Lecture 6: The Indifference Principle - Uni Erfurt€¦ · Dr. Michael Trost Microeconomics I: Game Theory Lecture 6 24 / 37 Finding all mixed action Nash equilibria The indifference

Microeconomics I: Game Theory

Lecture 6:The Indifference Principle

(see Osborne, 2009, Sect 4.3.3,4.3.4,4.8,4.10,4.11)

Dr. Michael TrostDepartment of Applied Microeconomics

November 29, 2013

Dr. Michael Trost Microeconomics I: Game Theory Lecture 6 1 / 37

Finding mixed action Nash equilibria

Figuring out directly the set of mixed action Nash equilibria isoften a tedious task.

In the following we present two alternative methods fordetecting mixed action Nash equilibria:

The first method is based on the players’ best responsecorrespondences, the second one on the indifference principle.


Method of best response correspondence

The method based on players’ best response correspondences(also termed as the two-step method) has been already appliedto figure out the set of pure action Nash equilibria.

As in the case of pure action Nash equilibria, we cancharacterize mixed action Nash equilibria by the players’ bestresponse correspondences.


B.r.c. and mixed action Nash equilibrium

Consider a vNM strategic game Γ := (I , (Ai)i∈I , (%i)i∈I ) anddenote by Bi the best response correspondence of player i ∈ I .

A profile α∗ := (α∗i )i∈I ∈ ×i∈I∆(Ai) of mixed actions is a mixedaction Nash equilibrium of Γ if and only if, for every player i ∈ I ,

α∗i ∈ Bi(α∗−i)

holds.


The two-step solution method

The characterization of the mixed Nash equilibrium by players’best response correspondences suggests following two-stepmethod for detecting mixed action Nash equilibria:

1 Find the best response correspondence Bi for each playeri ∈ I .

2 Find all profiles α∗ := (α∗i )i∈I of mixed actions that satisfyα∗i ∈ Bi(α

∗−i) for each player i ∈ I .


Example: Matching Pennies

Player Bq 1− q

Head Tail

Player Ap Head 1,-1 -1,1

1− p Tail -1,1 1,-1

In the following, we figure out the mixed action Nash equilibriaof the vNM strategic game MATCHING PENNIES by the two stepmethod.



To derive the players’ best response correspondences, proceedas follows for each player.

1 Calculate the player’s expected utility if both playersrandomize.

2 For each possible randomization by the opponents,describe the set of all mixed actions which maximizes theplayer’s expected utility given this randomization.



Suppose player B chooses head with probability q and tail withprobability 1− q.

If player A chooses head with probability p and tail withprobability 1− p, then

action profile (head,head) (head,tail) (tail,head) (tail,tail)

occurs withprobability p ·q p ·(1−q) (1−p)·q (1−p)·(1−q)

and A’sexpected utilityis = p ·q ·1 + p ·(1−q)·(−1) + (1−p)·q ·(−1) + (1−p)·(1−q)·1




If player A chooses head with probability p and tail withprobability 1− p, then her expected utility is

UA(p, q)

= pq1 + p(1− q)(−1) + (1− p)q(−1) + (1− p)(1− q)1

= pq − p + pq − q + pq + 1− q − p + pq

= 1− 2q − 2p + 4pq

= 1− 2q + 4p(q − 1

2)




If player A chooses head with probability p and tail withprobability 1− p, then A’s expected utility is

UA(p, q) = 1− 2q + 4p(q − 1

2).

Hence, the best response BA is given by

BA(q) =

{0} if q < 1

2,

[0, 1] if q = 1

2,

{1} if q > 1

2.


Example: Matching PenniesPlot of A’s best response correspondence

q

BA(q)

00

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Suppose player A chooses head with probability p and tail withprobability 1− p.

If player B chooses head with probability q and tail withprobability 1− q, then

action profile (head,head) (head,tail) (tail,head) (tail,tail)

occurs withprobability p ·q p ·(1−q) (1−p)·q (1−p)·(1−q)

and B’sexpected utilityis = p ·q ·(−1) + p ·(1−q)·1 + (1−p)·q ·1 + (1−p)·(1−q)·(−1)



Suppose player A choose head with probability p and tail withprobability 1− p.

