1
Chapter Six
Monopoly :
Market Power
垄断：市场力
? 2005 MOL
2
Chapter 6 includes:
? 6.1 Profit Max. and Pricing
Decision In Monopoly
? 6.2 Monopoly and Resource
Allocation
? 6.3 Price Discrimination(价格歧
视)
3
Overview of Last Week Class
? Perfectly Competitive Markets
? Choosing Output in the Short-Run
? The Competitive Firm’s Short-Run Supply Curve
? Choosing Output in the Long-Run
? The Industry’s Long-Run Supply Curve
? Evaluating the Gains and Losses from Government
Policies--Consumer and Producer Surplus
? The Efficiency of a Competitive Market
4
Outline of Today’s Class
? Definition of Monopoly
? What cause Monopoly?
? Profit Max. and Pricing Decision in
Monopoly
? There is No Monopoly Supply Curve
? Markup Pricing and Monopoly Power
? The Inefficiency of Monopoly
5
Readings about the part of
this chapter
? Zhang: Chapter 10,P304-333
? Nicholson: Chapter 18, P495-524
6
Review: Perfect Competition
? Large number of buyers and sellers
? Homogenous product
? Free Entry and Exit
? Perfect information
? So Firm is a price taker
? When the Optimal Output in the Long Run
?P = LMC = LAC=SAC=SMC=AR
? Normal profits or zero economic profits in
the long run
Review: Perfect Competition
Q
q
PPMarket Individual Firm
DS
Q
0
P
0
P
0
d = MR = P
q
0
LACLMC
8
What is Monopoly (垄断)
? A Monopoly is a single supplier to a market.
This firm may choose produce at any point on
the market demand curve. Price-maker.
? Characteristics of Monopoly
1) One seller - many buyers
2) One product (no good substitutes)
3) Barriers to entry(进入壁垒)
9
What causes monopolies?
? Legal restrictions(法律限制) -- copyrights & patents.
? Control of critical resources（主要资源的控制）
creates market power.
? Government-authorized franchises（政府授予的
特权）, such as provided to cable TV companies.
? Economies of size—Natural Monopoly allow larger
firms to produce at lower cost than smaller firms. Such as
railway, public goods
? Brand loyalty（对品牌忠诚）and extensive
advertising makes entry highly expensive.
10
Profit-Maximization in a Monopoly
? Suppose that a monopolist seeks to
maximize its economic profit,
? What output level q* maximizes profit?
0
.() () ().
q
Max q p q q c q
>
Π =?
11
Profit-Maximization
At the profit-maximizing output level q*
()
() ()
() 0
dq d dcq
pqq
dq dq dq
Π
= ?=
F.O.C
()
()
() .
ddcq
pq q
dq dq
?=
() ()MRq MCq=
12
Marginal Revenue and Marginal Cost
()
()
() () () .
ddpq
MR q p q q p q q
dq dq
==+
dp(q)/dq is the slope of the market inverse
demand function, so dp(q)/dy < 0. Therefore
()
() () ()
dp q
MRq pq q pq
dq
=+ <
for q > 0.
13
Marginal Revenue
E.g.
if p(q) = a - bq
then
TR(q) = p(q)q = aq - bq
2
and so
AR=a-bq
MR(q) = a - 2bq < a - bq =
p(q), for q> 0.
D=AR: p(q) = a - bq
a
q
a/b
MR(q) = a - 2bq
a/2b
P
14
Summary: Profit Max. in Monopoly
? The monopolist is the supply-side of the
market and has complete control over
the amount offered for sale.
? Profits will be maximized at the level of
output where marginal revenue equals
marginal cost.
15
Short-run Equilibrium
Under Monopoly
16
Short-run Equilibrium Under Monopoly: Total approach
? Pricing and output under monopoly follow basically the
same rules as under competition: MR=MC
? Major difference between competition and monopoly
lies in the shape of the total revenue curve (to sell
more units the monopolists must lower price)
? Monopolist will maximize profits where the difference
between TR and TC is maximum
? Greatest difference occurs where the slopes of TR and
TC are the same, since the slopes are the same
marginal revenue equals marginal cost
17
Monopolists Short Run Profit Maximization- Total Approach
TC
Breakeven
Profit Maximizing Point
0
Q*
Greatest difference
occurs where the
slopes of TR and TC
are the same, since
the slopes are the
same: MR=MC
FC
Breakeven
(收支相抵点)
$
TR
Q
18
Short-run Equilibrium Under Monopoly:
Marginal approach
? marginal approach more useful
? firm will produce where
? marginal revenue (MR)=marginal cost (MC)
? profit per unit equals
? P-SAC
19
Short-run Equilibrium Under Monopoly:
Marginal approach
Profit<0
P
1
Q
1
Profit>0
SMC
SAC
Quantity
$ per
unit of
output
D = AR
MR
P*
Q*
P
2
Q
2
profit per
unit (P-SAC)
20
Short-run Equilibrium Under Monopoly
SMC
SAVC
Quantity
$ per
unit of
output
D = AR
MR
P*
Q*
SAC
P>SAC→π>0
(excess profit)
21
There is no profit in Monopoly in Short Run
SAVC
SAC
SMC
P=SAC→π=0
(normal profit
exists)
Q*
MR=SMC
E
$
P*
D, AR
Q
0
MR
22
Monopoly, Short Run Loss Minimizing
D, AR
Q
SMC
P<SAC→π<0
Loss
Q*
MR=SMC
E
$
SAC
P*
SAVC
0
MR
23
Profit or Loss in SR Equilibrium
? monopoly does not guarantee profit in
the short run
? monopolist must typically, however, earn
normal profits in long run
? monopolists will try to maximize profits
or minimize losses
? If SAC lies above AR (demand curve) the
monopolist will suffer a loss
24
Long-run Equilibrium
Under Monopoly
25
Long-run Equilibrium Under
Monopoly
? In the Long Run
? All inputs are variable
? Maximizing level at
LMC = MR = SMC
? Also where LMC = SMC then
? SAC = LAC
26
Long Run Equilibrium under Monopoly
SMC1
SMC2
SAC3
LMC
LAC
SAC2
SAC1
P
P*
SMC
3
MR=LMC=SMC
2
E
Economics Profit
P
0
Q
Q
0
Q*
27
Second-best
Production Scale
Excess Production Scale
28
Best Size
Min LAC
29
Conditions for LR Equili. Under
Monopoly
?(1) MR=LMC=SMC
?(2)MR’<LMC’
?(3)P≥LAC
30
There is no Supply Curve under
Monopoly
? In perfectly competitive market, We
constructed the short-run supply curve
by allowing the market demand curve
to shift and observing the supply curve
that traced out by the series of
equilibrium price-quantity combination.
