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More Game Theory

Rob Seamans & Richard Wang

MBA 299: Strategy

April 18th, 2008

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More game theory

Warm Up Repeated games Sequential Games Signaling games

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Application of Prisoner’s Dilemma: Oil Production

$70M,$60M $180M,$50M

$160M,$100M$60M,$120M

Increase Production

RestrictOutput

Increase Production Restrict Output

Saudi Arabia

Iran

Nash Equilibrium:

Given what the other player is doing, you can’t do better by deviating

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More game theory

Warm Up Repeated games Sequential Games Signaling games

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What about repeated games?What happens if Coke and Pepsi play a price war, given the payoffs below?

0,0 4,-3

1,1-3,4

Fight

Accommodate

Fight Accommodate

Coke

Pepsi

What happens if Coke and Pepsi play a price war again and again?

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Thinking about price wars

Why do price wars start?

How do you credibly signal commitment to fight forever?

Why do price wars stop?

PV (nice payoffs) > PV (bitter competition)

What role will the future play in determining the outcome?

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Equilibrium in repeated play Consider the strategy: If you fought in the previous round I will fight forever . . . If you accommodated in the previous round I will accommodate until you fight

Assume no discounting for simplicity, then you get:

4+0+0+0+0 . . . . =4 if choose “fight” (call this “strategy f”, or Sf)

1+1+1+1+1 . . . . =n if choose “accommodate” (call this “strategy a”, or Sa)

What would you do if there were three periods? What would you do if there were five periods? (keep this answer in mind…)

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Infinitely repeated game with discounting Assumptions:

Receive x each period Discount factor is = 1/[1+r] < 1

Solve for NPV in terms of x and delta:NPV=x+x+2x+3x+…=SS=x+2x+3x+4x+…S-S=x S=x/(1- )

Compare payoffs between two strategies:For example, from last slide, x=1:Let Sa=1/(1- ) and Sf =4When is Sa>Sf? When is Sa<Sf?When does Sa=Sf? Set 1/(1- ) =4 =3/4>3/4 Accommodate; <3/4 Fight

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The Folk Theorem Result As long as delta is big enough (players are sufficiently patient) any outcome can be supported.

This is a nice result because it tells us we can support cooperation with trigger strategies in long-period repeated games where players are patient as long as neither player knows when the game will end.

Doesn’t rely exclusively on “doom triggers” but punishments must be “big”.

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Finitely repeated game

What happens if it’s not an infinite game? Would you ever cooperate? What if you do know when the game will end? What if you don’t know when the game will end?

We can use backward induction to solve the outcome for finitely repeated games.

So why might you not want to announce you intend to exit after a certain point?

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More game theory

Warm Up Repeated games Sequential Games Signaling games

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Sequential games In sequential games, players move in a pre-determined

order, and might observe moves of other players that happen before they move

This type of game is useful in developing predictions in situations where one firm moves first and others follow RyanAir vs. BA/AL on pricing

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Example: Capacity Expansion and Entry An established manufacturer is facing possible

competition from a rival

The established manufacturer can try to stave off entry by engaging in costly capacity expansion, which increases supply and lowers price charged to customers

Rival can observe whether incumbent expands capacity or not before deciding on entry

Example: Nutrasweet vs. HSC on capacities

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Incumbent decides to expand or not, then rival decides whether to enter

I

R

R

1,1

3,2

2,4

4,2

Expand

Do notexpand

Enter

Don’t Enter

Enter

Don’t Enter

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Game is solved using Backward Induction

Look to the end of the game tree and prune back (similar to working backwards through a decision tree)

Rationality assumption implies that players choose the strategy at each node that yields highest expected payoff

There’s no incomplete information in this game, so there’s no uncertainty in the prediction

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What will rival do?

I

R

R

1,1

3,2

2,4

4,2

Expand

Do notexpand

Enter

Don’t Enter

Enter

Don’t Enter

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Rival’s Choice

I

R

R

1,1

3,2

2,4

4,2

Expand

Do notexpand

Enter

Don’t Enter

Enter

Don’t Enter

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What will the Incumbent do?

