ECON6021 (Nov 2004)
Game Theory
Outlines
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Nash Equilibrium
Simultaneous games
dominant and dominated strategies
Prisoner dilemma
Meeting-competition clauses
Sequential games
Subgame perfect equilibrium
What is a game?
A game consists of
A set of players
Rules of the games
A payoff function, one for each player,
telling us his payoff given actions of all
players
Knowledge of the players (viz., whether
one firm’s MC is also known by the other, …)
Remark: Unless o. w. stated, we assume
common knowledge among players about
bullet points 1 – 3. If not, we are in a
domain of asymmetric information—a
forthcoming topic in our class
John Nash
Nash equilibrium
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A Nash equil. is a complete prescription of what
each player will do in every possible
contingency in such a way that no player will
have incentive to unilaterally deviate from his
own prescription. (John Nash--Nobel Laureate
1994)
Remarks:
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Nash equilibrium may not be unique (multiple equil.)
Nash equilibrium can be mixed (penalty kick game)
The game need not be symmetric. (Symmetric
games are special cases.)
Simultaneous Game
Output Decisions in Duopoly
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2 firms producing the same product
each can produce 0, 1, 2, 3 units of
output at a zero marginal cost
the inverse market demand function
is as follows:
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Q. demanded
6
price
1.5
2
3
4
5
16
10
6
3
Output Decisions in Duopoly
Payoff Table
firm 1's
output
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firm 2's output
1
2
3
1 (16, 16)
(10, 20)
(6. 18)
2 (20, 10)
(12, 12)
(6, 9)
3 (18, 6)
(9, 6)
(4.5, 4.5)
Each producing 1 unit maximizes their joint
profits
Each producing 2 units will be the outcome
without a binding agreement
Cover story competition
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Each needs to determine
what cover story to use
for the coming issue
70 percent of potential
are interested only in
EOC; the other 30
percent interested only
in Harbor fest; If same
story, buy either one
with equal chance
Each cares only about its
size of readership
The above is commonly
known between Next
and Oriental
Then, EOC being chosen
by both is a Nash
equilibrium
Oriental
EOC Harbor
fest
EOC
35, 70, 30
Next
35
Harbor 30, 15, 15
fest
70
Dominant & Dominated Strategies
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A strategy is a dominant strategy for a
player if that strategy gives a higher
payoff than other strategies of his do,
irrespective of the other player’s strategy.
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A dominant strategy may not exist
A strategy is a dominated strategy for
a player if that strategy gives a lower
payoff than some other strategy of the
player does, irrespective of the other
player’s strategy.
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A dominated strategy may not exist
How to find out a Nash equilibrium?
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If you have a dominant strategy,
use it.
If you have a dominated
strategy, don’t use it.
Assume similar behavior on the
part of your rivals
Cover story competition (cont.)
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Next and Oriental
each needs to
determine what cover
Oriental
story to use for the
coming issue
Next
EOC Harbor
70 percent of
potential are
fest
interested only in EOC;
EOC 42, 28 70, 30
the other 30 percent
interested only in
Harbo 30, 70 18, 15
Harbor fest
r fest
Same story, 60%
among the interested
readers will buy Next,
the other 40% buy
from Oriental
Prisoners’ Dilemma
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Two suspects were prosecuted for robbery. Now
they are under investigation separately and
simultaneously. What will each of them do to
minimize his/her years of imprisonment?
suspect 2
confess not confess
suspect 1 confess
10, 10
0, 20
not confess
20, 0
1, 1
Prisoners’ Dilemma
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Neither to confess is best for both
suspects when they are viewed as an
entity (collective rationality).
But each to confess is optimal regardless
of the other does (individual rationality)
Individual rationality may not lead to
social optimum.
Much research in oligopoly is on collusion,
I.e., how firms can resolve the prisoners’
dilemma.
Application: Meeting-Competition Clauses
firm 1 low price
high price
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firm 2
low price high price
5, 5
15, 0
0, 15
10, 10
The equilibrium outcome is (low price, low
price). Collusion of (high price, high price) is
unsuccessful, at least for a one-shot
relationship.
But collusion can be supported by meetingcompetition clauses.
Meeting-Competition Clauses
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A meeting-competition clause states
that if another firm offers a lower
price, the seller will match it or
release the buyer from the contract.
With meeting-competition clauses,
both firms choose high prices.
Hence MCCs might be anticompetitive, rather than procompetitive!!