31
Shift in Demand Leads to
Change in Price but Same Output
D
2
MR
2
D
1
MR
1
Quantity
MC
$/Q
P
2
P
1
Q
1
= Q
2
32
Shift in Demand Leads to
Change in Output but Same Price
D
1
MR
1
MC
$/Q
MR
2
D
2
P
1
= P
2
Q
1
Q
2
Quantity
33
There is no Supply Curve under
Monopoly: Summary
?In perfect competition, the market
supply curve is determined by
marginal cost.
?For a monopoly, output is determined
by marginal cost and the shape of the
demand curve.
34
Markup Pricing(加成定价)and
Monopoly Power（垄断力）
? Monopoly is rare.
? However, a market with several firms,
each facing a downward sloping
demand curve will produce so that price
exceeds marginal cost.
35
Markup Pricing(加成定价) in
Monopoly
? Markup pricing: Output price is the
marginal cost of production plus a
“markup.”
? How big is a monopolist’s markup and
how does it change with the own-price
elasticity of demand?
36
Monopolistic Pricing & Own-Price
Elasticity of Demand
()
()
() () ()
()
()1 .
()
ddpq
MR q p q q p q q
dq dq
qdpq
pq
pq dq
==+
??
=+
??
??
Own-price elasticity of demand is
,
()
()
DP
pq dq
e
qdpq
=?
37
Monopolistic Pricing & Own-
Price Elasticity of Demand
so
1
() ()1 .MR q p q
e
? ?
=?
? ?
? ?
38
Monopolistic Pricing & Own-
Price Elasticity of Demand
1
() ()1 .MR q p q
e
? ?
=?
? ?
? ?
Suppose the monopolist’s marginal cost of
production is constant, at $k/output unit.
For a profit-maximum,
which is
1
(*) (*)1MRq pq MC K
e
??
= ?= =
??
??
39
Monopolistic Pricing & Own-
Price Elasticity of Demand
(*) .
11
11
MCk
pq
ee
==
? ?
is the monopolist’s price.
E.g. if e = 3 then p(q*) = 3k/2,
and if e = 2 then p(q*) = 2k.
The markup rises as the own-price
elasticity of demand falls towards 1.
40
Markup Pricing：Summary
? Markup Pricing
? Pricing for any firm with monopoly power
? If E
d
is large, markup is small
? If E
d
is small, markup is large
()
11
d
M C
P
e
=
?
Elasticity of Demand and Price Markup
$/Q $/Q
Quantity Quantity
AR
MR
MR
AR
MC MC
Q* Q*
P*
P*
P*-MC
The more elastic is
demand, the less the
markup.
42
Measuring Monopoly Power:
Lerner Index (勒纳指数)
? From
()
11
d
MC
P
e
=
?
1
d
d
d
P
P MC
e
P
PMC
e
PMC
P e
?? =
?? =
?
?=
43
Measuring Monopoly Power:
Lerner Index (勒纳指数)
? Lerner’s Index of Monopoly Power
? E
d
is elasticity of demand for a firm, not
the market
? The larger the value of L (between 0 and
1), the greater the monopoly power.
1
d
PMC
L
Pe
?
= =
44
The Inefficiency（无效率）of
Monopoly
? A market is Pareto efficient（有效率）if
it achieves the maximum possible total
gains-to-trade（Total Welfare）.
? Otherwise a market is Pareto inefficient.
45
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
q
e
p(q
e
)
The efficient output level
q
e
satisfies p(q) = MC(q).
46
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
q
e
p(q
e
)
The efficient output level
q
e
satisfies p(q) = MC(q).
CS
47
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
q
e
p(q
e
)
The efficient output level
q
e
satisfies p(q) = MC(q).
CS
PS
48
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
q
e
p(q
e
)
The efficient output level
q
e
satisfies p(q) = MC(q).
Total gains-to-trade is
maximized.
CS
PS
49
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
MR(q)
q*
p(q*)
50
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
MR(q)
q*
p(q*)
CS
51
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
MR(q)
q*
p(q*)
CS
PS
52
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
MR(q)
q*
p(q*)
CS
PS
Total Welfare:
CS+PS
53
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
MR(q)
q*
p(q*)
CS
PS
54
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
MR(q)
q*
p(q*)
CS
PS
MC(q*+1) < p(q*+1) so both
seller and buyer could gain
if the (q*+1)th unit of output
was produced. Hence the
market is Pareto inefficient.
55
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
MR(q)
q*
p(q*)
DWL
Deadweight loss measures
the gains-to-trade(福利) not
achieved by the market.
56
The Inefficiency of Monopoly
$/output unit
q
MC(q)
p(q)
MR(q)
q*
p(q*)
q
e
p(q
e
)
DWL
The monopolist produces less than
the efficient quantity, making the
market price exceed the
efficient market price.
57
The End
58
Last Revised:
November. 12, 2005
1
Chapter Six
Monopoly :
Market Power
? 2005 MOL
2
Chapter 6 includes:
? 6.1 Profit Max. and Pricing
Decision In Monopoly
? 6.2 Monopoly and Resource
Allocation
? 6.3 Price Discrimination(价格歧
视)
3
Overview of Last Class
? Definition of Monopoly
? What cause Monopoly?
? Profit Max. and Pricing Decision in
Monopoly
? There is No Monopoly Supply Curve
? Markup Pricing and Monopoly Power
? The Inefficiency（非效率）of
Monopoly
4
Outlines of Today’s Class
? The Social Costs (DWL) of Monopoly Power
? Rent-seeking
? First-degree Price Discrimination
? Second-degree Price Discrimination
? Third-degree Price Discrimination
? Antitrust Laws（反托拉斯法）
5
Readings about the part of
this chapter
? Zhang: Chapter 10,P304-333
? Nicholson: Chapter 18, P495-524
6
Review: Evaluation of the Monopoly
The relationships between DWL and Monopoly Power
? Monopoly power results in higher prices
and lower quantities.
? However, does monopoly power make
consumers and producers in the
aggregate better or worse off?
? How much is DWL in Monopoly?
7
Deadweight Loss from Monopoly Power
B
A
Lost Consumer Surplus
Deadweight
Loss
Because of the higher
price, consumers lose
A+B and producer
gains A-C.
C
Quantity
AR
MR
MC
Q
C
P
C
P
m
Q
m
$/Q
8
The relationships between DWL and Monopoly Power
? The incentive to engage in monopoly
practices is determined by the profit to
be gained.
? The larger the transfer from consumers
to the firm, the larger DWL of monopoly
9
The relationships between DWL and Monopoly
Power
? In order to earn larger Profit, Monopoly
will do many thing which make more
DWL.
? Seeking Rent
? Patent
? Price Discrimination
10
Rent Seeking
? Rent seeking is any attempt to capture
consumer surplus, producer surplus, or
economic profit.
? Rent seeking is not confined to a monopoly.
? There are two forms of rent seeking activity to
pursue monopoly:
? Buy a monopoly—transfers rent to creator of
monopoly.