I

R

R

1,1

3,2

2,4

4,2

Expand

Do notexpand

Enter

Don’t Enter

Enter

Don’t Enter

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Incumbent’s Choice

I

R

R

1,1

3,2

2,4

4,2

Expand

Do notexpand

Enter

Don’t Enter

Enter

Don’t Enter

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Equilibrium Prediction

The prediction from this model is that the incumbent will expand capacity and this will effectively forestall entry

Notice that even in absence of actual entry, the potential competition from the rival eats into the incumbent’s profits.

By thinking dynamically, game theory allows a refinement of the typical economics monopoly prediction of production quantity at MR=MC

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Sequence of Play is Important

The preceding game assumed rival could move at the last moment, after seeing incumbent’s decision

Suppose that the rival must commit to enter or not before the capacity expansion decision of the incumbent

How would this affect the outcome of the game?

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Game Tree – Rival moves first, same payoffs

R

I

I

1,1

4,2

2,3

2,4

Enter

Don’t Enter

Expand

Not expand

Expand

Not expand

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Game Tree – Rival moves first, same payoffs

R

I

I

1,1

4,2

2,3

2,4

Enter

Don’t Enter

Expand

Not expand

Expand

Not expand

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Equilibrium Prediction

Game sequence can affect decision outcomes and payoffs!

This game has a first-mover advantage

Is it always true in sequential move games that there is a first-mover advantage?

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More game theory

Warm Up Repeated games Sequential Games Signaling games

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Signaling game set-up Imagine there are two types of people in the world, but that type is private information know only to the individual:

people who are good at business but not at art (business people) people who are good at art but not at business (artists)

Employers want to hire business people not artists

Employers pay very well such that artists would like to have business jobs

How should employers find business people?

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How do employers find business people? One solution is to ask people, “are you a business person or an artist?”

What will business people say? What will artists say?

Is there a signal business people can send? What if wearing a suit signals that one is a business

person? What will artists do? Is there a credible signal business people can send

that businesses will believe?

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Credible signals

A signal is credible if it is costly enough such that artists will not want to invest in signaling

One potential credible signal is going to business school

Interestingly the signal works even if business school does not affect business people’s productivity

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Signaling game set-up Assumptions:

½ the people in the world are business people and ½ are artists

Business people are worth $5 to employers, while artists are worth $4*

Only enough business jobs for ½ the people in the world

Employers pay $3 to anyone hired, regardless of type (since unobservable)

Business school is free. However, it costs $1 of effort from business people and $2 of effort from artists (artists dislike business school)

School does not change the productive capacity of the people

*Feel free to multiply these numbers by 100K to make more realistic!

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Signaling game

0,0

0,0

Business people

Artists

B School No B School

B School No B School

3,2

3,1

-1,0

-2,0

2,2

1,1

50%

50%

Nature

H

N

H

N

H

N

H

N

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Signaling game - Equilibrium

0,0

0,0

Business people

Artists

B School No B School

B School No B School

3,2

3,1

-1,0

-2,0

2,2

1,1

50%

50%

Nature

H

N

H

N

H

N

H

N

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Signaling game – No incentive to deviate by the business people

0,0

0,0

Business people

Artists

B School No B School

B School No B School

3,2

3,1

-1,0

-2,0

2,2

1,1

50%

50%

Nature

H

N

H

N

H

N

H

N

EV=(3+0)/2=1.5

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Signaling game – No incentive to deviate by the artists

0,0

0,0

Business people

Artists

B School No B School

B School No B School

3,2

3,1

-1,0

-2,0

2,2

1,1

50%

50%

Nature

H

N

H

N

H

N

H

N

EV = (1-2)/2 = -0.5

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Equilibrium Business people go to business school, artists do not and employers only hire business school graduates

This is the only equilibrium in this game, no one can do better by changing their strategy given what other players do

Note that if everyone goes to school the expected value for artists is -½.

Note the role of business school in this game Business people don’t learn anything useful in business

school in this set up. However business school is still a socially useful institution

since it allows business people to send credible signals to potential employers.

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A few comments A signal is only valuable if it is credible

Credible signals must be costly to send

This game is actually more complicated than what has been laid out here . . . If you want more take John Morgan’s game theory class

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Application of Game Theory In a prisoner’s dilemma set up, everyone loses. How can firms get away from this bad outcome?

How might the following game theory concepts help your team in the CSG?

Repeated play Sequential moves Signaling

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Next week

Oligopoly Games Value Net Transaction Cost Economics