Reviewing Bayes-Nash Equilibria

Two Questions from the midterm

An Auction Question

First-bidder, sealed bid auction. Two bidders.Object goes to high bidder. If there is a tie, coinflip decides who gets it. Each knows his own value, which is either 3, 5, or 8. Each believes that other’s value is 3 with probability .3, 5 with probability .3, and 8 with probability .4.Is there a symmetric Bayes-Nash equlibrium in which each bids 2 if his value is 3, 3 if his value is 4, and 4 if his value is 8?

What is probability my bid wins?

If I bid less than 2, I never win. If I bid 2, I win only if other bidder has value 3 and I win coin toss. So probability of winning if I bid 3 is always (1/2)x.3=.15If I bid 3, I win if other bidder bids 3 or if other bidder bids 4 and I win coin toss. This happens with probability .3 +(1/2)x.3=.45.If I bid 4, I win if other bids 2 or 3 or if he bids 4 and I win coin toss. This happens with probability .3+.3+(1/2)x.4=.8If I bid more than 4, I get it for sure.

Table of winningsMy Bid Prob I get it Expected payoff:

V=3Expected payoff:V=5

Expected payoff: V=8

1 0 0 0 0

2 0.15 1x.1=0.15* 3x.15=0.45 6x.15=0.9

3 .45 0x.45=0 2x.45=0.9* 5x.45=2.25

4 0.8 -1x.8=-0.8 1x.8=0.8 4x.8=3.20*

5 1 -2x1=-2 0x1=0 3x1=3.0

Checking it out

• Look at table on previous slide. • Note that if other guy is bidding 2 when his

value is 3, 3 when his value is 4, and 4 when his value is 8, your best choices are:– 2 if 3– 3 if 5 – 4 if 8

• So there is a symmetric Bayes Nash equilibrium where each bids this way.

Another Bayes Nash equilibrium?

• Suppose each bids 2 if 3, 3 if 5 and 5 if 8.What are your probabilities of winning?Same as before with bids of 1, 2, or 3, but not the same with bids of 4 or 5.

If other bids 2 if 3, 3 if 4 and 5 if 8My Bid Prob I get

itExpected payoff: V=3

Expected payoff:V=5

Expected payoff: V=8

1 0 0 0 0

2 0.15 1x.1=0.15 * 3x.15=0.45 6x.15=.9

3 .45 0x.45=0 2x.45=0.90* 5x.45=2.25

4 .6 -1x..6= -.6 1x.6=.6 4x.6 =2.4*

5 0.8 -2x1= -2 0x.45=0 3x.8=2.4*

6 1 -3x1= -3 -1x1=-1 2x1=2

Equilibrium

• So we see from the table that if the other bids 2 when value is 3, 3 when value is 5 and 5 when value is 8, that it is a best response for you to bid 2 when your value is 3, 3 when your value is 5 and 5 when your value is 8.

• Each is doing a best response to the other’s action. So this is a symmetric Nash equilibrium.

Tough or Weak?Nature

2 is strong 2 is weak

Player 2Player 2

Player 1

Fight

Fight Fight Fight Fight

Fight

YieldYield

YieldYield

Yield Yield

100-100

Top number is payoff to Player 2. Bottom is payoff to Player 1

1000

0100

-100100

1000

0100

00

00

p 1-p

What do we specify for SPNE

• Strategy for Player 1: Yield or Fight• Strategy for Player 2 in case he is strong • Strategy for Player 2 in case he is weak.

Is there a Nash Equilbrium where Player 1 yields?

If Player 1 yields, then Fight is the best response for Player 2, whether he is strong or weak.

If Player 2 always plays Fight, is Yield the best response for Player 1?

Expected payoff for Player 1: • From Yield is 0.• From Fight is -100p+100(1-p)=100(1-2p)

• Payoff from Yield is larger than from Fight if p>1/2

Bayes-Nash equilibrium if p>1/2

Player 1 Yields.Player 2 Fights if strong and Fights if weak.

Expected payoff to Player 1 is 0.

Is there a SPNE where Player 1 Fights?

If Player 1 fights, best response for Player 2 is to Fight if he is strong and Yield if he is weak.

If Player 2 fights when he is strong and yields when he is weak, expected payoff to Player 1 from Fight is -100p+100(1-p)=100(1-2p).