If player B chooses head with probability q and tail withprobability 1− q, then B ’s expected utility is

UB(p, q)

= pq(−1) + p(1− q)1 + (1− p)q1 + (1− p)(1− q)(−1)

= −pq + p − pq + q − pq − 1 + q + p − pq

= −1 + 2p + 2q − 4pq

= −1 + 2p + 4q(12− p)



Suppose player A choose head with probability p and tail withprobability 1− p.

If player B chooses head with probability q and tail withprobability 1− q, then B ’s expected utility is

UB(p, q) = −1 + 2p + 4q(12− p).

Hence, the best response BB is given by

BB(p) =

{1} if p < 1

2,

[0, 1] if p = 1

2,

{0} if p > 1

2.


Example: Matching PenniesGraph of B ’s best response correspondence

BB(p)

p0

00.1

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Example: Matching PenniesThe best response correspondences of A and B in one figure.

q

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The plot of the best response correspondences of A and B

suggests that the mixed action

((p, 1− p), (q, 1− q)) =((12, 12), (1

2, 12))

is a Nash equilibrium of MATCHING PENNIES.

Indeed, the best response correspondences of A and B satisfy

(12, 12) ∈ BA(1

2, 12) and (1

2, 12) ∈ BB(1

2, 12).


Exercise: Battle of Sexes

Player Bq 1− q

Bach Stravinsky

Player Ap Bach 2,1 0,0

1− p Stravinsky 0,0 1,2

EXERCISE: For above vNM strategic game BATTLE OF SEXES,determine the expected utility of A and B and their bestresponse correspondences! Plot them in one figure. Find the setof all mixed action Nash equilibria.


Exercise: Battle of SexesGraph of the best response correspondences of A and B

q

p0

00.1

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Finding mixed action Nash equilibria

Another method to figure out the set of mixed action Nashequilibria is to apply the so-called indifference principle, whichis stated next.

The indifference principle is an alternative characterization ofmixed action Nash equilibria.

Before stating this principle some useful notation is introduced.


Mixed action profiles containing pure actions

Consider vNM strategic game Γ := (I , (Ai)i∈I , (%i)i∈I ) and letai ∈ Ai be a definite action of player i .

The mixed action profile (ai , α−i) ∈ ×i∈I∆(Ai) means that

player i chooses the definite action (or synonymously, thedegenerated mixed action) ai ,

every player j different to i chooses the mixed action αj .


Mixed action profiles containing pure actions

The expected utility of player i obtained by mixed action profile(ai , α−i) is

Ui(ai , α−i) =∑a′∈A

(∏j∈I

αj(a′j)

)ui(a

′i , a′j)

=∑

a′−i∈A−i

∏j∈I\{i}

αj(a′j)

ui(ai , a′j)

where the second row follows from the fact that αi is thedegenerated lottery that ascribes probability of 1 to action ai .


Indifference principle

Theorem 6.1Consider a vNM strategic game Γ := (I , (Ai)i∈I , (%i)i∈I ).

A profile (α∗i )i∈I of mixed actions is a Nash equilibrium in mixedactions of Γ if and only if, for each player i ∈ I , the following twoconditions are satisfied:

1 Ui(ai , α∗−i) = Ui(a

′i , α∗−i) holds for every actions ai , a′i ∈ Ai to

which α∗i assigns positive probability,

2 Ui(ai , α∗−i) ≥ Ui(a

′i , α∗−i) holds for every action ai ∈ Ai to which

α∗i assigns positive probability and for every action a′i ∈ Ai towhich α∗i assigns zero probability.


Indifference principle

The indifference principle states that a profile α∗ := (α∗i ) ofmixed actions is a Nash equilibrium in mixed actions if and onlyif, for each player i ∈ I ,

1 all definite actions realizable under mixed action αi yieldthe same expected payoff for player i ,

2 all definite actions not realizable under mixed action αi

yield an expected payoff for player i that does not exceedthat of definite actions realizable under mixed action αi


Finding all mixed action Nash equilibria

The indifference principle opens up the following method tofind all mixed actions Nash equilibria of a vNM strategic game.

1 For each player i ∈ I , choose a non-empty subset Si of Ai

2 Check whether there is a mixed action profile (αi)i∈I where(1) supp(αi ) = Si holds for every player i ∈ I

(2) the two conditions of the indifference principle hold forevery player i ∈ I .

