Strategic Form Representation

1
Strategic Form Representation Simultaneous move
games Static games
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Strategic Uncertainty Example Two cars arrive
simultaneously at an intersection without a
traffic light. Problem Guessing whether the
other driver will not stop to decide whether I
will not stop. Guessing how the other driver
will guess my behavior to determine his/her
behavior, then I can make a decision. Guessing
how the other driver will guess my guess to make
his/her decision, and then I can make me
decision. An infinite ring of guessing!!!
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Representation Abstract 1. Who are
playing? 2. Possible strategies 3. Once actions
are determined, what are the outcomes?
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Some 2x2 Examples 2x2 Two players each
player has two strategies. Often represented by
a Box.
Player 2
Not Confess
Confess
Each players strategies are represented by the
names on each players side of the box.
Confess
Player 1
Not Confess
5
Player 2
Not Confess
Confess
The numbers in the box denote payoffs.
6 , 0
Confess
1 , 1
Player 1
The red numbers denotes Player 1s payoff. ...
Not Confess
0 , 6
3 , 3
6
Prisoners Dilemma
Your Prediction?
Player 2
Not Confess
Confess
6 , 0
Confess
1 , 1
Player 1
Not Confess
0 , 6
3 , 3
7
Chicken
Player 2
Dove
Hawk
0 , 2
Dove
1 , 1
Player 1
Hawk
2 , 0
-1 , -1
8
Battle of Sexes
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
9
A Business Example
A two-tier tender offer Robert Campeau used a
two-tier tender offer to bid for Federated
Stores (and its crown jewel Bloomingdales).
Suppose the market share price is 100.
Tender offer (1) Offers 105 to the first
tendered shareholders till half of the
outstanding shares are tendered. (2) The next
50 will be paid 90 only. (more details
later) Strategy interaction? Other shareholders
behavior.
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Solution 1 The case where strategic uncertainty
is not important. (Strictly) Dominant
Strategy Strategy yields the highest payoff
in every situation.
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Example Prisoners dilemma
Confess is a strictly dominant strategy
Player 2
Not Confess
Confess
6 , 0
Confess
1 , 1
Player 1
gt
gt
Not Confess
0 , 6
3 , 3
12
It is RATIONAL to use strictly dominant
strategies. Any objections? So we have
Rational Requirement 1 Use dominant
strategies, if there is any.
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Since confess is the dominant strategy to every
players, we will recommend the strategy profile
(confess, confess) as the outcome of the game.
Player 2
Not Confess
Confess
6 , 0
Confess
1 , 1
Not Confess
0 , 6
3 , 3
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Exercise (1) What are dominant strategies of
the following game?
Firm 2
Lower
Stay
Raise
Firm 1
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
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The two-tier tender offer Suppose the market
share price is 100. (1) Offers 105 to the
first tendered shareholders till half of the
outstanding shares are tendered. (2) The next
50 will be paid 90 only. Shares are not
places in the different tiers based on the order
in which they are tendered. Rather, everyone gets
a blend price all the shares tendered are placed
on a prorated basis in the two tiers. Ex If X
gt50 are tendered, then everyone get 105 50/X
90 (X-50)/X 90 15 50/X
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Analysis There are three relevant
situations (1) Less than 51 shares are
tendered. (2) Exactly 51 shares if you tender
tendered. (3) More than 51 shares are tendered.
Under (1), if not tender, get the market value
of shares 100 if tender, get
105 Choice? Under (2), if not tender, get
100. if tender , get 105. Under (3),
if not tender, get 90. if tender, get
105(50/X)90(X-50)/X gt 90.
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Therefore, tendering is a strictly dominant
strategy. Everyone will tender his/her shares!
(This is the predicted outcome. The average
price paid for buying the shares is 97.5! This
is lower than the market price of
shares! Legal case two-tier offer is
coercive, so it may violate some common law.
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Exercise A. Suppose that Macys also makes a
bid for the company. It will pay 102 for any
shares tendered if it gets a majority of shares.
Is tendering to Campeau still a dominant
strategy? B. Suppose that Macy makes an
unconditional offer that pays 102 regardless how
many shares are tendered. Is tendering still a
dominant strategy?
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Solution Question A There are five
situations (1) Both less than 51 shares are
tendered to Campeau and Macy. (2) Campeau gets
more than 51 shares. (3) Macy gets more than 51
shares. (4) Macy gets exactly 50 shares (5)
Campeau gets exactly 50 shares Under (1), will
tender to Campeau and get 105 (2),
will tender to Campeau, otherwise get 90
(3), will tender to Campeau and get 105
(4), will tender to Campeau and gets gt90,
or 105 (5), will tender to Campeau
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Solution Question B There are three relevant
situations (1) Less than 51 shares are
tendered to Campeau. (2) Exactly 51 shares if
you tender tendered. (3) More than 51 shares
are tendered. Under (1), if not tender, then
tender to Macy and get 100 if tender,
get 105 Under (2), if not tender, get
102. if tender , get 105. Under (3),
if not tender, get 102. if tender, get
105(50/X)90(X-50)/X gt 90. The choice depends
on the magnitude of X.
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Dominated Strategy

