, a

B2).

Each action pair will usually cause different payoffs for the players.

Best Responses

Suppose that A’s and B’s payoffs when the chosen actions are aA1 and a

B1 are

UA(aA1, aB1) = 6 and U

B(aA1, aB1) = 4.

Similarly, suppose that


B(aA1, aB2) = 5


B(aA2, aB1) = 3


B(aA2, aB2) = 7.

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Best Responses


B(aA1, aB1) = 4


B(aA1, aB2) = 5


B(aA2, aB1) = 3


B(aA2, aB2) = 7.

If B chooses action aB1 then A’s best

response is action aA1 (because 6 > 4).

Best Responses


B(aA1, aB1) = 4


B(aA1, aB2) = 5


B(aA2, aB1) = 3


B(aA2, aB2) = 7.

If B chooses action aB2 then A’s best

response is action aA2 (because 5 > 3).

Best Responses

If B chooses aB1 then A chooses aA1.

If B chooses aB2 then A chooses aA2.

A’s best‐response “curve” is therefore

A’s bestresponse aA

1

aA2

aB2aB

1B’s action

+

+

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Best Responses


B(aA1, aB1) = 4


B(aA1, aB2) = 5


B(aA2, aB1) = 3


B(aA2, aB2) = 7.

If A chooses action aA1 then B’s best

response is action aB2 (because 5 > 4).

Best Responses


B(aA1, aB1) = 4


B(aA1, aB2) = 5


B(aA2, aB1) = 3


B(aA2, aB2) = 7.

If A chooses action aA2 then B’s best

response is action aB2 (because 7 > 3).

Best Responses

If A chooses aA1 then B chooses aB2.


B’s best‐response “curve” is therefore

A’s action aA

1

aA2

aB2aB

1B’s best response

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Best Responses If A chooses aA1 then B chooses a

B2.


B’s best‐response “curve” is therefore

A’s action aA

1

aA2

aB2aB

1B’s best response

Notice that aB2 is a

strictly dominantaction for B.

Best Responses & Nash Equilibria

A’s response

aA1

aA2

aB2aB

1

aA1

aA2

aB2aB

1

+

+

A’s choice

B’s choice B’s response

How can the players’ best‐response curves beused to locate the game’s Nash equilibria?→ Put one curve on top of the other.

B A


A’s response

aA1

aA2

aB2aB

1

+

+

Is there a Nash equilibrium?Yes, (aA2, a

B2). Why?

aA2 is a best response to aB2.

aB2 is a best response to aA2.

B’s response

How can the players’ best‐response curves beused to locate the game’s Nash equilibria?→ Put one curve on top of the other.

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6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

aA2 is the only best response to aB2.

aB2 is the only best response to aA2.

Here is the strategicform of the game.

Is there a 2nd Nasheqm.? No, becauseaB2 is a strictlydominant actionfor Player B.


Now allow both players to randomize (i.e.,mix)over their actions.

6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

A1 is the prob. Achooses action aA1.B1 is the prob. Bchooses action aB1.


6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


Given B1, whatvalue of A1 is bestfor A?

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6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

EVA(aA1) = 6B1 + 3(1 ‐ B1) = 3 + 3B1.




6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A



EVA(aA1) = 6B1 + 3(1 ‐ B1) = 3 + 3B1.EVA(aA2) = 4B1 + 5(1 ‐ B1) = 5 ‐ B1.


EVA(aA1) = 3 + 3B1.EVA(aA2) = 5 ‐ B1.3 + 3B1 5 ‐ B1 as B1 ??

>=<>=<

A1 is the prob. A chooses action aA1.B1 is the prob. B chooses action aB1.

Given B1, what value of A1 is best for A?

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EVA(aA1) = 3 + 3B1.EVA(aA2) = 5 ‐ B1.3 + 3B1 > 5 ‐ B1 as B1 > ½.


