Todd and Steven - PowerPoint Presentation

Slide1

Todd and StevenDivide the EstateProblem 8.10Slide2

Bargaining over 100 pounds of goldRound 1: Todd makes offer of Division.

Steven accepts or rejects.

Round 2: If Steven rejects, estate is reduced to 100d pounds. Steven makes a new offer and Todd accepts or rejects.

Round 3: If Todd rejects, estate is reduced to 100d

2

pounds. Todd makes new offer and Steven accepts or rejects. If Steven rejects, both get zero. Slide3

Working backwards for SPNEIn last subgame, Steven must either accept or reject Todd’s offer. If he rejects, both get 0. If he accepts, he gets what Todd offered him.

If Todd offers any small positive amount

ε

, Steven’s best reply is to accept.

So in next to last

subgame

, Todd would offer Steven

ε

and take

100

d

2

-ε for himself.Slide4

Part of game tree

Steven

Todd

Todd

Todd

Steven

Steven

Propose

Propose

Propose

Accept

Accept

Accept

Reject

Reject

RejectSlide5

Back one more stepAt node where Steven has offered Todd a division, there are 100d units to divide. Todd would accept 100d

2

or more, would reject less.

So at previous node Steven would offer Todd 100

d

2

and would have 100(d-

d

2) for himself.Slide6

Back once moreNow consider the subgame where Todd makes his first proposal.

At this point there are 100 pounds of gold to divide.

Todd sees that Steven would accept anything greater than

100(d-d

2

).

So Todd would offer Steven 100(d-

d

2)+ε

and keep 100(1-d+d2)-ε for himself.Slide7

SPNE Equilibrium strategy for ToddFirst node: Offer Steven 100(d-d

2

)+

ε

Second node: If Steven rejects Todd’s offer and makes a counteroffer to Todd: Accept

100d

2

or more, reject less.

Third node: If Todd rejects Steven’s counter offer, make a new offer to Steven of a small ε.Slide8

SPNE strategy for StevenFirst node: Accept any offer greater than

100(d-d

2

), reject smaller offers.

Second node: If Steven rejects Todd’s first offer, then offer Todd

100d

2

Third node: Accept any positive offer. Slide9

PayoffsSuppose d =.9, then 100(1-d+d

2

)=91

If d=.5,

100(1-d+d

2

)= 75

In fact,

100(1-d+d2

) is minimized at d=.5.What happens with more rounds of bargaining? Slide10

Bayes-Nash equilibrium with Incomplete InformationSlide11

An oil lease auction Slide12

The set-upYou own an oil company. A new field has come up for lease.

There are two bidders. You and another firm.

Each of you has explored half of the oil field and knows the value of the half they explored.

The value of each side is either $3 million or 0,

which depended on the flip of a fair coin.

Total value of field is the sum of the two sides

You know what your side is worth, but not the other company’s side. Slide13

The AuctionThe lease for the entire field is up for auction. A bid must be an integer number possibly 0) of million $.

There are two bidders, you and the company that explored the other side.

You know what your side is worth.

Entire field will be leased to the higher bidder in a sealed bid auction. If there are tie bids, winner is chosen by coin flip.

If you win the auction, your profit or loss is the value of the total field minus your bid.Slide14

A strategyA strategy states the amount you will bid if your side is worth $0 and the amount you will bid if your side is worth $3 million.Slide15

What would you bid if your side is worth $0?A) $0

B) $1 million

C) $2 million

D) $3 million

E) $4 millionSlide16

What would you bid if your side is worth $3 million?A) $1 million

B) $2 million

C) $3 million

D) $4 million

E) $5 millionSlide17

Some things to think aboutWhat would be your expected profit if the company you bid against uses the same profit that you do?

If your side is worth $0 and you win the auction, what do you expect the total oilfield to be worth?Slide18

Finding a symmetric Bayes-Nash equilibriumSuppose both players bid higher when they see $3 million than when they see $0.

What is the best bid if you see $0?Slide19

Is (0,4) a symmetric Bayes-Nash equilibrium? suppose other guy bids 0 when he sees 0 and $4 million when he see $3 million on his own side.

My expected profit from the (0,4) strategy would be ½ x0+ ½(-1 x

½

+2 x

½)=1/4.

If instead I played (0,1), my expected payoff would

be ½ x0+ ½(0 x ½ +2 x ½)

=1/2

So (0,4) is not a symmetric Bayes-Nash equilibrium. Slide20

Is (0,3) a symmetric Bayes-N.E?If other guy is playing (0,3), the my expected payoff from (0,3) is

½x0 +½ (0+½ x½x3)=3/8.