• If Strategy As payoffs are always greater than
  Strategy Bs, then we say Strategy B is dominated
  by Strategy A, and Strategy B is a (strictly)
  dominated strategy.

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Example Firms 2s Strategy Raise is dominated
by Stay. Raise is a (strictly) dominated
strategy.
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
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Iterated Elimination of Dominated Strategies

• It is rational not to use strictly dominated
  strategies.
• Suppose everybody knows everybody is rational.
  Then everybody knows no strictly dominated
  strategy will be used.

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Firm 1 knows Firm 2 would not use Raise.
Therefore, the Raise column is irrelevant to
his/her consideration.
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
25
The game becomes
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
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In this new game, Firm 1s Strategy East is
strictly dominated by Midwest.
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
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If Firm 2 knows Firm 1 knows Firm 2 would not use
Raise. Then Firm 2 can deduct that Firm 1 would
not use East.
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
28
The relevant game becomes
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
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Then Lower is strictly dominated by Stay.
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
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Then Firm 1 knows Firm 2 knows Firm 1 knows
The relevant game becomes
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
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Then Firm 2s strategy is solved. When Firm 2
uses Stay only, Firm 1 will use West. (It is
trivially a strictly dominant strategy.
Firm 2
Lower
Stay
Raise
West
1, 1
3, 4
2, 1
Midwest
2, 4
2, 5
8, 1
East
3, 3
0, 4
0, 9
The predicted outcome is (West, Stay).
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This is called Iterated Elimination of Dominated
Strategies.

• Obtain this solution through the following
  assumptions
• 1. It is rational not to use strictly dominated
  Strategies.
• 2. This rationality requirement is common
  knowledge.
• Common Knowledge Everybody knows everybody knows
  everybody knows ...

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Exercise Iterated Elimination
L
R
1, 4
T
3, 5
C
2, 6
4, 5
B
1, 2
0, 3
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Solution Iterated Elimination
2nd
L
R
1, 4
T
3, 5
3rd
C
2, 6
4, 5
B
1, 2
0, 3
1st
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Declining Industries

• Table of Prices
• David Alone Goliath Alone David and Goliath
• 1/1988 3 2 0.5
• 4 2.75 1.75 0.25
• 7 2.5 1.5 0
• 10 2.25 1.25 -0.25
• 1/1989 2 1 -0.5
• 4 1.75 0.75 -0.75
• 7 1.5 0.5 -1
• 10 1.25 0.25 -1.25

