lecture 4: best response correspondence - uni erfurt · microeconomics i: game theory lecture 4:...

Microeconomics I: Game Theory

Lecture 4:Best Response Correspondence

(see Osborne, 2009, Sect 2.8)

Dr. Michael TrostDepartment of Applied Microeconomics

November 8, 2013

Dr. Michael Trost Microeconomics I: Game Theory Lecture 4 1 / 22

Detecting the set of Nash equilibria

Up to this point, Nash equilibria in a strategic game have beendetected by checking each possible action profile whether someplayer has an incentive to deviate from this profile.

However, to detect the set of Nash equilibria in a strategic game,it is often better to work with a concept that is known as thebest response correspondence (sometimes also called reactioncorrespondence).


Best response correspondence

Consider a strategic game Γ := (I , (Ai)i∈I , (%i)i∈I ).

The best response correspondence Bi is the correspondencethat assigns to every other players’ action a−i ∈ A−i set

Bi(a−i) := {ai ∈ Ai : Ui(ai , a−i) ≥ Ui(a′i , a−i) for every a′i ∈ Ai} .

Verbally, best response correspondence Bi gives, for every otherplayers’ action a−i ∈ A−i , the set of player i ’s actions that yieldthe highest possible utility to her provided that the otherplayers choose action a−i .


Best response function

The best response correspondence Bi of player i is said to besingle-valued if set Bi(a−i) contains only one element for everyaction a−i ∈ A−i of the other players.

If the best response correspondence Bi is single-valued, then afunction bi , known as the best response function, is definablewhich gives the single member of Bi(a−i) for every a−i ∈ A−i , i.e.,

Bi(a−i) = {bi(a−i)} .

holds for every a−i ∈ A−i .


B.r.c. of Prisoner’s Dilemma

Suspect B

quiet fink

Suspect Aquiet 2,2 0,3fink 3,0 1,1

Best response correspondence of

suspect A: BA(quiet) = {fink}, BA(fink) = {fink},suspect B: BB(quiet) = {fink}, BB(fink) = {fink}.


B.r.c. of Matching Pennies

Individual B

Head Tail

Individual AHead 1,-1 -1,1Tail -1,1 1,-1

EXERCISE: Determine the best response correspondences ofMATCHING PENNIES!


B.r.c. of Battle of Sexes

Individual B

Bach Stravinsky

Individual ABach 2,1 0,0Stravinsky 0,0 1,2

EXERCISE: Determine the best response correspondences ofBATTLE OF SEXES!


B.r.c. of the two player Guessing Game

Gue

ssof

play

er2

Guess of player 1

00

10

10

20

20

30

30

40

40

50

50

60

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70

70

80

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90

90

100

100


B.r.c. of the two player Guessing Game

In the two player guessing game, the best responsecorrespondence of player 1 is

B1(a2) =

{{0}, if a2 = 0

{a1 ∈ [0, 100] : a1 < a2}, if a2 > 0

and best response correspondence of player 2 is

B2(a1) =

{{0}, if a1 = 0

{a2 ∈ [0, 100] : a2 < a1}, if a1 > 0 .


B.r.c. and Nash equilibrium

The following theorem sets forth how the concept of bestresponse correspondence is related to the Nash equilibriumconcept.

This relationship paves the way for a two-step method fordetecting the Nash equilibria of a strategic game.


B.r.c. and Nash equilibrium

Theorem 4.1Consider a strategic game Γ := (I , (Ai)i∈I , (%i)i∈I ) and denote by Bi

the best response correspondence of player i ∈ I .

An action profile a∗ ∈ ×i∈IAi is a Nash equilibrium of Γ if and only if,for every player i ∈ I ,

a∗i ∈ Bi(a∗−i)

holds.


B.r.f. and Nash equilibrium

Suppose the players’ best response correspondences Bi aredescribable by best response functions bi .

According to Theorem 4.1 action profile a∗ is a Nash equilibriumif and only if equation

a∗i = bi(a∗−i)

is satisfied for every player i .


A two-step solution method

The characterization of Nash equilibrium by best responsecorrespondences in Theorem 4.1 suggests following two-stepmethod for detecting Nash equilibria:

1 Find the best response correspondence for each player.2 Find the actions profiles a∗ that satisfy a∗i ∈ Bi(a

∗−i) for every

player i ∈ I .


Two-step method in Prisoner’s Dilemma

Suspect B

quiet fink


The action profile (quiet,quiet) is not a Nash equilibrium,because quiet /∈ BA(quiet) (and even quiet /∈ BB(quiet)) holds.



Suspect B

quiet fink


The action profile (quiet,fink) is not a Nash equilibrium, becausequiet /∈ BA(fink) holds.



Suspect B

quiet fink


The action profile (fink,quiet) is not a Nash equilibrium, becausequiet /∈ BB(fink) holds.



Suspect B

quiet fink


The action profile (fink,fink) is a Nash equilibrium because

fink ∈ BA(fink) and

fink ∈ BB(fink)

are satisfied.


Two-step method in Matching Pennies

Individual B

Head Tail

Individual AHead 1,-1 -1,1Tail -1,1 1,-1

EXERCISE: Find the Nash equilibria of MATCHING PENNIES bythe two-step solution method!


Two-step method in Battle of Sexes

Individual B

Bach Stravinsky

Individual ABach 2,1 0,0Stravinsky 0,0 1,2

EXERCISE: Find the Nash equilibria of BATTLE OF SEXES by thetwo-step solution method!


Nash equilibrium of the Guessing Game

The profile (a1, a2) of guesses of the two player GUESSING

GAME for which

1 a1 = a2 > 0 holds, is not a Nash equilibrium becausea1 /∈ B1(a2) (and even a2 /∈ B2(a1)) applies.

2 a1 > a2 holds, is not a Nash equilibrium because a1 /∈ B1(a2)

applies.

3 a2 > a1 holds, is not a Nash equilibrium because a2 /∈ B2(a1)

applies.

4 a1 = a2 = 0 holds, is a Nash equilibrium because 0 ∈ B1(0)

and 0 ∈ B2(0) apply.


Fixed point and Nash equilibrium

Let B the correspondence that gives, for every action profilea ∈ A, the Cartesian product of the best responses of all players,i.e., for every a ∈ A

B(a) := ×i∈IBi(a−i) .

By Theorem 4.1 an action profile a∗ ∈ A turns out to be a Nashequilibrium if and only if a∗ is a fixed point of correspondenceB, i.e., if and only if

a∗ ∈ B(a∗)

holds.


Fixed point and Nash equilibrium

The last remark describes the Nash equilibria as a fixed points.Thus, the issue of existence of a Nash equilibrium amounts tothe issue of finding a fixed point of correspondence B .

Fixed point theorems (e.g., KAKUTANI’S FIXED POINT

THEOREM) state conditions on correspondence B so that B has afixed point.


lecture 4: best response correspondence - uni erfurt · microeconomics i: game theory lecture 4:...

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