Game Theory
Topic 3
Sequential Games
“It is true that life must be
understood backward, but …
it must be lived forward.”
- Søren Kierkegaard
Review
Understanding the outcomes of games
 Sometimes easy

 Dominant

Sometimes more challenging
 “I

strategies
know that you know …”
What if a game is sequential?
 Market
entry
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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
The Game
$0,
$0
– $1 billion,
– $1 billion
Airbus
$0.3 billion,
– $3 billion
Boeing
– $4 billion,
– $4 billion
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Looking Forward …

Airbus makes the first move:
 Must

consider how Boeing will respond
If stay out:
$0 billion
Boeing
– $1 billion

Boeing stays out
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Looking Forward …

Airbus makes the first move:
 Must

consider how Boeing will respond
– $3 billion
If enter:
Boeing
– $4 billion

Boeing accommodates, stays out
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… And Reasoning Back

Now consider the first move:
out
$0,
$0
out
$0.3 billion,
– $3 billion
Airbus
Boeing

Only ( In, Out ) is sequentially rational
 In
is not credible (for Boeing)
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What if Boeing Can Profit?
The Game
$0,
$0
– $1 billion,
+ $1 billion  ?
Airbus
$0.3 billion,
– $3 billion
Boeing
– $4 billion,
– $4 billion
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Mike Shor
Nash Equilibria Are Deceiving
Boeing
Airbus
Out
In
Out
0
, 0
0.3 , -3
In
-1 , 1
-4 , -4

Two equilibria (game of chicken)

But, still only one is sequentially rational
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Airbus vs. Boeing
October 2007
A380 enters commercial service
Singapore to Sydney
List price: $350 million
September 2011
Four year anniversary: 12,000,000 seats sold
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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.
2.
3.
4.
5.
6.
7.



Pick a last move
What player is making the decision?
What decisions are available to that player?
What are that player’s payoffs from each decision?
Select the highest
Place an arrow on the selected branch
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
Player 2
Player 1




Mike Shor
Less
More
X
10 , 0
20 , 20
Y
30 , 30
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?
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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 Speaker
Revisited
Graduation speaker
Bernie Sanders, Donald Trump, or Hillary Clinton?

Four committee members prefer:
Bernie

(B>D>H)
to Hillary
to Bernie
( D > H > B)
Two committee members prefer:
Hillary

to Hillary
Three committee members prefer:
Donald

to Donald
to Bernie
to Donald
(H>B>D)
Voting by Majority Rule
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Graduation Speaker
Revisited
Graduation speaker
Bernie Sanders, Donald Trump, or Hillary Clinton?


Member preferences:
4 (B>D>H)
3 (D>H>B)
Majority rule results:


B beats D
;
D beats H
2 (H>B>D)
;
H beats B
Voting results (example):

B beats D
then winner versus H

H
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Voting as a Sequential Game
B
B vs. H
H
B vs. D
D
D vs. H
H
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Looking Forward …
B
B vs. H
A majority prefers H to B
H
D
A majority prefers D to H
D vs. H
H
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… And Reasoning Back
Four committee members prefer B to D to H.
How should they vote in the first round?
B vs. H
H
B vs. D
D
D vs. H
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Sequential Rationality
Look forward and reason back.
Anticipate what your rivals will do
tomorrow
in response to your actions
today
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Importance of Rules

Outcome is still predetermined:



B
D
H
vs.
vs.
vs.
D
H
B
then winner versus H

then winner versus B

then winner versus D

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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 Challenge
 There
are 21 flags
 Players alternate removing 1, 2, or 3 flags
 The player to take the last flag wins
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Unraveling
1
grow
2
take
3, 1
take
3
grow
4
take
2, 6
9, 3
grow

take
4, 12
97
98
99
100
291,
97
98,
294
297,
99
100,
300
gro
w
take
Mike Shor
grow
gro
w
take
gro
w
take
gro
w
take
202,
202
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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
E3
E2
E1
M
M
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Looking Forward …

In the last period:
$0, $100 + previous
E
50,
50 + previous
M
–50, –50 + previous

No reason to fight final entrant, thus
( In, Accommodate )
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… And Reasoning Back

The Incumbent will not fight the last entrant




Only one sequential equilibrium




But then, no reason to fight the previous entrant
…
But then, no reason to fight the first entrant
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
(in thousands)
600
sales
product development costs:
$1.2M per product
100
500
400
300
200
000
1
less
sweet
2
3
4
5
6
7
8
9
10
11
more
sweet
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Breakfast Cereals
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(in thousands)
600
sales
Breakfast Cereals
100
500
400
300
200
000
1
less
sweet
2
3
4
5
6
7
8
9
10
11
more
sweet
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Mike Shor
First Product Entry
SCENARIO 1
300
(in thousands)
600
sales
Profit = ½ 5(600) – 1200 =
100
500
400
300
200
000
1
less
sweet
2
3
4
5
6
7
8
9
10
11
more
sweet
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Mike Shor
Second Product Entry
SCENARIO 2
600
(in thousands)
600
sales
Profit = 2 x 300 =
100
500
400
300
200
000
1
less
sweet
2
3
4
5
6
7
8
9
10
11
more
sweet
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Mike Shor
Third Product Entry
SCENARIO 3
420
(in thousands)
600
sales
Profit = 300 x 3 – 240 x 2 =
100
500
400
300
200
000
1
less
sweet
2
3
4
5
6
7
8
9
10
11
more
sweet
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Competitor Enters
SCENARIO 4
360
(in thousands)
600
sales
Profit = 300 x 2 - 240 =
100
500
400
300
200
000
1
less
sweet
2
3
4
5
6
7
8
9
10
11
more
sweet
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Mike Shor
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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Mike Shor
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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