041913

Game Theory: Refinements

April 19, 2013

Announcements

• Homework 3 is due before class on Monday• Next week we’ll begin talking about

externalities

Last Class

• Considered more example of Nash equilibria and strictly dominant strategies

• Played a game: Write down ½ of the average of others’ guesses. Choose from [0,100].– Nash equilibrium: Everyone writes down zero

How did we solve this problem?

• If everyone guessed 100, the best number to write down (best response!) would be 50.

• But if everyone wrote 50, the BR is 25.• If all 25, BR is 12.5, … etc• If all 0, BR is 0. Only Nash Equilibrium• This iterative process for finding equilibrium is

a ``refinement’’ of Nash equilibrium.

Refinements

• Extensions of the basic version of Nash Equilibrium we have seen.

• The game we played demonstrates a concept called iterated dominance.

• Another refinement has to do with the timing of the game; called sequential rationality.

Games in Which Timing MattersOpening a New Restaurant

Bill

Dinner Breakfast

AliceDinner $1000 (A)

$1000 (B)$1600 (A)$1400 (B)

Breakfast $1400 (A) $1600 (B)

$800 (A)$800 (B)

Alice and Bill are each considering opening a restaurant in their local neighborhood . . . But what kind?

Decision Tree & Backward Induction

Bill Decides

Alice Decides

Outcome

D

D

DB

B

B

$1000 (A)$1000 (B)

$1600 (A)$1400 (B)

$1400 (A)$1600 (B)

$800 (A)$800 (B)

A. To Bed, TantrumB. To Bed, Not TantrumC. Not to Bed, TantrumD. Not to Bed, Not Tantrum

To Bed

Not To Bed

Tantrum

Not Tantrum

Tantrum

Not Tantrum

-10

-10

+ 10

-5

-15

0

-5

+ 10

Games in Which Timing MattersOpening a Convenience Store

Freeway

1 mile1,200 people

A B

C⅓ mile400 people

⅓ mile400 people

⅓ mile400 people

1 mile1,200 people

Store A opens first . . . Where should the owner of a second convenience store choose to locate?

Monopolistic competition

Tendency for Monopolistic Competitors to Cluster

Hotelling’s (1929) insights help explain• Why convenience stores and gas stations are often clustered

on the same corner.• Why political candidates often “moderates”.• Why Coca-Cola and Pepsi are so similar.• Why Kellogg’s and Post both have purple raisin bran boxes.

Commitment Problems

• In many of the games we’ve looked at players cannot achieve the desired outcome because of a commitment problem—that is they cannot make credible threats or promises about their future actions.

• Sometimes, it’s possible to adopt commitment devices

‒ Omerta: anyone who testifies against a fellow mob member is killed.

‒ Legally binding contracts‒ Commissions to salespeople

Commitment Problems• Sometimes we want to commit ourselves . . .

– Bring a fixed amount of cash when you go out.– Leave your laptop at work so you don’t work from home.– Brush your teeth to avoid snacking later.– Cut up your credit cards.

Commitment Problems

• Sometimes psychological incentives can solve commitment problems– Guilt– Sympathy towards others

Stranger

A

Give $

Keep the $

Don’t Give $

Buy Tacos

-20 for you20 for the stranger

5 for you10 for stranger

0 for you0 for stranger

You’re hungry but too busy to go buy lunch, so you give a stranger $20 to go buy you two fish tacos (which you value at $25).

A. Give, KeepB. Give, BuyC. Don’t give

Stranger

A

Give $

Keep the $

Don’t Give $

Buy Tacos

-20 for you-10 for your friend

5 for you10 for your friend

0 for you0 for your friend

You’re hungry but too busy to go buy lunch, so you give your best friend $20 to go buy you two fish tacos (which you value at $25).

A. Give, KeepB. Give, BuyC. Don’t Give

Lesson

• Having a trustworthy trading partner can make both parties better off.

• The hard part is identifying people who you can trust.

• Reputation matters.

Splitting a cake

Donnie

Marie

50/50

>.5

Big/Small

<.5

0.5 (D)0.5 (M)

<0.5 (D)>0.5 (M)

>0.5 (D)<0.5 (M)