More game theory

Today: Some classic games in game theory

Last time…

Introduction to game theory Games have players, strategies, and

payoffs Based on a payoff matrix with

simultaneous decisions, we can find Nash equilibria (NE)

In sequential games, some NE can be ruled out if people are rational

Today, some classic game theory games Games with inefficient equilibria

Prisoner’s Dilemma Public Goods game

Coordination games Battle of the Sexes Chicken

Zero-sum game Matching pennies

Animal behavior Subordinate pig/Dominant pig

Prisoner’s dilemma Why is this

game called prisoner’s dilemma?

Think about a pair of criminals that have a choice of whether or not to confess to a crime

Yes No

Yes –1, –1 +3, –6

No –6, +3 +1, +1

Player 1

Player 2

Prisoner’s dilemma What is the

NE? Let’s

underline

Yes No

Yes –1, –1 +3, –6

No –6, +3 +1, +1

Player 1

Player 2

Prisoner’s dilemma What is the

NE? Let’s

underline Each player

has a dominant strategy of choosing Yes

However, both players get a better payout if each chooses No

Yes No

Yes –1, –1 +3, –6

No –6, +3 +1, +1

Player 1

Player 2

Prisoner’s dilemma and cartels

Cartels are usually unstable since each firm has a dominant strategy to charge a lower price and sell more

See Table 11.4 (p. 327) for an example

Public goods game

You can decide whether or not you want to contribute to a new flower garden at a local park If you decide Yes, you will lose $200, but

every other person in the city you live in will gain $10 in benefits from the park

If you decide No, you will cause no change to the outcome of you or other people

Public goods game

What is each person’s best response, given the decision of others?

We need to look at each person’s marginal gain and loss (if any) Choose yes Gain $10, lose $200 Choose no Gain $0, lose $0

Public goods game Which is the better choice?

Choose no (Gain nothing vs. net loss of $190)

NE has everybody choosing no Efficient outcome has everybody

choosing yes Why the difference?

Each person does not account for others’ benefits when making their own decision

Battle of the Sexes

Two people plan a date, and each knows that the date is either at the bar or a play

Neither person knows where the other is going until each person shows up

If both people show up at the same place, they enjoy each other’s company (+1 for each)

Bar Play

Bar +3, +1

+0, +0

Play

+0, +0

+1, +3

Player 1

Player 2

Battle of the Sexes:Other things to note

Player 1 gets additional enjoyment from the bar if Player 2 is there too, since Player 1 likes the bar more

Player 2 enjoys the play more than Player 1 if both show up there

As before, we underline the best strategy, given the strategy of the other player

Bar Play

Bar +3, +1

+0, +0

Play

+0, +0

+1, +3

Player 1

Player 2

Battle of the Sexes Two NE

(Bar, Bar) (Play, Play)

As in cases before when there are multiple NE, we cannot determine which outcome will actually occur

Bar Play

Bar +3, +1

+0, +0

Play

+0, +0

+1, +3

Player 1

Player 2

Battle of the Sexes Battle of the

Sexes is known as a coordination game Both get a

positive payout if they show up to the same place

Bar Play

Bar +3, +1

+0, +0

Play

+0, +0

+1, +3

Player 1

Player 2

Chicken

Two cars drive toward each other If neither car swerves, both drivers

sustain damage to themselves and their cars

If only one person swerves, this person is known forever more as “Chicken”

Chicken

Next step: Underline as before

Swerve

Straight

Swerve +0, +0

–1, +1

Straight +1, –1 –10, –10

Player 1

Player 2

Chicken

Notice there are 2 NE One player swerves and the other goes straight

This game is sometimes referred to as an “anti-coordination” game

NE results from each player making a different decision

Swerve

Straight

Swerve +0, +0

–1, +1

Straight +1, –1 –10, –10

Player 1

Player 2

Matching pennies

Two players each choose Heads or Tails

If both choices match, Player 1 wins

If both choices differ, Player 2 wins

This is an example of a zero-sum game, since the sum of each box is zero

Heads Tails

Heads +1, –1 –1, +1

Tails –1, +1 +1, –1

Player 1

Player 2

Matching pennies

Underlining shows no NE

A characteristic of zero-sum games Whenever I win,

the other player must lose

Heads Tails

Heads +1, –1 –1, +1

Tails –1, +1 +1, –1

Player 1

Player 2

Subordinate pig/Dominant pig

Two pigs are placed in a cage Left end of cage: Lever to release

food 12 units of food released when lever

is pressed Right end of cage: Food is

dispensed here

Subordinate pig/Dominant pig If both press lever at the same time, the

subordinate pig can run faster and eat 4 units of food before the dominant pig “hogs” the rest

If only the dominant pig presses the lever, the subordinate pig eats 10 of the 12 units of food

If only the subordinate pig presses the lever, the dominant pig eats all 12 units

Pressing the lever exerts a unit of food

Subordinate pig/Dominant pig

Who do you think will get more food in equilibrium? Who thinks ?

Who thinks ?

Subordinate pig/Dominant pig

Next: Underline test

The numbers on the previous slide translate to the payoff matrix seenYes No

Yes 3, 7 –1, 12

No 10, 1 0, 0

sub

ord

inate

pig

dominant pig

Subordinate pig/Dominant pig

Exactly 1 NE The dominant

pig presses lever

In Nash equilibrium, the dominant pig always gets the lower payout

Why? The subordinate pig has a

dominant strategy: No The dominant pig, knowing

that the subordinate pig will not press the lever, will want to press the lever

Yes No

Yes 3, 7 –1, 12

No 10, 1 0, 0

sub

ord

inate

pig

dominant pig

Do people always play as Nash equilibrium predicts?

No Many papers have shown that people

often are not selfish, and donate into public goods

Norms are often established to make sure that people are encouraged to act in the best interest of society

Summary

Today, we looked at some well-known games

Some games have NE; others do not

However, people do not always behave as NE would predict