Expected payoff to Player 1 from Yield is 0.Fight is best response if p<1/2.

Bayes-Nash Equilibrium if p<1/2

• Player 1 chooses Fight• Player 2 chooses Fight if he is Strong and Yield

if he is weak.

• Expected payoff to Player 1 in this equilibrium is 100(1-2p).

The game if Player 2 is strong

Fight YieldFight 100, -100 100,0Yield 0,100 0,0

Player 2

Player 1

We see that Fight is a dominant strategy for Player 2 if Player 2 is strong.So Player 1 can conclude that Player 2 will Fight if he is strong.

Hiring a Spy

• How much is it worth to Player 1 to find out whether Player 2 is strong or weak?

• If Player 1 knows whether 2 is strong or weak,Player 1 will Yield when 2 is strong and Fight

when Player 2 is weak. In this case, Player 1 will get a payoff of 0 when

2 is strong and 100 when Player 2 is weak.Expected payoff is then 0p+100(1-p)=100(1-p).

What is a spy worth?

• If p>1/2, knowing Player 2’s type increases profits from 0 to 100(1-p), so Player 1 would be willing to pay a spy up to 100(1-p).

• If p<1/2, knowing Player 2’s type would increase Player 1’s profits from 100-2p to 100-p, so Player 1 would be willing to pay a spy up to

100(1-p)- 100(1-2p)=100p.

Signaling Games

Econ 171

General form

• Two players– a sender and receiver.• Sender knows his type. Receiver does not. It is

not necessarily in the sender’s interest to tell the truth about his type.

• Sender chooses an action that receiver observes

• Receiver observes senders action, which may influence his belief about receiver’s type.

• Receiver takes action

Perfect Bayes Nash equilibrium for signaling game

• Sender’s strategy specifies an action for each type that she could be. Her action maximizes her expected payoff for that type, given the way the receiver will respond.

• For each action of the sender, receiver’s strategy specifies an action that maximizes his expected payoff.

• Receiver’s beliefs about sender’s type, conditional on actions observed are consistent.

Types of equilibria.

• Separating equilibria. Different types of senders take different actions.

• Pooling equilibria Different types of senders take same actions.

Breakfast: Beer or quiche?A Fable *

*The original Fabulists are game theorists, David Kreps and In-Koo Cho

Breakfast and the bully

• A new kid moves to town. Other kids don’t know if he is tough or weak.

• Class bully likes to beat up weak kids, but doesn’t like to fight tough kids.

• Bully gets to see what new kid eats for breakfast.

• New kid can choose either beer or quiche.

Preferences

• Tough kids get utility of 1 from beer and 0 from quiche.

• Weak kids get utility of 1 from quiche and 0 from beer.

• Bully gets payoff of 1 from fighting a weak kid, -1 from fighting a tough kid, and 0 from not fighting.

• New kid’s total utility is his utility from breakfast minus w if the bully fights him and he is weak and utility from breakfast plus s if he is strong and bully fights him.

Nature

New Kid New Kid

Tough Weak

Beer BeerQuiche

QuicheFight

Fight

Fight

Fight

Don’t

Don’t

Don’t

Don’t

B

Bully

Bully1+s-1

10

s-1

00

-w1

00

1-w1

10

How many possible strategies are there for the bully?A) 2B) 4C) 6D) 8

What are the possible strategies for bully?

Fight if quiche, Fight if beerFight if quiche, Don’t if beerFight if beer, Don’t if quicheDon’t if beer, Don’t if quiche

What are possible strategies for New Kid

• Beer if tough, Beer if weak• Beer if tough, Quiche if weak• Quiche if tough, Beer if weak• Quiche if tough, Quiche if weak

Separating equilibrium?

• Is there an equilibrium where Bully uses the strategy Fight if the New Kid has Quiche and Don’t if the new kid has Beer.

• And the new kid has Quiche if he is weak and Beer if he is strong.

• For what values of w could this be an equilibrium?

Best responses?

• If bully will fight quiche eaters and not beer drinkers:

• weak kid will get payoff of 0 if he has beer, and 1-w if he has quiche. – So weak kid will have quiche if w<1.