? Create a monopoly—uses resources in political
activity.
11
Rent Seeking
? The resources used in
rent seeking can exhaust
the monopoly’s economic
profit and leave the
monopoly owner with only
normal profit.
? Figure shows the normal
profit that results from
rent seeking.
12
Rent Seeking
? Average total cost
increases and the
profits disappear to
become part of the
enlarged
deadweight loss
from rent seeking.
13
Rent Seeking
? Firms may spend to gain monopoly power
? Lobbying(游说)
? Advertising（广告）
? Building excess capacity(建立多余生产能力)
14
Inefficiency of a Monopoly:
Patents
? Patents（专利）in the US grant
monopoly to a company over an
innovative product or process for 17
years.
? Company will sell the product for 17
years at monopoly prices: deadweight
loss.
15
Inefficiency of a Monopoly:
Patents
? Are Patents necessary for our society? How
Long is it proper for patents protecting?
? No patent protection
little incentive to innovate.
? Too strong patent protection
1) Deadweight loss for longer time
2) Low incentives to innovate
? Optimal patent life balances these conflicting
effects.
Price Discrimination
? So far a monopoly has been thought of as a
firm which has to sell its product at the same
price to every customer. This is uniform
pricing（单一价格）.At this condition, All
consumers are consider as Homogeneous or
identical(同质的)
? Can price-discrimination（价格歧视）earn a
monopoly higher profits if the consumers are
considered as heterogeneous(异质的)?
17
Price Discrimination
? Price discrimination is the practice of
selling different units of a good or service
for different prices.
? To be able to price discriminate, a
monopoly must:
? Identify and separate different buyer types
? Sell a product that cannot be resold
? No arbitrage
? Price differences that arise from cost
differences are not price discrimination.
18
Price Discrimination
? Price Discrimination and Consumer Surplus
? Price discrimination converts consumer surplus
into economic profit.
? A monopoly can discriminate
? Among units of a good. Quantity discounts are an
example.
? Among groups of buyers. (Advance purchase and
other restrictions on airline tickets are an example.)
19
Price Discrimination
? As a single-price
monopolist, this
firm maximized
profit by producing
8 units, where MR
= MC and selling
them for $1,200
each.
20
Price Discrimination
? By price
discriminating, the
firm can increase
its profit.
? In doing so, it
converts consumer
surplus into
economic profit.
Types of Price Discrimination
? 1st-degree（一级价格歧视）: Each output
unit is sold at a different price. Prices may
differ across buyers.
? 2nd-degree（二级价格歧视）: The price paid
by a buyer can vary with the quantity
demanded by the buyer. But all customers
face the same price schedule. (block pricing成
批定价) E.g. bulk-buying discounts（批量购买
折扣）.
Types of Price Discrimination
? 3rd-degree（三级价格歧视）: Price
paid by buyers in a given group is the
same for all units purchased. But price
may differ across buyer groups.
E.g., senior citizen（老年人）and
student discounts vs. no discounts for
middle-aged persons.
First-degree Price
Discrimination
? Each output unit is sold at a different price.
Price may differ across buyers.
? It requires that the monopolist can
discover the buyer with the highest
valuation of its product, the buyer with the
next highest valuation, and so on.
? The Monopolist have perfect information
about the consumer’s preference
First-degree Price
Discrimination
p(q)
q
$/output unit
MC(q)
(')pq
Sell the q’th unit for $
(').p q
q’
First-degree Price
Discrimination
p(q)
q
$/output unit
MC(q)
'q
(')pq
"q
(")pq
Sell the th unit for $ Later on
sell the th unit for $
'q
(').pq
"q (").pq
First-degree Price
Discrimination
(').pq
p(q)
q
$/output unit
MC(q)
'q
"q
q
′′′
()p q
′′′
Sell the q’th unit for $ Later on
sell the q”th unit for $ P(q”) Finally
sell the q”’th unit for marginal
cost, $
(').pq
().pq
′′′
(").pq
First-degree Price
Discrimination
(').p q
p(q)
q
$/output unit
MC(q)
'q "q
q
′′′
The gains to the monopolist
on these trades are:
and zero.
(') ('),(") (")pq MCq pq MCq? ?
The consumers’ gains are zero.
(").pq
P(q”’)
First-degree Price
Discrimination
p(q)
$/output unit
MC(q)
So the sum of the gains to
the monopolist on all
trades is the maximum
possible total gains-to-trade.
PS
q
′′′
q
First-degree Price
Discrimination
p(q)
q
$/output unit
MC(q)
q
′′′
The monopolist gets
the maximum possible
gains from trade.
PS
First-degree price discrimination
is Pareto-efficient.
First-degree Price
Discrimination
? First-degree price discrimination gives a
monopolist all of the possible gains-to-
trade, leaves the buyers with zero
surplus, and supplies the efficient
amount of output.
Summary: First-degree Price Discrimination
（Perfect Price Discrimination）
? Perfect price discrimination
refers to the situation when the
monopolist knows exactly the
willingness to pay of each
customer and can charge each
customer a different price.
32
Second-degree Price Discrimination
The price paid
by a buyer can
vary with the
quantity
demanded by
the buyer.
Third-degree Price
Discrimination(三级价格歧视)
? Price paid by buyers in a given group is
the same for all units purchased. But
price may differ across buyer groups.
? Conditions for 3
rd
Degree price
discrimination:
? The market can be separated
? The different elasticity of demand at every
separated market
Third-degree Price
Discrimination
? A monopolist manipulates market price
by altering the quantity of product
supplied to that market.
? So the question “What discriminatory
prices will the monopolist set, one for
each group?” is really the question “How
many units of product will the monopolist
supply to each group?”
A Third-degree Price
Discrimination Model
? Two markets, 1 and 2.
? q
1
is the quantity supplied to market 1.
Market 1’s inverse demand function is
p
1
=p
1
(q
1
).
? q
2
is the quantity supplied to market 2.
Market 2’s inverse demand function is
p
2
=p
2
(q
2
).
Third-degree Price
Discrimination
? For given supply levels q
1
and q
2
the
firm’s profit is
? What values of q
1
and q
2
maximize
profit?
12 111 222 1 2
(, ) () () ( ).qq pqq pqq cq qΠ= + ?+
Third-degree Price
Discrimination
The profit-maximization conditions are
12 111 222 1 2
(, ) () () ( ).qq pqq pqq cq qΠ= + ?+
()
12 12
111
11 12 1
()()
()
0
cq q q q
pqq
qq qq q
? ???
?? ? ?
Π ++
=?×
+
=
()
12 12
222
22 12 2
()()
()
0
cq q q q
pqq
qq qq q
? ???
?? ? ?
Π ++
=?×
+
=
Third-degree Price
Discrimination
12
1
()
1
qq
q
?
?
+
=
12
2
()
1
qq
q
?
?
