# Game Theory

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```					                    Game Theory

“Life must be understood backward,
but … it must be lived forward.”
- Soren Kierkegaard

Mike Shor
Lecture 4
Review
   Recognize dominant and
dominated strategies
   Dominant strategies are always played
   Dominated strategies are never played

   Equilibrium: likely outcome of a game
• Dominance solvable
• Iterated deletion of dominated strategies

Game Theory - Mike Shor           2
Cell-by-Cell Inspection
   Games of Assurance
• The “good” equilibrium is risky
• Need assurances
   Games of Coordination
• Failure to agree leads to no profits
• Can’t agree on “what to agree on”
   Games of Chicken

Game Theory - Mike Shor     3
Games of Chicken
   A monopolist faces a potential entrant
   Monopolist can accommodate or fight
   Potential entrant can enter or stay out

Monopolist
Accommodate               Fight
Potential
Entrant

In      50 , 50               -50 , -50
Out      0 , 100                0 , 100

Game Theory - Mike Shor          4
Equilibrium
to find equilibria
Monopolist
Accommodate                  Fight
Potential
Entrant

In      50 , 50                  -50 , -50
Out      0 , 100                   0 , 100

Game Theory - Mike Shor               5
Importance of Order
   Two equilibria exist
• ( In, Accommodate )
• ( Out, Fight )
   Only one makes temporal sense
• Fight is a threat, but not credible
• Not sequentially rational
   Simultaneous outcomes may not
make sense for sequential games.

Game Theory - Mike Shor    6
Sequential Games
The Extensive Form

0 , 100
E
-50 , -50
M

50 , 50

Game Theory - Mike Shor        7
Looking Forward…
   Entrant makes the first move:
• Must consider how monopolist will respond
   If enter:                         -50 , -50
M

50 , 50

   Monopolist accommodates

Game Theory - Mike Shor         8
… And Reasoning Back
   Now consider entrant’s move
0 , 100
E

M
acc
50 , 50

   Only ( In, Accommodate ) is
sequentially rational
Game Theory - Mike Shor     9
Sequential Rationality

COMMANDMENT
Look forward and reason back.

Anticipate what your rivals will do
tomorrow
today

Game Theory - Mike Shor    10
Solving Sequential Games
 Determine what that player will do
 Trim the tree
• Eliminate the dominated strategies
 This results in a simpler game
 Repeat the procedure

Game Theory - Mike Shor    11
Voting Revisited
   Majority rule results:
• B beats G   ;    G beats R            ;   R beats B
   What if you want “Remand” to Win?
• B vs. G then winner vs. R                    R
   Problem:
• Everyone knows you want “R”
B vs. G then winner vs. R? Good Luck!
• Better chance:
R vs. G, then winner versus B

Game Theory - Mike Shor                   12
Extensive Form
B
B vs. R

R vs. G                               R

B

B vs. G
G
Game Theory - Mike Shor       13
Looking Forward
B
B vs. R
A majority prefers R to B

R

B

A majority prefers B to G
B vs. G
G
Game Theory - Mike Shor             14
Trim The Tree

B vs. R

R vs. G                               R

B

B vs. G

Game Theory - Mike Shor       15
Reasoning Back
   First stage, in effect vote between R & B
   Gore supporters prefer G>R>B, vote R

B vs. R

R vs. G                               R

B

B vs. G
Game Theory - Mike Shor       16
What Happened?
   Gore supporters have preferences
•Gore > Remand > Bush
   In first round, vote between R and G
   Gore supporters prefer Gore
But vote for G is in effect a vote for B!
   So Gore supporters vote for remand.
Guarantee themselves second best choice

Game Theory - Mike Shor   17
Rollback in Voting and
“Being Political”
   Not necessarily good to vote
   Amendments to make bad bills worse
   Crossing over in open primaries
   “Centrist” voting in primaries
   STILL – Outcome predetermined
   AGENDA SETTING!

Game Theory - Mike Shor   18
Predatory Pricing
   An incumbent firm operates in three
markets, and faces entry in each
• Market 1 in year 1, Market 2 in year 2, etc.

   Each time, I can slash prices, or
accommodate the new entry
   What should I do the first year?

Game Theory - Mike Shor        19
Predatory Pricing

E2                         E3

E1

M                   M

Game Theory - Mike Shor        20
Predatory Pricing
    The end of the tree: year 3
0 , 100 + previous
E3
-50 , -50 + previous
M

50 , 50 + previous
    In year 3: ( In, Accommodate )
Game Theory - Mike Shor         21
Rollback
   Trim the tree:
E3          M acc
in                0 , 100 + previous

E2
in      acc
-50 , -50 + previous
M

in     acc
50 , 50 + previous
Game Theory - Mike Shor              22
Predatory Pricing
 Since the Incumbent will not fight
Entrant 3, he will not fight Entrant 2
 Same for Entrant 1
 Only one “Rollback Equilibrium”
• All entrants play In
• Incumbent plays Accommodate
   Why do we see predatory pricing?

Game Theory - Mike Shor   23
Game Theory

Example

Sequential Entry
Market Opportunity Analysis

“Assesses the potential of a
geographic market for a specific set
of products, providing a
prioritization of market coverage
voids and recommending market
entry strategies.”

Game Theory - Mike Shor   25
“Market Analysis”

700
SBCs
400
SBCs

Game Theory - Mike Shor      26
Extended Market Analysis
   If enter 400 SBC market:
 Next entrant, to break even,
must expect market share of 300/400
 Must expect market share of 75%

   If enter 700 SBC market:
 Next entrant, to break even,
must expect market share of 300/700
 Must expect market share of 43%

   Real decision:
   All of 400 or half of 700
Game Theory - Mike Shor   27
Breakfast Cereals
vertical axis:
sales (in thousands)
600

500
product development costs:
\$1.2M per product
400

300

200

100

000

1   2   3   4      5    6    7    8       9    10     11
less                                           more
sweet                                          sweet
Game Theory - Mike Shor                    28
1
First Product Entry
Profit = ½ bh – F

600
= ½ 5(600) – 1200
500                                   = 1500 – 1200
400
= 300
300

200

100

000

1      2   3   4     5    6    7    8    9    10     11

less                                          more
sweet                                         sweet
Game Theory - Mike Shor                    29
2
Second Product Entry
Profit = 2 x 300

600
= 600
500

400

300

200

100

000

1      2    3   4     5    6    7    8    9    10     11

less                                           more
sweet                                          sweet
Game Theory - Mike Shor                    30
3
Third Product Entry
Profit = 600 + 1500 – 480 – 1200

600
= 420
500

400

300

200

100

000

1      2    3   4     5    6    7    8    9    10     11

less                                           more
sweet                                          sweet
Game Theory - Mike Shor                    31
4
Competitor Enters
Profit = 600 - 240

600
= 360
500

400

300

200

100

000

1      2    3   4     5    6    7    8    9    10     11

less                                           more
sweet                                          sweet
Game Theory - Mike Shor                    32

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