EVA(aA1) = 3 + 3B1.EVA(aA2) = 5 ‐ B1.3 + 3B1 > 5 ‐ B1 as B1 > ½. A’s best response is:

aA1 if B1 > ½aA2 if B1 < ½aA1 or a

A2 if B1 = ½




EVA(aA1) = 3 + 3B1.EVA(aA2) = 5 ‐ B1.3 + 3B1 > 5 ‐ B1 as B1 >½. A’s best response is:

aA1 (i.e. A1 = 1) if B1 > ½aA2 (i.e. A1 = 0) if B1 < ½aA1 or a

A2 (i.e. 0 A1 1) if B1 = ½



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0

A1

1 B10

A’s best response

½

A’s best response is: aA1 (i.e. A

1 = 1) if B1 > ½

aA2 (i.e. A

1 = 0) if B1 < ½

aA1 or aA

2 (i.e. 0 A1 1) if

B1 = ½

1


0

A1

1 B10

A’s best response

½

1


1 = 1) if B1 > ½

aA2 (i.e. A

1 = 0) if B1 < ½

aA1 or aA

2 (i.e. 0 A1 1) if

B1 = ½


0

A1

1 B10

A’s best response

½

1


1 = 1) if B1 > ½

aA2 (i.e. A

1 = 0) if B1 < ½

aA1 or aA

2 (i.e. 0 A1 1) if

B1 = ½

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0

A1

1 B10

A’s best response

½

1 This is A’s best responsecurve when players areallowed to mix over theiractions.


1 = 1) if B1 > ½

aA2 (i.e. A

1 = 0) if B1 < ½

aA1 or aA

2 (i.e. 0 A1 1) if

B1 = ½


6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


Given A1, whatvalue of B1 is bestfor B?


6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

EVB(aB1) = 4A1 + 3(1 ‐ A1) = 3 + A1.



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6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

EVB(aB1) = 4A1 + 3(1 ‐ A1) = 3 + A1.EVB(aB2) = 5A1 + 7(1 ‐ A1) = 7 ‐ 2A1.





Given A1, what value of B1 is best for B?

EVB(aB1) = 3 + A1.EVB(aB2) = 7 ‐ 2A1.3 + A1 7 ‐ 2A1 as A1 ??>=<

>=<


EVB(aB1) = 3 + A1.EVB(aB2) = 7 ‐ 2A1.3 + A1 < 7 ‐ 2A1 for all 0 A1 1.



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EVB(aB1) = 3 + A1.EVB(aB2) = 7 ‐ 2A1.3 + A1 < 7 ‐ 2A1 for all 0 A1 1.B’s best response is:

aB2 always (i.e. B1 = 0 always).




0

A1

1 B10

B’s best response½

B’s best response is aB2 always (i.e. B

1 = 0 always).

1 This is B’s best responsecurve when players areallowed to mix over theiractions.


0

A1

1 B10


1

0

A1

1 B10

A’s best response

½

1

B A

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0

A1

1 B10


1

0

A1

1 B10

A’s best response

½

1

B A

Is there a Nash equilibrium?


0

A1

1 B10


1

0

A1

1 B10

A’s best response

½

1

B A



0

A1

1 B10


1

A’s best response


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0

A1

1 B10


1

A’s best response

Is there a Nash equilibrium? Yes. Just one.(A1, B1) = (0,0);i.e. A chooses aA2 only& B chooses aB2 only.


6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

Let’s change the game.

3,1


6,4 3,5

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

Here is a new2×2 game.

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3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

Here is a new2×2 game. Againlet A1 be the prob.that A chooses aA1and let B1 be theprob. that B choosesaB1. What are the NEof this game?

Notice that Player B no longer has a strictly dominant action.

3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A

A1 is the prob. that Achooses aA1.B1 is the prob. that Bchooses aB1.

EVA(aA1) = ??EVA(aA2) = ??

3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


EVA(aA1) = 6B1 + 3(1 ‐ B1) = 3 + 3B1. EVA(aA2) = ??