+
=
and
so
the profit-maximization conditions are
()
12
111
112
()
()
()
cq q
pqq
qq
? ?
??
+
=
+
()
12
222
212
()
() .
()
cq q
pqq
qq
? ?
??
+
=
+
and
Third-degree Price
Discrimination
()()
12
111 2 2 2
12 12
()
() ()
()
cq q
pqq pqq
qq
? ??
?? ?
+
==
+
Third-degree Price
Discrimination
()()
12
111 2 2 2
12 12
()
() ()
()
cq q
pqq pqq
qq
? ??
?? ?
+
==
+
???
MR
1
(q
1
) = MR
2
(q
2
) says that the allocation
q
1
, q
2
maximizes the revenue from selling
q
1
+ q
2
output units.
E.g. if MR
1
(q
1
) > MR
2
(q
2
) then an output unit
should be moved from market 2 to market 1
to increase total revenue.
Third-degree Price
Discrimination
()()
12
111 2 2 2
12 12
()
() ()
cq q
pqq pqq
qq
? ??
?? ?
+
==
+
?
??
The marginal revenue common to both
markets equals the marginal production
cost if profit is to be maximized.
Third-degree Price
Discrimination
MR
1
(q
1
)MR
2
(q
2
)
q
1
q
2
q
1
*q
2
*
p
1
(q
1
*)
p
2
(q
2
*)
MC MC
p
1
(q
1
)
p
2
(q
2
)
Market 1 Market 2
MR
1
(q
1
*) = MR
2
(q
2
*) = MC
Third-degree Price
Discrimination
MR
1
(q
1
)MR
2
(q
2
)
q
1
q
2
q
1
*q
2
*
p
1
(q
1
*)
p
2
(q
2
*)
MC MC
p
1
(q
1
)
p
2
(q
2
)
Market 1
Market 2
MR
1
(q
1
*) = MR
2
(q
2
*) = MC and p
1
(q
1
*) ≠ p
2
(q
2
*).
Third-degree Price
Discrimination
? In which market will the monopolist set
the higher price?
Third-degree Price
Discrimination
? In which market will the monopolist
cause the higher price?
? Recall that
and
11 11
1
1
() ()1MR q p q
e
? ?
=?
? ?
? ?
22 22
2
1
() ()1 .MR q p q
e
? ?
=?
? ?
? ?
Third-degree Price
Discrimination
? In which market will the monopolist
cause the higher price?
? Recall that
? But,
11 11
1
1
() ()1MR q p q
e
? ?
=?
? ?
? ?
22 22
2
1
() ()1 .MR q p q
e
? ?
=?
? ?
? ?
and
****
11 2 2 1 2
() () ( )MRq MRq MCq q==+
Third-degree Price
Discrimination
So
**
11 2 2
12
11
()1 ()1 .pq pq
ee
? ???
?= ?
? ???
? ???
48
Third-degree Price
Discrimination
**
11 2 2
() ()p qpq>
Therefore, only if
12
12
11
11 .ee
ee
? <? ? <
The monopolist sets the higher price in
the market where demand is least
own-price elastic.
49
Application: Third-degree Price
Discrimination
Examples of Price Discrimination
? Movie tickets
? Airline prices
? Discount coupons(折扣优
惠券)
? Financial aid
? Quantity discounts
Two-Part Tariffs（两部定价法）
? A two-part tariff is a lump-sumfee（固
定费）, p
1
, plus a price p
2
for each unit
of product purchased.
? Thus the cost of buying x units of
product is
p
1
+ p
2
x.
Two-Part Tariffs （两部定价法）
? Should a monopolist prefer a two-part
tariff to uniform pricing, or to any of the
price-discrimination schemes discussed
so far?
? If so, how should the monopolist design
its two-part tariff?
Two-Part Tariffs
? p
1
+ p
2
x
? Q: What is the largest that p
1
can be?
Two-Part Tariffs
? p
1
+ p
2
x
? Q: What is the largest that p
1
can be?
? A: p
1
is the “entrance fee”（入场费）so
the largest it can be is the surplus the
buyer gains from entering the market.
? Set p
1
= CS and now ask what should
be p
2
?
Two-Part Tariffs
p(q)
y
$/output unit
MC(q)
'q
2
(')ppq=
Should the monopolist
set p
2
above MC?
Two-Part Tariffs
p(q)
q
$/output unit
'q
CS
Should the monopolist
set p
2
above MC?
p
1
= CS.
MC(q)
2
(')p pq=
Two-Part Tariffs
p(q)
q
$/output unit
'q
CS
Should the monopolist
set p
2
above MC?
p
1
= CS.
PS is profit from sales.
MC(q)
PS
2
(')ppq=
Two-Part Tariffs
2
(')p pq=
p(q)
q
$/output unit
CS
Should the monopolist
set p
2
above MC?
p
1
= CS.
PS is profit from sales.
MC(q)
PS
Total profit
'q
Two-Part Tariffs
p(q)
q
$/output unit
"q
2
(")ppq=
Should the monopolist
set p
2
= MC?
MC(q)
Two-Part Tariffs
"q
p(q)
q
$/output unit
Should the monopolist
set p
2
= MC?
p
1
= CS.
CS
MC(q)
2
(")p pq=
Two-Part Tariffs
2
(")p pq=
p(q)
q
$/output unit
Should the monopolist
set p
2
= MC?
p
1
= CS.
PS is profit from sales.
MC(q)
CS
PS
"q
Two-Part Tariffs
2
(")p pq=
p(q)
q
$/output unit
Should the monopolist
set p
2
= MC?
p
1
= CS.
PS is profit from sales.
MC(q)
CS
Total profit
PS
"q
Two-Part Tariffs
2
(")p pq=
p(q)
q
$/output unit
Should the monopolist
set p
2
= MC?
p
1
= CS.
PS is profit from sales.
MC(q)
CS
PS
"q
Two-Part Tariffs
$/output unit
"q
p(q)
q
Should the monopolist
set p
2
= MC?
p
1
= CS.
PS is profit from sales.
MC(q)
CS
Additional profit from setting p
2
= MC.
PS
2
(")p pq=
q”
Two-Part Tariffs
? The monopolist maximizes its profit
when using a two-part tariff by setting
its per unit price p
2
at marginal cost and
setting its lump-sum fee p
1
equal to
Consumers’ Surplus.
Two-Part Tariffs
? A profit-maximizing two-part tariff gives
an efficient market outcome in which
the monopolist obtains as profit the
total of all gains-to-trade.
67
Limiting Market Power: The Antitrust Laws
? Antitrust Laws（反托拉斯法）:
? Promote a competitive economy
? Rules and regulations designed to promote
a competitive economy by:
? Prohibiting actions that restrain or are likely to
restrain competition
? Restricting the forms of market structures that
are allowable
68
Antitrust Laws
Sherman Act（谢尔曼法案）of 1890:
? Section 1: prohibits contracts and
conspiracies（共谋）, explicit or
implicit, to restraint trade by fixing
prices or restrict output（限产）.