• Tough kid will get payoff of 1 if he has beer and s if he has quiche.– So tough kid will have beer if s<1 – Tough kid would have quiche if s>1. (explain)

Suppose w<1 and s<1

• We see that if Bully fights quiche eaters and not beer drinkers, the best responses are for the new kid to have quiche if he is weak and beer if he is strong.

• If this is the new kid’s strategy, it is a best response for Bully to fight quiche eaters and not beer drinkers.

• So the outcome where Bully uses strategy “Fight if quiche, Don’t if beer “ and where New Kid uses strategy “Quiche if weak, Beer if tough” is a Nash equilibrium.

Clicker question

• The equilibrium in which the new kid has quiche if weak and beer if tough and where the bully fights quiche-eaters but doesn’t fight beer-drinkers is

A) A separating equilibriumB) A pooling equilibriumC) Neither of these

If w>1

Then if Bully uses strategy “Fight if quiche, Don’t if beer”, what will New Kid have for breakfast if he is weak?

Pooling equilibrium?

• If w>1, is there an equilibrium in which the New Kid has beer for breakfast, whether or not he is weak.

• If everybody has beer for breakfast, what will the Bully do?

• Expected payoff from Fight if quiche, Don’t if beer depends on his belief about the probability that New Kid is tough or weak.

Payoff to Bully

• Let p be probability that new kid is tough.• If new kid always drinks beer and bully

chooses Don’t Fight if Beer, Fight if Quiche, Bullie’s payoff is 0.

• If Bully chooses a strategy that Fight if Beer, (anything) if Quiche, Bullie’s expected payoff is -1xp+1x(1-p)=1-2p. If p>1/2, Fight if Beer, Don’t fight if Quiche is a best response for Bully.

Pooling equilibrium

• If p>1/2, there is a pooling equilibrium in which the New Kid has beer even if he is weak and prefers quiche, because that way he can conceal the fact that he is weak from the Bully.

• If p>1/2, a best response for Bully is to fight the New Kid if he has quiche and not fight him if he has beer.

What if p<1/2 and w>1?

There won’t be a pure strategy equilibrium.There will be a mixed strategy equilibrium in which a weak New Kid plays a mixed strategy that makes the Bully willing to use a mixed strategy when encountering a beer drinker.

What if s>1?

• Then tough New Kid would rather fight get in a fight with the Bully than have his favorite breakfast.

• It would no longer be Nash equilibrium for Bully to fight quiche eaters and not beer drinkers, because best response for tough New Kid would be to eat quiche.

An Education Fable

• Imagine that the labor force consists of two types of workers: Able and Middling with equal proportions of each.

• Employers are not able to tell which type they are when they hire them.

• A worker is worth $1500 a month to his boss if he is Able and $1000 a month if he is Middling.

• Average worker is worth • $ ½ 1500 + ½ 1000=$1250 per month.

Competitive labor market

• The labor market is competitive and since employers can’t tell the Able from the Middling, all laborers are paid a wage equal to the productivity of an average worker: $1250 per month.

Enter Professor Drywall

Drywall claims

• My 10-lecture course raises worker productivity by 20%!

• One employer believes that Drywall’s lectures are useful and requires its workers attend 10 monthly lectures by Professor Drywall and payswages of $100 per month above the average wage.– Middling workers find Drywall’s lectures excruciatingly

dull. Each lecture is as bad as losing $20.– Able workers find them only a little dull. To them, each

lecture is as bad as losing $5.• Which laborers stay with the firm?• What happens to the average productivity of

laborers?

Other firms see what happened

• Professor Drywall shows the results of his lectures for productivity at the first firm.

• Firms decide to pay wages of about $1500 for people who have taken Drywall’s course.

• Now who will take Drywall’s course? • What will be the average productivity of

workers who take his course? Do we have an equilibrium now?

Professor Drywall responds

• Professor Drywall is not discouraged.• He claims that the problem is that people have

not heard enough lectures to learn his material.

• Firms believe him and Drywall now makes his course last for 30 hours a month.

• Firms pay almost $1500 wages for those who take his course and $1000 for those who do not.

Separating Equilibrium

• Able workers will prefer attending lectures and getting a wage of $1500, since to them the cost of attending the lectures is $5x30=$150 per month.

• Middling workers will prefer not attending lectures since they can get $1000 if they don’t attend. Their cost of attending the lectures would be $20x30=$600, leaving them with a net of $900.

So there we are.