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3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


EVA(aA1) = 6B1 + 3(1 ‐ B1) = 3 + 3B1. EVA(aA2) = 4B1 + 5(1 ‐ B1) = 5 ‐ B1.

3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


EVA(aA1) = 6B1 + 3(1 ‐ B1) = 3 + 3B1. EVA(aA2) = 4B1 + 5(1 ‐ B1) = 5 ‐ B1.3 + 3B1 5 ‐ B1 as B1 ½.>=<

>=<



>=<

0

A1

1 B10

A’s best response

½

1

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0

A1

1 B10

A’s best response

½

1


>=<

3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


EVB(aB1) = ??EVB(aB2) = ??

3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


EVB(aB1) = 4A1 + 3(1 ‐ A1) = 3 + A1. EVB(aB2) = ??

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3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


EVB(aB1) = 4A1 + 3(1 ‐ A1) = 4 + A1. EVB(aB2) = A1 + 7(1 ‐ A1) = 7 ‐ 6A1.

3,1


6,4

5,74,3

aA1

aA2

aB1 aB

2

Player B

Player A


EVB(aB1) = 4A1 + 3(1 ‐ A1) = 3 + A1. EVB(aB2) = A1 + 7(1 ‐ A1) = 7 ‐ 6A1.3 + A1 7 ‐ 6A1 as A1>=<

>=< 4 7/


0

A1

1 B10

1

47/

B’s best response


>=< 4 7/

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0

A1

1 B10

1

47/

B’s best response


>=< 4 7/


0

A1

1 B10

B’s best response

1

0

A1

1 B10

A’s best response

½

1

B A

47/


0

A1

1 B10

B’s best response

1

0

A1

1 B10

A’s best response

½

1

B A

47/


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0

A1

1 B10

B’s best response

1

B A

47/

0

A1

1 B10

A’s best response

½

1



0

A1

1 B10

B’s best response

1

47/

A’s best response

½



0

A1

1 B10

B’s best response

1

47/

A’s best response

½

Is there a Nash equilibrium? Yes. 3 of them.

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0

A1

1 B10

B’s best response

1

47/

A’s best response

½

Is there a Nash equilibrium? Yes. 3 of them.(A

1, B1) = (0,0)


0

A1

1 B10

B’s best response

1

47/

A’s best response

½


1, B1) = (0,0)

(A1, B

1) = (1,1)



0

A1

1 B10

B’s best response

1

47/

A’s best response

½


1, B1) = (0,0)

(A1, B

1) = (1,1)(A

1, B1) = ( , )½4

7/


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Two more examples

Games of coordination

Games of coexistence

Coordination Games

Simultaneous play games in which the payoffs to the players are largest when they coordinate their actions. Famous examples are:

– The Battle of the Sexes Game

– The Prisoner’s Dilemma Game

– The Chicken Game

Coordination Games; The Battle of the Sexes

Sissy prefers watching ballet to watching ultimate fighting.

Jock prefers watching ultimate fighting to watching ballet.

Both prefer watching something together to being apart.

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1,28,4

4,82,1

B

UF

B UFJock

Sissy

SB is the prob. thatSissy chooses ballet.JB is the prob. thatJock chooses ballet.

What are the players’best‐responsefunctions?

EVS(B) = 8JB + 1(1 ‐ JB) = 1 + 7JB.EVS(UF) = 2JB + 4(1 ‐ JB) = 4 ‐ 2JB.1 + 7JB 4 ‐ 2JB as JB .1 3/


1,28,4

4,82,1

B

UF

B UFJock

Sissy


What are the players’best‐responsefunctions?

