Game Theory

“It is true that life must be understood backward, but …

it must be lived forward.”- Søren Kierkegaard

Topic 3Sequential Games

Review Understanding the outcomes of games Sometimes easy

Dominant strategies Sometimes more challenging

“I know that you know …”

What if a game is sequential? Market entry

2Mike Shor

Very Large Airplanes:Airbus vs. Boeing Industry background

“ The problem is the monopoly of the 747 … They have a product. We have none. ” - Airbus Executive

Industry feasibility studies: Room for at most one megaseater

Airbus Initiated plans to build a super-jumbo jet

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Very Large Airplanes:Airbus vs. Boeing Boeing reaction

“Boeing, the world’s top aircraft maker, announced it was building a plane with 600 to 800 seats, the biggest and most expensive airliner ever.”

- BusinessWeek

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Sequential Games

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The Game

Airbus

Boeing – $4 billion, – $4 billion

$0.3 billion, – $3 billion

out

inout

in

– $1 billion, – $1 billion

$0, $0out

in

Looking Forward … Airbus makes the first move:

Must consider how Boeing will respond

If stay out:

Boeing stays out6

Boeing

$0 billion

– $1 billion

out

in

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Looking Forward … Airbus makes the first move:

Must consider how Boeing will respond

If enter:

Boeing accommodates, stays out7

Boeing

– $3 billion

– $4 billion

out

in

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… And Reasoning Back Now consider the first move:

Only ( In, Out ) is sequentially rational In is not credible (for Boeing)

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Airbus

Boeing

$0, $0out

inout $0.3 billion,

– $3 billion

out

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What if Boeing Can Profit?

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The Game

Airbus

Boeing – $4 billion, – $4 billion

$0.3 billion, – $3 billion

out

inout

in

– $1 billion, + $1 billion

$0, $0out

in

?

Nash Equilibria Are Deceiving

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BoeingOut In

AirbusOut 0 , 0 -1 , 1

In 0.3 , -3 -4 , -4

Two equilibria (game of chicken)

But, still only one is sequentially rational

Airbus vs. Boeing

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October 2007A380 enters commercial service Singapore to SydneyList price: $350 millionSeptember 2011Four year anniversary: 12,000,000 seats sold

Solving Sequential Games Intuitive Approach:

Start at the end and trim the tree to the present Eliminates non-credible future actions

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Solving Sequential Games Steps:

1. Pick a last move2. What player is making the decision?3. What decisions are available to that player?4. What are that player’s payoffs from each decision?5. Select the highest6. Place an arrow on the selected branch7. Delete all other branches Now, treat the next-to-last player to act as last Continue in this manner until you reach the root

Equilibrium: the “name” of each arrow

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Subgame Perfect Equilibrium Subgame:

A decision node and all nodes that follow it

Subgame Perfect Equilibrium:(a.k.a. Rollback, Backwards Induction)

The equilibrium specifies an action at every decision node in the game

The equilibrium is also an equilibrium in every subgame

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Nash Equilibria Are Deceiving

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Player 2X Y

Player 1Less 10 , 0 30 , 30More 20 , 20 40 , 10

Does either player have a dominant strategy? What is the equilibrium?

What if Player 1 goes first? What if Player 2 goes first?

Solving Sequential Games Thinking backwards is easy in game trees

Start at the end and trim the tree to the present

Thinking backwards is challenging in practice

Outline: Strategic moves in early rounds The rule of three (again) Seeing the end of the game

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Graduation SpeakerRevisited

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Graduation speakerBernie Sanders, Jeb Bush, or Hillary Clinton?

Four committee members prefer: Bernie to Jeb to Hillary ( B > J > H )

Three committee members prefer:Jeb to Hillaryto Bernie ( J > H > B)

Two committee members prefer:Hillary to Bernie to Jeb ( H > B > J )

Voting by Majority Rule

Graduation SpeakerRevisited

Graduation speakerBernie Sanders, Jeb Bush, or Hillary Clinton?