3 Repeat the analysis for every collection of non-emptysubsets of the players’ sets of actions.



Player Bq 1− q

Head Tail

Player Ap Head 1,-1 -1,1

1− p Tail -1,1 1,-1

QUESTION: Figure out all mixed action Nash equilibria of thevNM strategic game MATCHING PENNIES by the procedurebased on the indifference principle.



The following combinations of supports of mixed actions mustbe checked.

Support of player B ’smixed action

{H} {T} {H,T}

Support of playerA’s mixed action

{H}{T}

{H,T}



CASE: The supports of mixed actions αA and αB aresuppA(αA) := {Head} and suppB(αB) := {Head}, respectively.

REASONING: Obviously, αA = Head and αB = Head apply.Since

UB(Head,Head) = −1 < 1 = UB(Head,Tail)

holds, condition 2 of the indifference principle is violated forplayer B .

Hence, no mixed action Nash equilibrium with such supportsexists.



CASE: The supports (suppA(αA), suppB(αB)) of mixed actionsαA and αB belong to one of the combinations ({Head}, {Tail}),({Tail}, {Head}) or ({Tail}, {Tail}).

REASONING: Due to arguments similar to that of the previouscase condition 2 of the indifference principle is violated by oneplayer.




CASE: The supports of mixed actions αA and αB aresuppA(αA) := {Head,Tail} and suppB(αB) := {Head}, resp.

REASONING: Obviously, 0 < p < 1 and αB = Head apply. Since

UA(Head,Head) = 1 > −1 = UA(Tail,Head)

holds, condition 1 of the indifference principle is violated forplayer A.




CASE: The supports (suppA(αA), suppB(αB)) of mixed actionsαA and αB are either ({Head,Tail}, {Tail}), ({Tail}, {Head,Tail})or ({Tail}, {Head,Tail}).

REASONING: Due to arguments similar to that of the previouscase condition 1 of the indifference principle is violated by oneplayer.




CASE: The supports of mixed actions αA and αB aresuppA(αA) := {Head,Tail} and suppB(αB) := {Head,Tail}),respectively.

REASONING: For player A, condition 1 of the indifferenceprinciple implies that

UA(Head, αB) = UA(Tail, αB) ,

or equivalently,

q · 1 + (1− q) · (−1) = q · (−1) + (1− q) · 1,

which results in q = 1

2.




REASONING: For player B , condition 1 of the indifferenceprinciple implies that

UB(αA,Head) = UB(αA,Tail) ,

or equivalently,

p · (−1) + (1− p) · 1 = p · 1 + (1− p) · (−1),

which results in p = 1

2.




REASONING: Summing up the previous discussion, the mixedaction profile

((αA(Head), αA(Tail)), (αB(Head), αB(Tail))) :=((

1

2, 12

),(1

2, 12

))is a mixed action Nash equilibrium.



The following combinations of supports of mixed actions areconsistent with a mixed Nash equilibrium.

Support of player B ’smixed action

{H} {T} {H,T}

Support of playerA’s mixed action

{H} × × ×{T} × × ×

{H,T} × × ((1

2, 12

),(1

2, 12

))


Exercise: Rock Paper Scissors

Player BqR qP qS

Rock Paper Scissors

Player ApR Rock 0,0 -1,1 1,-1pP Paper 1,-1 0,0 -1,1pS Scissors -1,1 1,-1 0,0

QUESTION: Figure out all mixed action Nash equilibria of thevNM strategic game ROCK PAPER SCISSORS by the procedurebased on the indifference principle.


Exercise: Rock Paper Scissors

The following combinations of supports of mixed actions mustbe checked.

Support of player B ’s mixed action{R} {P} {S} {R,P} {R,S} {P,S} {R,P,S}

Supp

orto

fpla

yerA

’sm

ixed

acti

on

{R}{P}{S}

{R,P}{R,S}{P,S}

{R,P,S}


Lecture 6: The Indifference Principle - Uni Erfurt€¦ · Dr. Michael Trost Microeconomics I: Game Theory Lecture 6 24 / 37 Finding all mixed action Nash equilibria The indifference

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