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Table of Prices David Alone Goliath
Alone David and Goliath 1/1990 1 0 -1.5 4 0.
75 -0.25 -1.75 7 0.5 -0.5 -2 10 0.25 -0
.75 -2.25 1/1991 0 -1 -2.5 4 -0.25 -1.2
5 -2.75 7 -0.5 -1.5 -3 10 -0.75 -1.75 -3
.25
Strategies When to exit.
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Solution Iterated Elimination
Table of Prices David Alone Goliath
Alone David and Goliath 1/1990 1 0 -1.5 4 0.
75 -0.25 -1.75 7 0.5 -0.5 -2 10 0.25 -0
.75 -2.25 1/1991 0 -1 -2.5 4 -0.25 -1.2
5 -2.75 7 -0.5 -1.5 -3 10 -0.75 -1.75 -3
.25
It it clear both firms will exit at 1/91, since
staying incurs loss regardless whether their
opponents has exited.
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Solution Iterated Elimination
Table of Prices David Alone Goliath
Alone David and Goliath 1/1990 1 0 -1.5 4 0.
75 -0.25 -1.75 7 0.5 -0.5 -2 10 0.25 -0
.75 -2.25 1/1991 0 -1 -2.5 4 -0.25 -1.2
5 -2.75 7 -0.5 -1.5 -3 10 -0.75 -1.75 -3
.25
By the same token, Goliath will exit at
1/90. Notice that David earns 2.5 if he did not
exit before 1/91
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Table of Prices David Alone Goliath
Alone David and Goliath 1/1988 3 2 0.5 4 2.75
1.75 0.25 7 2.5 1.5 0 10 2.25 1.25 -0
.25 1/1989 2 1 -0.5 4 1.75 0.75 -0.75 7
1.5 0.5 -1 10 1.25 0.25 -1.25 (2.25 if
not exited)
Consider Davids problem now if Goliath do not
exit, he earns -1.252.251 since Goliath will
definitely exit in the next period. If Goliath
exits, he earns 1.252.253.25 from 10/89 to
10/90.
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Table of Prices David Alone Goliath
Alone David and Goliath 1/1988 3 2 0.5 4 2.75
1.75 0.25 7 2.5 1.5 0 10 2.25 1.25 -0
.25 1/1989 2 1 -0.5 4 1.75 0.75 -0.75 7
1.5 0.5 -1 10 1.25 0.25 -1.25 (2.25 if
not exited)
Therefore not exiting at 10/89 is a dominant
strategy for David. Given this, Goliaths
exiting is dominant.
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Table of Prices David Alone Goliath
Alone David and Goliath 1/1988 3 2 0.5 4 2.75
1.75 0.25 7 2.5 1.5 0 10 2.25 1.25 -0
.25 1/1989 2 1 -0.5 4 1.75 0.75 -0.75 7
1.5 0.5 -1 10 1.25 0.25 -1.25 (2.25 if
not exited)
By the same token, Goliath will exit at
10/88. Exercise Why?
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Nash Equilibrium

• A Game may not have dominant or dominated
  strategies.

Player 2
Movie
Opera
Battle of Sexes
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
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Nash invented a solution concept called Nash
equilibrium. Nashs question Ask what outcome
(strategic profile) would be observed the most
frequently if the game situation appears
numerous times. That is which outcome is
stable.
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Is (Movie, Opera) stable?
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
45
Is (Movie, Opera) stable? No, since it gets a
higher payoff for Player 1 by using
Opera, assuming Player 2 does not change his/her
strategy.
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
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Is (Movie, Opera) stable? Assuming Player 1 does
not change his/her behavior, Player 2s using
Opera is not stable.
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
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To conclude, (Movie, Opera) is not stable it
cannot be observed over the long run.
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
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Nash Equilibrium We are studying
non-cooperative games. It means players (1)
cannot communicate with each other (ex.
Prisoners dilemma), (2) can communicate, but
cannot write contracts, or (3) can communicate,
can write contracts, but cannot enforce
the contracts. Therefore, they will seek
unilaterally to improve their payoffs.
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Nash Equilibrium (Stability) An outcome is a
Nash equilibrium if there is no unilateral
profitable deviation. Definition Best
Response, Best Reply (????) Strategies that
gives the highest payoff, keeping
other playerss strategy unchanged. Nash
equilibrium Playerss strategies are mutually
best responses.
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Nash Equilibrium (Movie, Movie) and (Opera,
Opera) Check Stability requirement.
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
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Nash Equilibrium (Movie, Movie) and (Opera,
Opera) Or use the notion of Best
Response. Movie is the best response to Movie
for Player 1.
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
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Nash Equilibrium (Movie, Movie) and (Opera,
Opera) Or use the notion of Best
Response. Opera is the best response to Opera
for Player 1.
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
53
Nash Equilibrium (Movie, Movie) and (Opera,
Opera) For Player 2, Movie is the best response
to Movie, too. Opera is the best
response to Opera.
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
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Nash Equilibrium (Movie, Movie) and (Opera,
Opera) Mark the best response rules in the
box. (Movie, Movie) and (Opera, Opera) are
mutually best Responses.
Player 2
Movie
Opera
0 , 0
Movie
3 , 1
Player 1
Opera
0 , 0
1 , 3
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Another Example
Chicken
Player 2
Dove
Hawk
0 , 2
Dove
1 , 1
Player 1
Hawk
2 , 0
-1 , -1
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Another Example
Chicken
Nash Equilibrium
Player 2
Dove
Hawk
0 , 2
Dove
1 , 1
Player 1
Hawk
2 , 0
-1 , -1
57
Exercise Find Nash Equilibria of the following
Game
L
R
9, 3
7, 5
T
M
12, 0
8, 4
6, 6
B
10, 2
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Solution Find Nash Equilibria of the following
Game
L
R
9, 3
7, 5
T
M
12, 0
8, 4
No Nash equilibrium Exists
6, 6
B
10, 2