? Section 2: illegal to monopolize or
attempt to monopolize a market.
69
Antitrust Laws
Clayton Act（克莱顿法案）of 1914:
? Illegal for a firm with a large market
share to require a buyer not to buy
from a competitor.
? Prohibits mergers（合并）and
acquisitions（收购）if they
substantially lessen competition.
? Illegal to sell a product at different
prices to different buyers, if this
injures competition.
Natural Monopoly（自然垄断）
? A natural monopoly arises when the
firm’s technology has economies-of-
scale（规模经济）。
71
Regulation: Natural Monopoly
? Some monopolies can be regulated:
government sets price equal to
marginal cost.
? Problems with this policy:
1) incentives to invest in research and
innovation decrease.
2) At that price monopoly could be
making negative profits!
72
Regulation：government sets
price equal to marginal cost
Regulating a Natural Monopoly
? So a natural monopoly cannot be forced
to use marginal cost pricing. Doing so
makes the firm exit, destroying both the
market and any gains-to-trade.
? Regulatory schemes（规制方案）can
induce the natural monopolist to produce
the efficient output level without exiting.
74
Regulation: Natural Monopoly
q
D
p
M
q
()
C
ACq
0
()AC q
()MCq
( )
D
pq
C
p
M
p
C
q
P
m
Unregulated, the monopolist
would produce q
m
and
charge P
m
.
If the price were regulate to be P
C
,
the firm would lose money
and go out of business.
Setting the price at AC(q
C
)
r
yields the largest possible
output; profit is zero.
75
Regulating Natural Monopolies
? Examples: phone companies, gas
companies, public utilities in general.
? Regulations:
1. Let monopolist charge price equal to
average cost. What is a firm’s cost
function?
2. Government operates service: price
equal marginal cost and subsidy to the
firm to cover losses.
Summary about chapter 6
? A monopoly is a firm that is the sole
seller in its market.
? It faces a downward-sloping
demand curve for its product.
? A monopoly’s marginal revenue is
always below the price of its good.
Summary about chapter 6
? Like a competitive firm, a monopoly
maximizes profit by producing the
quantity at which marginal cost and
marginal revenue are equal.
? Unlike a competitive firm, its price
exceeds its marginal revenue, so its
price exceeds marginal cost.
Summary about chapter 6
? A monopolist’s profit-maximizing level
of output is below the level that
maximizes the sum of consumer and
producer surplus.
? A monopoly causes deadweight losses
similar to the deadweight losses
caused by taxes.
Summary about chapter 6
? Policymakers can respond to the
inefficiencies of monopoly behavior
with antitrust laws, regulation of prices,
or by turning the monopoly into a
government-run enterprise.
? If the market failure is deemed small,
policymakers may decide to do nothing
at all.
Summary about chapter 6
? Monopolists can raise their profits by
charging different prices to different
buyers based on their willingness to
pay.
? Price discrimination can raise
economic welfare and lessen
deadweight losses.
81
The End
82
Last Revised:
November 15, 2005
北京大学经济学院1
Chapter 7
Monopolistic
Competition
(垄断竞争市场)
? 2005 MOL
北京大学经济学院2
Outline of Today’s Class
? Characteristics of Monopolistic
Competition.
? Monopolistic Competition in the short-
run.
? Monopolistic Competition in the long-
run.
? Advertising and Brand names.
北京大学经济学院3
Readings about the part of
this chapter
? Zhang: Chapter 9,P270-284
? Nicholson: Chapter 19, P537-545
北京大学经济学院4
The Spectrum（分布）of
Market Structure
Perfect
Competition
Pure
Monopoly
Imperfect
Competition
北京大学经济学院5
Imperfect Competition
? Two types of imperfectly competitive
markets:
cMonopolistic Competition（垄断竞争市
场）
Many firms selling products that are similar but
not identical (e.g. movies).
dOligopoly（寡头市场）
Only a few sellers, each offering a similar or
identical product to the others (e.g. hockey
skates).
北京大学经济学院6
Monopolistic Competition...
? A market structure between perfectly
competitive and monopolistic.
? Departs from the perfectly competitive
because each seller offers a somewhat
different product.
? Departs from a monopoly because there
are many sellers, each of which is small
compared to the market.
北京大学经济学院7
Monopolistic Competition
? Definition of Monopolistic Competition
? Markets that have some features of
competition and some features of
monopoly.
? Many firms selling a slightly
differentiated product in a market with
relatively easy entry / exit.
北京大学经济学院8
Attributes of Monopolistic
Competition(垄断竞争市场的特性)
c Many Sellers
d Product Differentiation
e Free Entry/Exit into/from
industry
北京大学经济学院9
Attribute 1: Many Sellers
? Many firms competing for
the same group of
customers(顾客群) or
product group(产品组).
? Examples:
? books, CDs, movies,
computer games,
restaurants, piano
lessons, cookies,
furniture, etc.
北京大学经济学院10
Definition of Product Group
? The output of a set of firms constitute a
product group if the substitutability in
demand among the products is very high
relative to the substitutability between those
firms’ outputs and other goods generally.
? the substitutability in demand among the
products is measured by the cross-price
elasticity
北京大学经济学院11
Attribute 2: Product Differentiation
? Alternative forms of differentiation:
?quality differences; additional service;
location; and packaging.
? Results in firm facing a downward-
sloping demand curve.
?Demand curve is highly, but not
perfectly, elastic.
北京大学经济学院12
Attribute 3: Free Entry or Exit
? Firms can enter or exit the
market without restriction.
? no overly restrictive licensing
（许可）requirements.
? no major capital
requirements.
? The number of firms in the
market adjusts until
economic profits are zero.
北京大学经济学院13
Demand Curve faced by
Monopolistic Competition’s Firm
Proportional
Demand Curve
（比例需求曲
线）（客观需
求曲线）
Own Demand Curve
Subjective Demand Curve
Expected Demand Curve
(主观需求曲线)
北京大学经济学院14
Monopolistic Competition - Elastic
Demand Curve
Quantity
0
Price
Demand
北京大学经济学院15
Monopolistic Competition -
Demand Curve is Distinct from MR
Curve
Q
0
P
Demand
MR
北京大学经济学院16
Short-Run Operation in
Monopolistic Competition
? In the short-run, the monopolistically
competitive firm:
?Follows a monopolist’s rule for profit-
maximization.
?MR = SMC, MR’<SMC’
?Price > SAC?π>0
?Price < SAC ?π<0
北京大学经济学院17
Monopolistic Competition - Short
Run
? Graph looks similar to that of a
monopolist, but demand is flatter (more
elastic)
? Profit maximization is at output where
MC = MR
北京大学经济学院18
Monopolistic Competitor - Short
Run
北京大学经济学院19
Monopolistic Competitor - Short
Run
北京大学经济学院20
Monopolistic Competitor - Short Run
Q
0
P
Demand
MR
SAC
P>SAC ?π>0
(Economic Profit)
SMC
Profit-
maximizing quantity
P
SAC
北京大学经济学院21
Monopolistic Competitors in the Short-Run
Q
SMC
SAC
Q
Profit Max.
