a few problems

A few problems

Problem 7, Chapter 9Find the subgame perfect Nash equilbria(um)

Problem 7, Chapter 9Find 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.

One SPNE is a1/c1/d1/d1/d1 for Player 1 anda2/b2 for Player 2In this equilibrium, the outcome is 1 plays a1,2 plays a2, and then 1 plays c1. Payoffs are4 for Player 1 and 3 for Player 2.

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.

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?

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?

Problem 10, page 287

A subgame between 2 and 3

a3 b3a2 2,1 2,0b2 0,2 3,4

Does this game have pure strategy Nash equilibria?

A) Yes, there are two of themB) No there are none.C) Yes there is one of them D) Yes there are three of them.

Truncated game with a2, a3 the Nash equilibrium in subgame between 2 and 3

2423

3213

Player 1

a1b1

Truncated game if b2, b3 in subgame between 2 and 3

2423

0111

Player 1

a1b1

One SPNE

• Player 1 uses b1• Player 2 uses b2/a2• Player 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

3,2,1,3

Another SPNE

• Player 1 chooses a1• Player 2 chooses b2/b2• Player 3 chooses b3 • Player 4 chooses a4/b4/b4/a4

• Course of play is now 1 chooses a1, 2 chooses b2. Payoffs 2,4,2,3

Mixed strategy eq in 2,3 game

a3 b3a2 2,1 2,0b2 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=2.66

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/a4

Bayes-Nash equilibrium with Incomplete Information

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.

She loves me, she loves me not? (Bob moves before Alice)

Go to A Go to B

Go to A

AliceAlice

Go to B Go to A

Go to B

23

00

11

32

She loves him

Nature

She scorns him

Go to A

Go to A Go to AGo to BGo to B

Go to B

21

02

13

30

Bob

Alice

Bob

Alice

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.

She loves me, she loves me not? (Bob moves before Alice)

Go to A Go to B

Go to A

AliceAlice

Go to B Go to A

Go to B

23

00

11

32

She loves him

Nature

She scorns him

Go to A

Go to A Go to AGo to BGo to B

Go to B

21

02

13

30

Bob

Alice

Bob

Alice

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.

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.

Does she or doesn’t she?Simultaneous Play

Go to A Go to B

Go to A

AliceAlice

Go to B Go to A

Go to B

23

00

11

32

She loves him

Nature

She scorns him

Go to A

Go to A Go to AGo to BGo to B

Go to B

21

02

13

30

Bob

Alice

Bob

Alice

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?

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 is3p+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.

Is there a Bayes-Nash equilibrium in pure strategies if p<1/4?

A) Yes, Alice goes to B if she loves Bob and A if she scorns him and Bob goes to B.

B) Yes, Alice goes to A if she loves Bob and B if she scorns him and Bob goes to B.

C) Yes there is one, where Alice always goes to A.

D) No there is no Bayes-Nash equilibrium in pure strategies.

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?

The game if Alice hates Bob

A BA 1,2 3,1B 2,0 0,3

Bob

Alice

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.

Making Alice indifferent

• If Bob goes to B with probabilty b:– Expected payoff to Alice from going to AIs 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.

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.

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)=3/2

Wyatt Earp and the Gun Slinger

A Bayesian gunslinger game

The gunfight game when the stranger is (a) a gunslinger or (b) a cowpoke

What are the strategies?

• Earp– Draw– Wait

• Stranger– Draw if Gunslinger, Draw if Cowpoke– Draw if Gunslinger, Wait if Cowpoke– Wait if Gunslinger, Draw if Cowpoke– Wait if Gunslinger, Wait if Cowpoke

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 equilibrium

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.

• 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.