>=<>=<

EVS(B) = 8JB + 1(1 ‐ JB) = 1 + 7JB.EVS(UF) = 2JB + 4(1 ‐ JB) = 4 ‐ 2JB.1 + 7JB 4 ‐ 2JB as JB .1 3/


1,28,4

4,82,1

B

UF

B UFJock

Sissy


>=<>=<

SB

JB1

1

00

1 3/

Sissy

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EVJ(B) = 4SB + 1(1 ‐ SB) = 1 + 3SB.EVJ(UF) = 2SB + 8(1 ‐ SB) = 8 ‐ 6SB.1 + 3SB 8 ‐ 6SB as SB .7 9/


1,28,4

4,82,1

B

UF

B UFJock

Sissy


>=<>=<

JB

SB1

1

00

7 9/

Jock


SB

JB1

1

00

1 3/

Sissy SB

JB1

1

00

7 9/

Jock

The game’s NE are ??


SB

JB1

1

00

1 3/

Sissy

Jock

7 9/

The game’s NE are: (JB, S

B) = (0, 0); i.e., (UF, UF)(J

B, SB) = (1, 1); i.e., (B, B)

(JB, S

B) = (1/3, 7/9); i.e., bothwatch the ballet with prob. 7/27, both watch the ultimate fighting with prob. 4/27, Sissy watches UF and Jock

watches ballet with prob. 2/27 and with prob. 14/27 they watch their preferred event alone.

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Coexistence Games

Simultaneous play games that can be used to model how members of a species act towards each other.

An important example is the hawk‐dove game.

Coexistence Games; The Hawk-Dove Game

“Hawk” means “be aggressive.”

“Dove” means “don’t be aggressive.”

Two bears come to a fishing spot. Either bear can fight the other to try to drive it away to get more fish for itself but suffer battle injuries, or it can tolerate the presence of the other, share the fishing, and avoid injury.

Coexistence Games;The Hawk-Dove Game

8,0

4,40,8

Hawk

Dove

Hawk DoveBear 2

Bear 1

Are there NE in pure strategies?

-5,-5

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8,0

4,40,8

Hawk

Dove

Hawk DoveBear 2

Bear 1

Are there NE in pure strategies?Yes (Hawk, Dove) and (Dove, Hawk).Notice that purely peaceful coexistence is not a NE.

-5,-5


8,0

4,40,8

Hawk

Dove

Hawk DoveBear 2

Bear 1

Is there a NE in mixed strategies?

-5,-5


8,0

4,40,8

Hawk

Dove

Hawk DoveBear 2

Bear 1

Is there a NE in mixed strategies?

1H is the prob. that

1 chooses Hawk.2

H is the prob. that2 chooses Hawk.What are the players’best-responsefunctions?

-5,-5

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8,0

4,40,8

Hawk

Dove

Hawk DoveBear 2

Bear 1


1 chooses Hawk.2


EV1(H) = -52H + 8(1 - 2

H) = 8 - 132H.

EV1(D) = 4 - 42H.

-5,-5


8,0

4,40,8

Hawk

Dove

Hawk DoveBear 2

Bear 1


1 chooses Hawk.2


EV1(H) = -52H + 8(1 - 2

H) = 8 - 132H.

EV1(D) = 4 - 42H.

8 - 132H 4 - 42

H as 2H 4/9.>=<

<=>

-5,-5


8,0

4,40,8

Hawk

Dove

Hawk DoveBear 2

Bear 1


1 chooses Hawk.2

H is the prob. that2 chooses Hawk.

Bear 11H

2H1

1

00

4/9

-5,-5

EV1(H) = -52H + 8(1 - 2

H) = 8 - 132H.

EV1(D) = 4 - 42H.

8 - 132H 4 - 42

H as 2H 4/9.>=<

<=>

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Bear 1 Bear 22H

1H1

1

00

4/9

1H

2H1

1

00

4/9


Bear 1 Bear 22H

1H1

1

00

4/9

1H

2H1

1

00

4/9

4/9


Bear 1 Bear 21H

2H1

1

00

4/9

1H

2H1

1

00

4/9

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1H

2H1

1

00

4/9

The game has a NE in mixed-strategies in whicheach bear plays Hawk with probability 4/9.

best response curves and nash equilibria_3pp

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