P
P=SAC
MR
Demand
P=SAC?π=0
北京大学经济学院22
Monopolistic Competitor - Short Run
Q
0
P
Demand
MR
P<SAC (Losses)
SMC
SAC
SAC
Loss-
minimizing
quantity
P
北京大学经济学院23
Long-Run Operation in Monopolistic
Competition
? If firms are making economic profits in the short-
run, new firms are encouraged to enter the
market. This results in:
? Increases the number of products offered.
? Reduces demand faced by firms already in the
market.
? Incumbent firms’(在位厂商) demand curves shift
to the left.
? Demand for the incumbent firms’ products fall,
and their profits decline.
北京大学经济学院24
Monopolistic Competition in the Long Run
? Firm Demand Shifts In and Gets More
Elastic.
? Demand Shifts Back Because More Firms
Enter the Industry, So There is Less
Demand Per Firm.
? Demand Gets More Elastic Because More
Firms Means More Substitutes and More
Substitutes Means More Elastic
北京大学经济学院25
Monopolistic Competition in the
Long Run
? Like Perfect Competition, Firms Will
Continue to Enter Until There are
No More Profits to Attract Them.
北京大学经济学院26
Long-Run Operation in
Monopolistic Competition
? Firms will enter and exit until the firms
are making exactly zero economic
profits.
? Two characteristics of monopolistic
competition in the long-run:
?Price exceeds marginal cost
?Price equals average total cost
北京大学经济学院27
Monopolistic Competition in the Short Run
SMC
P
Q
D
0
MR
SAC
SAC*
p*
Firms will
Enter
Economic
Profit
北京大学经济学院28
Monopolistic Competition in the Long Run
LMC
P
Q
D
2
0
MR2
LAC
LAC*P*=
D
1
Zero Economic
Profit
北京大学经济学院29
Q
D
MR
SMC
P
SAC
Price and Costs
Price and Costs
Q
Economic
Losses
Firms will
Exit
Monopolistic Competition in the Short Run
ATC
北京大学经济学院30
Monopolistic Competition in the Long
Run
Q
MR2
LMC
P
LAC
Price and Costs
Price and Costs
Q
Zero Economic
Profit
C
D
1
D
2
北京大学经济学院31
Monopolistic Competition in the Long
Run
北京大学经济学院32
A Monopolistic Competitor in the Long-Run
Q
LMC
LAC
Q
Profit Max.
P
P=LAC
D
MR
北京大学经济学院33
Monopolistic Competition vs.
Perfect Competition
? Two differences arise in the long-run
between monopolistic competition and
perfect competition:
?Excess Capacity（超额能量）
?Markup(成本加成幅度)
北京大学经济学院34
Monopolistic Competition:
Excess Capacity
?In perfect competition, firms
produce at the efficient scale, i.e.
the point where average total cost
is minimized.
?Free entry in competitive markets
drive firms to produce at the
minimum of average total cost.
北京大学经济学院35
The Competitive Firm’s Output in the Long-Run
Q
LMC
LAC
P=MR=AR
Q
Efficient Scale
P
P=MC
北京大学经济学院36
Monopolistic Competition:
Excess Capacity(超额产能)
? In monopolistic competition, the quantity
of output is less than the “efficient scale”
of perfect competition.
? A monopolistically competitive firm could
decrease the quantity it produces and
increase the average total cost of
production.
北京大学经济学院37
Monopolistically Competitive Output in the Long-Run
Q
LMC
LAC
Q
Produced
P
MC
P
MR
Demand
北京大学经济学院38
Monopolistically Efficient Output in the Long-Run
Q
LMC
LAC
Q
P
MC
P
MR
Demand
Q
Efficient Scale
北京大学经济学院39
Monopolistically Efficient Output in the Long-
Run
Q
MC
SAC
Q
P
MC
P
MR
Demand
Q
Efficient Scale
Excess
Capacity
北京大学经济学院40
Monopolistic vs. Perfect Competition
(a) Monopolistically Competitive Firm (b) Perfectly Competitive Firm
Quantity
Quantity
Price
P = MR
(demand
curve)
LMC
LAC
Quantity
produced
Efficient
scale
Price
P
Demand
LMC
LAC
P = MC
Excess capacity
Marginal
cost
Markup
MR
Quantity produced =
Efficient scale
北京大学经济学院41
Monopolistic Competition vs. Monopoly
? In monopoly there can be long run
economic profits, but not in M. Comp.
? In the short run, both monopolists and
M. Comp. can make economic profit.
? Monopolist probably has higher average
total costs (SAC).
? Monopolist in the long run probably has
a higher price and lower output.
北京大学经济学院42
Monopolistic Competition and Monopoly
Q
MC
SAC
Q
M
P
P
MC
MR
Demand
Q
MC
P
M
北京大学经济学院43
Monopolistic Competition vs.
Monopoly
? Both should operate where MC = MR.
? Neither operate in long run at low point
of average total cost (LAC) curve.
北京大学经济学院44
Comparing Monopoly,
Competition, and Monopolistic
Competition
Model
Type
Produce
Where?
AC
Minimum?
LR
Profit?
D/W
Loss?
Competition P=MC Yes No No
Monopolistic
Competition
MR=MC No No Yes
Monopoly MR=MC No Yes Yes
北京大学经济学院45
Monopolistic Competition:
Advertising and Brand Names
（品牌）
? Some critics of monopolistic competition
contend that advertising and brand
names exploit consumers and reduce
competition.
? Defenders argue that advertising
increases competition by offering a
greater variety of products and prices.
北京大学经济学院46
Monopolistic Competition: Advertising
? Firms that sell highly differentiated
consumer goods typically spend
between 10 and 20 percent of revenue
for advertising.
? As a whole (total economy) about 2
percent of total firm revenue is spent on
advertising.
北京大学经济学院47
Monopolistic Competition: Advertising
and Brand Names
? Brand Names may provide two benefits
to consumers:
? Provide consumers information about
quality when quality cannot be easily
judged in advance of purchase.
? Give firms an incentive to maintain high
quality.
北京大学经济学院48
Conclusion
? Monopolistically competitive markets
are characterized by many firms each
producing a differentiated product with
freedom of market entry.
? In equilibrium, monopolistically
competitive markets produce with some
excess capacity and each firm charges a
price above marginal cost.
北京大学经济学院49
Conclusion
? The selling price of a monopolistic
competitive market results in some
deadweight losses and resource
misallocation.
? Product differentiation leads to
advertising and brand names.
