Problem 7 Chapter 9 Find the subgame perfect Nash equilbria um Problem 7 Chapter 9 Find the subgame perfect Nash equilbria um One SPNE List entire strategies for both players Player 1 has 5 information sets and Player 2 has 2 information sets ID: 559371
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Slide1
A few problemsSlide2
Problem
7,
Chapter
9
Find the
subgame
perfect Nash
equilbria
(um)Slide3
Problem
7,
Chapter
9
Find the
subgame
perfect Nash
equilbria
(um)Slide4
One SPNE
List entire strategies for both players/ Player 1 has 5 information sets and Player 2 has 2 information sets.
One SPNE is a1/c1/d1/d1/d1 for Player 1 and
a2/b2 for Player 2
In this equilibrium, the outcome is 1 plays a1,
2 plays a2, and then 1 plays c1. Payoffs are
4 for Player 1 and 3 for Player 2.Slide5
Another SPNE
Another
SPNE is a1/d1/d1/d1/d1 for Player 1 and b2/b2 for Player 2
.
With these strategies, the course of play is Player 1 goes a1, then Player 2 goes b2. Then Player 1 goes d1.
The payoffs are 5 for Player 1 and 2 for Player 2.Slide6
Also some mixed strategies
At the second info set, since Player 1 is indifferent between c1 and d1, he could also use a mixed strategy.
If
prob
of c1 is p,
The expected payoff to 2 from going a2 is
3p+1-p=1+2p and the expected payoff from going b2 is 2.
What happens with p<1/2? p>1/2? p=1/2?Slide7
Looking back
We see that in any
subgame perfect equilibrium, if Player 1 plays b1 on his first move, 2 will play b2 and 1 will then play d1, so the outcome if Player 1 plays b1 will have payoffs of 3 for 1 and 3 for 2.
We see that if player 1 plays a1 on his first move, he can guarantee himself at least 4 by going right.
So 1’s first move must be a1.
If 2 is going to do a2 when it is his turn, then 1 is indifferent between c1 and d1. Either action is consistent with SPNE.
SPNE does not tell us that the 5, 2 outcome is more likely than the 4,1. Does it seem like it should be?Slide8
Problem 10, page 287Slide9
A subgame between 2 and 3
a3
b3
a2
2,1
2,0
b2
0,2
3,4Slide10
Does this game have pure strategy Nash equilibria
?
Yes, there are two of them
No there are none.
Yes there is one of them
Yes there are three of them. Slide11
Truncated
game with a2, a3 the Nash equilibrium
in
subgame
between 2 and 3
2
4
2
3
3
2
1
3
Player 1
a1
b1Slide12
Truncated
game if b2, b3 in subgame between 2 and 3
2
4
2
3
0
1
1
1
Player 1
a1
b1Slide13
One SPNE
Player 1 uses b1
Player 2 uses b2/a2Player 3 uses a3
Player 4 uses a4/b4/b4/a4
Course of play is then 1 chooses b1, 2 chooses a2, 3 chooses a3, 4 chooses a4. Payoffs are
3,2,1,3Slide14
Another SPNE
Player 1 chooses a1
Player 2 chooses b2/b2Player 3 chooses b3
Player 4 chooses a4/b4/b4/a4
Course of play is now 1 chooses a1, 2 chooses b2. Payoffs 2,4,2,3Slide15
Mixed strategy eq in 2,3 game
a3
b3
a2
2,1
2,0
b2
0,2
3,4
2/3
1/3
1/3
2/3
Payoff to Player 1 from this equilibrium:
3(2/9)+4(4/9)+2(1/9)+0=2.66Slide16
A third SPNE
Players 1 plays b1
Player 2 plays b2 if 1 plays a1 and plays a mixed strategy of a2 with probability 2/3 and b2 probability 1/3 if 1 plays b1.
Player 3 plays a mixed strategy a2 with probability 1/3 and b2 with probability 2/3.
Player 4 plays a4/b4/b4/a4Slide17
Bayes-Nash equilibrium with Incomplete InformationSlide18
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
”. Players
Knew each others payoffs, but might not know each other’s
moves.
Slide19
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
AliceSlide20
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.Slide21
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
AliceSlide22
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.Slide23
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.Slide24
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
AliceSlide25
Bayes’ Nash equilibrium
Is there a Bayes’ Nash equilibrium where Bob goes to B and 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?
Slide26
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.Slide27
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. Slide28
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? Slide29
The game if Alice hates Bob
A
B
A
1,2
3,1
B
2,0
0,3
Bob
AliceSlide30
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.Slide31
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.Slide32
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= 3/4
So Alice would go to her favorite movie ¾ of the time
Then Bob would meet her at A with probability
¾ x ¾=9/16 and at B with probability ¼ x ¼ =1/16.Slide33
Expected payoff
In the mixed strategy equilibrium, where Alice scorns 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)=3/2Slide34
Wyatt Earp and the Gun SlingerSlide35
A Bayesian gunslinger gameSlide36
The gunfight game when the stranger is (a) a gunslinger or (b) a cowpokeSlide37
What are the strategies?
Earp
DrawWait
Stranger
Draw if Gunslinger, Draw if Cowpoke
Draw if Gunslinger, Wait if Cowpoke
Wait if Gunslinger, Draw if Cowpoke
Wait if Gunslinger, Wait if CowpokeSlide38
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=2.75
Earp’s Payoff from drawing, given these strategies for the other guys is (¾)2+(1/4) 4=2.5
So this is a Bayes Nash equilibriumSlide39
There is another equilibrium
Lets see if there is an equilibrium where everybody draws.
If Earp always draws, both cowpoke and gunslinger are better off drawing.
Let p be probability stranger is gunslinger.
If both types always draw, payoff to Earp from draw is 2p+5(1-p)=5-3p and payoff to Earp from wait is p+6(1-p)=6-5p
Now 5-3p>6-5p if p>1/2.Slide40
If Earp always draws, best response for stranger of either type is to draw.If stranger always draws, best response for
Earp is to always , whenever he thinks stranger is a gunslinger with p>1/2.
Note that this is so, even though if he knew stranger was a cowpoke, it would be dominant strategy to wait.