Member preferences:

4 (B>J>H) 3 (J>H>B) 2 (H>B>J) Majority rule results:

B beats J ; J beats H ; H beats B

Voting results (example): B beats J then winner versus H H

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Voting as a Sequential Game

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B vs. J

B vs. H

J vs. H

B

J

B

J

H

H

B

H

J

H

Looking Forward …

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A majority prefers H to B

A majority prefers J to H

B vs. H

J vs. H

B

J

H

H

B

H

H

J

… And Reasoning BackFour committee members prefer B to J to H.How should they vote in the first round?

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B vs. J

B vs. H

J vs. H

B

JJ

H

H

J

Sequential Rationality

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Look forward and reason back.

Anticipate what your rivals will do tomorrow

in response to your actions today

Importance of Rules

Outcome is still predetermined: B vs. J then winner versus H J vs. H then winner versus B H vs. B then winner versus J

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Accommodating a Potential Entrant

Do you enter?

Do you accommodate entry?

What if there are fifty potential entrants?

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Survivor Immunity ChallengeThere are 21 flagsPlayers alternate removing 1, 2, or 3 flagsThe player to take the last flag wins

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Unraveling

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291, 97

take take take take

grow

grow

grow grow

98, 294

297, 99

100, 300

202, 202

97 98 99 100

3, 1

take take take take

grow grow grow grow

2, 6 9, 3 4, 12

1 2 3 4

Unraveling Equilibrium:

take , take , take , take , take , … take , take , take , take , take , …

Remember: An equilibrium specifies an action at every

decision node Even those that will not be reached in equilibrium

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Sequential Games You have a monopoly market in every state There is one potential entrant in each state

They make their entry decisions sequentially Florida may enter today New York may enter tomorrow etc.

Each time, you can accommodate or fight What do you do the first year?

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The Game

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E1 out

in M fight

acc

E2

out

in fight

acc

M

E3

Looking Forward … In the last period:

No reason to fight final entrant, thus ( In, Accommodate )

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E

M

$0, $100 + previous

–50, –50 + previous

50, 50 + previousout

in acc

fight

… And Reasoning Back The Incumbent will not fight the last entrant

But then, no reason to fight the previous entrant … But then, no reason to fight the first entrant

Only one sequential equilibrium All entrants play In Incumbent plays Accommodate

But for long games, this is mostly theoretical People “see” the end two to three periods out!

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Breakfast Cereals A small sampling of the Kellogg’s portfolio

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Breakfast Cereals

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product development costs: $1.2M per product

600

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100

000

less sweet

more sweet

1 2 3 4 5 6 7 8 9 10 11

sale

s (i

n th

ousa

nds)

Breakfast Cereals

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Breakfast Cereals

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600

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less sweet

more sweet

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sale

s (i

n th

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nds)

First Product Entry

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Profit = ½ 5(600) – 1200 = 300600

500

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000

less sweet

more sweet

1 2 3 4 5 6 7 8 9 10 11

SCENARIO 1

sale

s (i

n th

ousa

nds)

Second Product Entry

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600

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100

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less sweet

more sweet

1 2 3 4 5 6 7 8 9 10 11

Profit = 2 x 300 = 600SCENARIO 2

sale

s (i

n th

ousa

nds)

Third Product Entry

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600

500

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200

100

000

less sweet

more sweet

1 2 3 4 5 6 7 8 9 10 11

Profit = 300 x 3 – 240 x 2 = 420SCENARIO 3

sale

s (i

n th

ousa

nds)

Competitor Enters

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600

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200

100

000

less sweet

more sweet

1 2 3 4 5 6 7 8 9 10 11

Profit = 300 x 2 - 240 = 360SCENARIO 4

sale

s (i

n th

ousa

nds)

Strategic Voting We saw that voting strategically rather

than honestly can change outcomes

Other examples? Amendments to make bad bills worse Crossing over in open primaries “Centrist” voting in primaries

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Strategic Voting Maybe majority rule causes this. Can we eliminate “strategic voting” with

other rules?

Ranking of all candidates Proportional representation Run offs Etc.

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Arrow’s Impossibility Theorem Consider a voting rule that satisfies:

If everyone prefers A to B, B can’t win If A beats B and C in a three-way race,

then A beats B in a two way race

The only political procedure that always guarantees the above is a dictator No voting system avoids strategic voting

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Summary Thinking forward misses opportunities

Make sure to see the game through to the logical end

Don’t expect others to see the end until it is close The rule of three steps

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