北京大学经济学院50
The End
北京大学经济学院51
Last Revised:
November 20, 2005
北京大学经济学院1
Chapter 8
Oligopoly
(寡头市场)
? 2005 MOL
北京大学经济学院2
Chapter 8 includes:
? 8.1 What is Oligopoly?
? 8.2 Quantity Competition
? Cournot Duopoly（古诺双寡头模型）
? Stackelberg Duopoly（斯泰克伯格双寡头模型）
? Kinked Demand Curve Model（折弯的需求曲线模型）
? 8.3 Price Competition
? Bertrand Duopoly（伯特兰双寡头模型）
? Price Leaderships(价格领导者模型)
? 8.4 Differentiated Product Market
? Cournot & Bertrand Duopoly Model
? Linear City Model(线形城市模型)
北京大学经济学院3
Overview of former Classes about
market structure
? Perfect Competition
? Monopoly
? Monopolistic Competition
? Short-Run Equilibrium for Each Type of
Market
? Long-Run Equilibrium for Each Type of
Market
? Evaluation for Each Type of Market
? Application for Each Type of Market
北京大学经济学院4
Outlines of Today’s Class
? What is Oligopoly?
? Quantity Competition
? Cournot Duopoly（古诺双寡头模型）
? Stackelberg Duopoly（斯泰克伯格双寡头
模型）
北京大学经济学院5
Readings about the part of this chapter
? Zhang: Chapter 10,P285-311
? Nicholson: Chapter 19, P528-537
北京大学经济学院6
The Spectrum of Market Structure
Perfect
Competition
Monopoly
Monopolistic
Competition
Oligopoly
Imperfect
Competition
Imperfect
Competition
北京大学经济学院7
How to Measure the degree of
concentration in Imperfect Markets?
? Economists use concentration ratios(市场集
中率) to measure the degree of concentration
in a market.
? A four-firm concentration ratio（CR
4
）is
the percentage of the market output
produced by the 4 largest firms.
北京大学经济学院8
measuring market dominance
? 4-firm concentration ratio(CR
4
)
? % sales from 4 largest firms
? > 40% then oligopoly
? < 40% then monopolistic comp.
北京大学经济学院9
? 4 Firm Concentration Ratios is the percentage
of total industry sales of the 4 largest firms in
the industry.
? For Example: there are 10 firms in a TV sets
industry
How to Measure the degree of concentration in
Imperfect Market?
Firm A = 20%
Firm D = 2%
Firm C = 6%
Firm G = 3%
Firm F = 35%
Firm J = 11%
Firm H = 7% Firm I = 3%
Firm E = 8%
Firm B = 5%
北京大学经济学院10
Out of 10 firms in the TV industry
the leading 4 constitutes 74% of
total sales
An example:
北京大学经济学院11
A measure of industry concentration,
calculated as the sum of the
squares of the market shares held
by each firm in the industry
Another Way of Measuring the degree of
concentration in Oligopoly
北京大学经济学院12
The Herfindahl-Hirschman Index(H指数):
........SSSSHHI
2
4
2
3
2
2
2
1
++++=
HHI = 20
2
+ 5
2
+ 6
2
+ 2
2
+ 8
2
+ 35
2
+ 3
2
+ 7
2
+ 3
2
+ 11
2
=1942
In this case 1,000 < HHI < 10,000
Another Way of Measuring the degree of
concentration in Imperfect Market
A measure of industry concentration,
calculated as the sum of the squares of
the market shares held by each firm in the
industry
北京大学经济学院13
Herfindahl-Hirschman Index (HHI)
? largest 50 firms
? sum square of % market share
? used by Justice Department(司法部门)
? if monopoly
= (100)
2
= 10,000
北京大学经济学院14
HHI (cont.)
? if < 1000
? market is competitive
? if > 1800
? market is uncompetitive
北京大学经济学院15
北京大学经济学院16
Oligopoly: Introduction
? Alternative Models of Imperfect Competition
? Monopoly and monopolistic competition
? Duopoly(双寡头厂商) - two firms in industry
? Oligopoly - a few (>2) firms in industry
? Essential Features of Oligopoly
? Nature of interaction between firms (beyond
those captured in price) is essence of theories
? No single “grand theory”（无单一理论）
北京大学经济学院17
What is Oligopoly?
? An Oligopoly is a market served by a relatively few firms.,
usually less than 10.
? Duopoly（双寡头市场）-two firms
? Triopoly(三寡头厂商) - three firms
? Multipoly(多寡头市场)
? The products firms offer can be either differentiated（异质性）
or homogeneous（同质性）.
? Firms in an oligopoly are interdependent(相互依存
相互依存
).
The actions of one firm affect the profits of the
other firms.
? So the key feature of an oligopoly is that firms act
strategically.
北京大学经济学院18
Examples of Oligopolies
? Tennis Balls: Wilson, Penn, Dunlop and Spalding.
? Cars: GM, Ford, DaimlerChrysler.
? Cereal(食品): Quaker, Ralston Food, Kellogg,
Post and General Mills.
? Airlines: American and Delta with US Airways,
Northwest and TWA struggling along.
? Aircraft: Boeing (+McDonnell Douglas) and
Lockheed Martin
? Mobile Phone: Nokia and Motorola
北京大学经济学院19
? steel
? cigarettes
? oil
? automobiles
? chemicals
? PC operating systems
? copy machines
Examples of Oligopolies
北京大学经济学院20
Oligopoly - Assumptions
? Assumption 1: There are a relatively few
firms., usually less than 10
? Each seller is large enough to influence price, it
means each seller faces a downward sloping
demand curve
? Assumption 2: firms are interdependent
? The actions of any one seller in the market can
have a large impact on the profits of all the
other sellers.
北京大学经济学院21
Oligopoly - Assumptions
? Assumption 3: The products firms offer can
be either differentiated or homogeneous.
? Product may be the same, such as aluminum
and crude oil。Pure Oligopoly
? product may be different, such as copy
machines and cigarettes. Differentiated
Oligopoly
? but competing products are similar.
北京大学经济学院22
Oligopoly - Assumptions
? Assumption 4: not easy for entry and Exit.
? But there are Competitions among sellers
? ranges from collusion to cutthroat
competition
Cartel
Cutthroat
competition
（残酷的竞争)
Open
collusion
Covert
collusion
（隐蔽的
串谋）
Price
leadership
北京大学经济学院23
Oligopoly: Traditional Models
? Cooperative Models(合作型模型)
? Cartels
? Tacit Coordination（默契合谋）
? Price Leadership Model（价格领导模型）
? Non-Cooperative Models (非合作型模型)
? Kinked Demand Curve Model（折弯的需求曲线模
型）
? Cournot Duopoly（古诺双寡头模型）
? Stackelberg Duopoly（斯泰克伯格双寡头模型）
? Bertrand Duopoly（伯特兰双寡头模型）
北京大学经济学院24
Oligopoly: Traditional Analysis
? Simplest Model of Oligopoly: Duopoly
? Assume only two firms (to limit interactions)
? Assume homogeneous output
? No product differentiation
? Single market price
? No competition in quality
? Equilibrium: Solve for output, price of each firm
北京大学经济学院25
Cournot Model
1
（古诺模型）
1
Augustin Cournot. Research Into the Mathematical Principles of
the Theory of Wealth, 1838
? Illustrates the principle of mutual
interdependence among sellers in
tightly concentrated markets--even
where such interdependence is
unrecognized by sellers.
