Whats New here Incomplete information Example Battle of the sexes gameBut Bob doesnt know what Alice wants ie her payoffs from possible outcomes In previous examples we had ID: 396502
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Slide1
Bayes-Nash equilibrium with Incomplete InformationSlide2
What’s New here?
Incomplete information:
Example:
Battle of the sexes game,But Bob doesn’t know what Alice wants (i.e. her payoffs from possible outcomes)In previous examples we had “Imperfect Information”. PlayersKnew each others payoffs, but might not know each other’s moves. Slide3
She loves me, she loves me not?
(
Bob moves before Alice)
Go to A
Go to B
Go to A
Alice
Alice
Go to B
Go to A
Go to B
2
3
0
0
1
1
3
2
She loves him
Nature
She scorns him
Go to A
Go to A
Go to A
Go to B
Go to B
Go to B
2
1
0
2
1
3
3
0
Bob
Alice
Bob
AliceSlide4
How we handle this story
Nature moves first—Tells Alice whether she loves Bob or despises him.
Nature doesn’t tell Bob.
Bob has probabilistic beliefs about Alice’s inclination.
Whatever Bob does, Alice knows how she feels and acts accordingly.Bob is aware of this, but doesn’t know how she feels.Slide5
She loves me, she loves me not?
(
Bob moves before Alice)
Go to A
Go to B
Go to A
Alice
Alice
Go to B
Go to A
Go to B
2
3
0
0
1
1
3
2
She loves him
Nature
She scorns him
Go to A
Go to A
Go to A
Go to B
Go to B
Go to B
2
1
0
2
1
3
3
0
Bob
Alice
Bob
AliceSlide6
Bayes-Nash Equilibrium
Alice could be one of two types. “loves Bob”
“scorns Bob
Whichever type she is, she will choose a best response.
Bob thinks the probability that she is a ``loves Bob’’ type is p. He maximizes his expected payoff, assuming that Alice will do a best response to his action.Slide7
Expected payoffs to Bob
If he goes to movie A, he knows that Alice will go to A if she loves him, B if she scorns him.
His expected payoff from A is
2p+0(1-p)=2p.
If he goes to movie B, he knows that Alice will go to B if she loves him, A if she scorns him. His expected from B is then 3p+1(1-p)=2p+1.For any p, his best choice is movie B since 2p+1>2p for all p.Slide8
Does she or doesn’t she?
Simultaneous Play
Go to A
Go to B
Go to A
Alice
Alice
Go to B
Go to A
Go to B
2
3
0
0
1
1
3
2
She loves him
Nature
She scorns him
Go to A
Go to A
Go to A
Go to B
Go to B
Go to B
2
1
0
2
1
3
3
0
Bob
Alice
Bob
AliceSlide9
Bayes’ Nash equilibrium
Is there a Bayes’ Nash equilibrium where Bob goes to B and Alice
Alice
goes to B if she loves Bob,
and to A if she scorns him?This is a best response for both Alice types.What about Bob? Slide10
Bob’s Calculations
If Bob thinks the probability that Alice loves him is p and Alice will go to B if she loves him and A if she scorns him:
His expected payoff from going to B is
3p+1(1-p)=1+2p.
His expected payoff from going to A is 2(1-p)+0p=2-2p.Going to B is Bob’s best response to the strategies of the Alice types if 1+2p>=2-2p. Equivalently p>=1/4.Slide11
Is there a Bayes-Nash equilibrium in pure strategies if p<1/4?
Yes, Alice goes to B if she loves Bob and A if she scorns him and Bob goes to B.
Yes, Alice goes to A if she loves Bob and B if she scorns him and Bob goes to B.
Yes there is one, where Alice always goes to A.
No there is no Bayes-Nash equilibrium in pure strategies. Slide12
What about a mixed strategy equilibrium?
If p<1/4, can
we find a mixed strategy for Bob that makes one or both types of Alice willing to do a mixed strategy
?
What if Bob knows Alice scorns him?Consider the Alice type who scorns Bob. If Bob goes to movie A with probability q, When will Alice be indifferent between going to the two movies? Slide13
The game if Alice hates Bob
A
B
A
1,23,1B2,00,3Bob
AliceSlide14
Mixed strategy equilbrium:
Bob the stalker
If Bob knows Alice hates him, then if he uses a pure strategy, he knows Alice would always avoid him.
If he uses a mixed strategy, he would catch her sometimes.
In mixed strategy Nash equilibrium, each would be indifferent about the two strategies.Slide15
Making Alice indifferent
If Bob goes to B with
probabilty
b:Expected payoff to Alice from going to A
Is 3b+(1-b)Expected payoff to Alice from going to B is 2(1-b)These are equal if 2b+1=2-2b or b=1/4.So Stalker Bob would go to Alice favorite movie ¾ of the time.Slide16
Making Bob indifferent
If Alice goes to movie A with probability a
Bob’s expected payoff from going to A would be 2a+0
Bob’s expected payoff from going to B would be a a+3(1-a)
Bob would be indifferent if 2a=3-2a which means a= 3/4So Alice would go to her favorite movie ¾ of the timeThen Bob would meet her at A with probability ¾ x ¾=9/16 and at B with probability ¼ x ¼ =1/16.Slide17
Expected payoff
In the mixed strategy equilibrium, where Alice scorn him, Bob’s expected payoff
is 2(9/16)+1(3/16)+0(3/16)+3(1/16)=3/2.
and expected payoff for Alice is
1(9/16)+3(3/16)+2(3/16)+0(1/16)=3/2Slide18
Wyatt Earp and the Gun SlingerSlide19
A Bayesian gunslinger gameSlide20
The gunfight game when the stranger is (a) a gunslinger or (b) a cowpokeSlide21
What are the strategies?
Earp
Draw
Wait
StrangerDraw if Gunslinger, Draw if CowpokeDraw if Gunslinger, Wait if CowpokeWait if Gunslinger, Draw if CowpokeWait if Gunslinger, Wait if CowpokeSlide22
One Bayes Nash equilibrium
Suppose that Earp waits and the other guy draws if he is a gunslinger, waits if he is a cowpoke.
Stranger in either case is doing a best response.
If stranger follows this rule, is waiting best for Earp?
Earp’s Payoff from waiting is 3/4x1+1/4x8=2.75Earp’s Payoff from drawing, given these strategies for the other guys is (¾)2+(1/4) 4=2.5So this is a Bayes Nash equilibriumSlide23
There is another equilibrium
Lets see if there is an equilibrium where everybody draws.