If I play (0,1), my expected payoff is

½x0+ ½(0 x ½ +2 x ½) =1/2

Slide21

Alice and Bob RevisitedSlide22

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

AliceSlide23

What are their strategies?For BobGo to A

Go to B

Alice has four information sets.

I Love him and he’s at A

I Love him and he’s at B

I Scorn him and he’s at A

I Scorn him and he’s at B

In each

information set, she can go to either A or B. This gives her 2x2x2x2=16 possible strategies.

Slide24

A weakly dominant strategy for AliceGo to A if you love he goes to A. Go to B if you love him and he goes B. Go to B if you scorn him and he goes to A. Go to A if you scorn him and he goes to B. (We write this as A/B/B/A)

This is weakly dominant but not strictly dominant. Explain.

Let’s look for a

subgame

perfect Nash equilibrium where Alice goes A/B/B/ASlide25

Checking equilibriumSuppose Alice goes where Bob is if she loves him and goes where he is not if she scorns him. (A/B/B/A)

Payoff to Bob from A is 2p. Payoff from B is

3p+1(1-p)=2p+1. Since 2p+1>2p, for all p>=0, B is his best response to (A/B/B/A).

Also A/B/B/A is a best response for Alice to Bob’s B.

So we have a Bayes-Nash equilibrium.Slide26

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

AliceSlide27

Alice’s (pure) strategies

Alice doesn’t know what Bob did, so she can’t make her action depend on his choice. She can go to either A or B.

She does know whether she loves him or scorns him when she chooses.

She has 4 possible strategies

A if love, A if scorn

A if love, B if scorn

B if love, A if scorn

B if love, B if scornSlide28

Bayes’ Nash equilibriumIs 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?

Slide29

Bob’s CalculationsIf 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.Slide30

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. Slide31

If p<1/4We showed there is no pure strategy Bayes-Nash equilibrium where Bob goes to B.

What if Bob goes to A?

The only possible Nash equilibrium would have Alice go to A if she loves him and B if she scorns him.

Then payoff to Bob from A would be 2p+0 and payoff from B would be 3p+1(1-p)=2p+1>2p.

So if p<1/4, there can’t be a pure strategy Bayes-Nash equilibrium where Bob goes to A.Slide32

Mixed strategy equilbrium: Bob the stalker

If Bob thinks it likely that Alice scorns 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.

Let’s look for a mixed strategy for Bob such that Alice, if she scorns Bob would be indifferent between Movies A and B.Slide33

What about a mixed strategy equilibrium?If p<1/4, can we find a mixed strategy for Bob such that Alice is indifferent

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? Slide34

The game if Alice hates Bob

A

B

A

1,2

3,1

B

2,0

0,3

Bob

AliceSlide35

Making Scornful Alice indifferentIf Bob goes to Movie A with probability q and Alice Scorns Bob:Alice’s payoff from A is 1q+3(1-q) =3-2q

Alice’s payoff from B is 2q+0(1-q)=2q

Alice will be indifferent if 3-2q=2q, which implies q=3/4.Slide36

When will Bob do a mixed strategy?Note that if Bob goes to A with probability ¾, and if Alice loves him, her best response is to go to Movie A.

If there is an equilibrium where Bob uses a mixed strategy, he must be indifferent between going to A and going to B.

Can we find a mixed strategy for Alice to use if she scorns him so that Bob will be indifferent between A and B?Slide37

Making Bob indifferentLet r be the probability that Alice goes to Movie A if she scorns Bob and suppose that Alice always goes to A if she loves Bob.

Expected payoffs for Bob are

If he goes to A, 2p+(1-p)(2r+0(1-r))=2p-2pr+2r

If he goes to B, 1p+(1-p)(1r+3(1-r))=2p+3+2pr-2r

He is indifferent between A and B if these are equal.

This implies r=(3-4p)/4-4p.

Now r is between 0 and 1 if and only if p<=3/4Slide38

Summing upWe previously found that if p>=1/4, there is a Bayes-Nash equilibrium in which Bob goes to B and Alice goes to B if she loves him and A if she scorns him.

Now we found that whenever p<=3/4, there is a mixed strategy Bayes-Nash equilibrium in which Bob goes to movie A with probability ¾, Alice goes to Movie A if she loves Bob and she goes to movie A with probability r=(3-4p)/(4-4p) if she scorns him.

So over the range of p between ¼ and ¾, there are two distinct Bayes-Nash

equilibria

. Slide39

Maybe, later?