Meeting-Competition Clauses
5, 5
0, 15
15, 0
10, 10
firm 1 low price
high price
With MCC
firm 2
low price high price
5, 5
5, 5
5, 5
10, 10
Most-Favored-Nation Clauses
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A most-favored-Nation clause guarantees
the buyer that the seller would not offer a
lower price to any other buyer, current or
future, without offering the same price
decrease to the initial buyer.
This rebate system creates a penalty for
cheating on the cartel:
If either firm deviates from the
agreement by cutting its price, it would
have to cut prices to all previous buyers
as well!
Sequential Game
Entry Game
Accommodate
$100,000 to
New Firm
$100,000 to
Incumbent
Enter
Incumbent
Fight Price War
New
Firm
Keep
Out
$0 to New Firm
$300,000 to
Incumbent
-$200,000 to
New Firm
-$100,000 to
Incumbent
Entry Game
Traditionally, a sequential game is represented in a matrix form like
this (as if it is a simultaneous game):
incumbent
accommodate
new
firm
fight price
war
enter
100k,100k
-200k,-100k
keep
out
0,300k
0,300k
Although (keep out, fight price war) is a Nash equilibrium. But we
wonder if the incumbent will really implement the fighting plan if
the new firm has really entered. It is not in its interest to do so…
Subgame Perfection (Selten)
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For sequential games, an equilibrium
consists of instructions regarding players’
prescribed actions even for the point which
is never reached should the equilibrium
strategies are followed.
Refinement/Restriction: the agents at the
point is still required to act optimally even
if the equilibrium does not prescribe the
point to be reached. Equilibria survive such
a refinement is called subgame perfection.
Subgame perfect equilibria rule out
incredible threats.
Poison Pills against Hostile takeovers
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Sharon – 1 million shares, $2 each
Shareholder rights plan – allowing each
shareholder to buy from the company
new shares equal to his current stake at
half a price.
If any party acquires 100,000 or more
Sharon’s shares, then management can
activates the plan – but the one whose
acquisition activated the rights, however,
is depleted of the rights
It increases the costs of acquisition,
making successful hostile takeover more
difficult
Poison Pills
activates rights
acquires 100,000
shares
Hilda
Sharon
does not
doesn’t acquire
Hilda loses on
initial stake +
cost of
takeover rises
Poison Pills (cont)
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Suppose the raider – Hilda – has acquired 100,000
shares and the management activates the plan.
Then price of share adjusts from $2 to $v.
Suppose all eligible shareholders use their rights
 It is dominant strategy for each shareholder to
exercise his rights--the market price is v, while
you can buy it at v/2
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The worth of the firm = $2M + 900,000(v/2)
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The number of share = 1.9M
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Hence, each share is worth (2 + 0.45v)/1.9 =v.
Hence, v = 1.38.
Poison Pills
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Hostile takeover is more difficult to
succeed for two reasons
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Acquiring the whole firm is more costly
-- After the plan is activated, the
remaining shares is worth 1.38 x 1.8m
= $2.48m, rather than 2 x 0.9m =
$1.8m – the worth in the absence of
poison pills
The worth of the 100,000 shares is
reduced – Hilda paid 2 x 100,000 =
$0.2m for it. But after the plan is
activated, it is worth 1.38 x 100,000 =
$0.138m only.
Are Poison Pills beneficial to
Shareholders?
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Poison Pills are takeover defenses
They remove the ability of the owners
(shareholders) of the firm to sell their
shares to a buyer that their nominal
agents – management and the board – do
not like.
Takeover defenses, if not strong enough to
make takeovers impossible, may improve
the stockholders’ bargaining position and
raise the price they ultimately receive
Judicial Procedures
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A defendant’s fate rests in the hands of three Judges
Three possible outcomes: acquittal, life in prison,
and death sentence
Judge A’s
ranking
Judge B’s
ranking
Judge C’s
ranking
Best
Death
Sentence
Life in
Prison
Acquittal
Middle
Life in
Prison
Acquittal
Death
Sentence
Worst
Acquittal
Death
Sentence
Life in
Prison
Judicial Procedures (cont.)
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The so called status quo Judicial Procedure is
used:
Stage 1: determine (by majority vote) innocence
or guilt by majority rule
Stage 2: If innocence, acquittal automatically; if
guilty, determine (by majority rule) the
appropriate punishment.
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What will the judges do in equilibrium?
Judicial Procedures (cont.)
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What will the judges do under different judicial
procedures?
Roman Tradition: First decide if the death penalty
should be imposed for this case. If not, then
decide whether a life sentence is justified. If,
after proceeding down the list, no sentence is
imposed, then the defendant is acquitted.
Mandatory Sentencing: First specify the sentence
for the crime. Then determine whether the
defendant should be convicted.