北京大学经济学院26
Cournot Model：Assumptions
1. Two sellers
2. TC
1
=TC(q
1
), TC
2
=TC(q
1
)
3. Homogeneous product
4. Output q is the “decision
variable”
5. Maximizing behavior (Profit)
北京大学经济学院27
Cournot Model
? Assume output is strategic variable, each firm
chooses output to max profits, given output
level of competitor
? So “Firms compete in outputs”
? Firm 1: q
1
units; Firm 2: q
2
units
? total quantity supplied is q
1
+ q
2
? market price will be p(q
1
+ q
2
)
? total cost functions are c
1
(q
1
) and c
2
(q
2
)
北京大学经济学院28
Cournot Model: Quantity Competition
? Firm 1 maximizes profit, given q
2
? Firm 1 profit function:
Π
1
(q
1
; q
2
) = p(q
1
+ q
2
)q
1
–c
1
(q
1
)
? Firm 1 “Reaction Function”(反应函数)
? What output q
1
maximizes firm 1 profit?
? Given q
2
(expected or observed)
? Solve for reaction function q
1
=R(q
2
)
北京大学经济学院29
Quantity Competition: Example
? Let market inverse demand function be
p(Q) = 60 - Q
Q= q
1
+ q
2
? Let firms’ (different) total cost functions be
c
1
(q
1
) = q
1
2
c
2
(q
2
) = 15q
2
+ q
2
2
北京大学经济学院30
Quantity Competition: Responses
2
112 1 21 1
(; ) (60 )qq qqqqΠ=???
Firm 1 profit function is
So, given q
2
, solve for firm 1 profit-maximizing q
1
1
12 1
1
602qq2q0
q
?
?
Π
= ???=
Firm 1’s reaction function (best response) is
112 2
1
qR(q)15 q
4
==?
北京大学经济学院31
Quantity Competition: Graph
q
2
q
1
60
15
Firm 1’s “Reaction Curve” R
1
(q
2
)
112 2
21
1
()15
4
(604)
qRq q
orq q
==?
=?
北京大学经济学院32
Quantity Competition: Responses
2
221 1 22 2 2
(;)(60 ) 15qq q qq q qΠ =?? ? ?
Similarly, given q
1
, Firm 2’s profit function is
To get Firm 2’s profit-maximizing output
2
12 2
2
60 2 15 2 0qq q
q
?
?
Π
= ?? ?? =
Firm 2’s reaction function (best response) is
1
221
45
()
4
q
qRq
?
==
北京大学经济学院33
Quantity Competition: Graph
q
2
q
1
Firm 2’s “Reaction Curve” R
2
(q
1
)
1
221
45
()
4
q
qRq
?
==
45/4
45
北京大学经济学院34
Quantity Competition: Equilibrium
? Equilibrium is a Cournot-Nash equilibrium
? Each firm’s output level is best response to other
firm’s output level
? Stable: neither firm wants to change output
? Thus, (q
1
*,q
2
*) such that
? q
1
* = R
1
(q
2
*) and
? q
2
* = R
2
(q
1
*)
北京大学经济学院35
Quantity Competition: CN Equilibrium
** *
112 2
1
()15
4
qRq q==?
*
**
1
221
45
()
4
q
qRq
?
==
and
Substitute for q
2
* to get
*
**
1
11
145q
q15 q13
44
???
=? ?=
??
??
*
2
45 13
8
4
q
?
= =
Cournot-Nash equilibrium is
**
12
(,)(13,8)qq =
北京大学经济学院36
Quantity Competition: Graph
q
2
q
1
Firm 2’s “reaction curve”
60
15
Firm 1’s “reaction curve”
112 2
1
()15
4
qRq q==?
1
221
45
()
4
q
qRq
?
==
45/4
45
北京大学经济学院37
Quantity Competition: Graph
q
2
q
1
48
60
8
13
Cournot-Nash equilibrium
( )
( )
**
12
,13,8qq =
Firm 2’s “reaction curve”
Firm 1’s “reaction curve”
112 2
1
()15
4
qRq q==?
1
221
45
()
4
q
qRq
?
==
北京大学经济学院38
Cournot Duopoly Concluded
? Let the market (inverse) demand function:
? p = a – bQ
? where Q = q
1
+ q
2
? MC
1
= MC
2
= c
The Linear Demand Curve
The Linear Demand Curve
北京大学经济学院39
Oligopoly
? An Example of the Cournot Equilibrium
11 1
Total Revenue, T ( )R Pq a bQ q= =?
1121
2
11 21
()aq b q q q
aq bq bq q
= ?+
=??
The Linear Demand Curve
The Linear Demand Curve
北京大学经济学院40
Oligopoly
111 12
11
12
21
2
Firm 1's Reaction Curve
1
22
Firm 2's Reaction Curve
1
22
MRTRqabqbq
MR MC c
ac
qq
b
ac
qq
b
=? ? = ? ?
==
?
=?
?
=?
The Linear Demand Curve
The Linear Demand Curve
北京大学经济学院41
An Example of the Cournot Equilibrium
12
21
22
22
12
12
Cournot Equilibrium:
*,*
33
2( )
3
3
ac b
qq
ac b
qq
Cournot
Cournot
ac ac
qq
bb
ac
Qqq
b
ac
PabQc
?
=?
?
=?
?
?
?
??
==
?
=+=
?
=? =+
北京大学经济学院42
q of Firm 1
q of Firm 2
(a – c)/b
(a – c)/2b
(a – c)/b
(a – c)/2b
Cournot equilibrium (q
1
*
, q
2
*
)
Firm 1’s Reaction function
q
1
= (a – c)/2b –q
2
/2
Firm 2’s Reaction function
q
2
= (a – c)/2b –q
1
/2
q
2
*
q
1
*
q
1
* = q
2
* = (a – c)/3b.
北京大学经济学院43
? Monopoly output would be (MR = a –
2bq
m
= c)
? q
m
= (a – c)/2b.
? Competitive output would be (p = a – bq
c
= c)
? q
c
= (a – c)/b.
? Hence, q
m
< q
Cournot
< q
c
.
北京大学经济学院44
? p
Cournot
= c + (a – c)/3.
? Monopoly price would be
? p
m
= c + (a – c)/2.
? Competitive price would be
? p
c
= c.
? Hence,
? p
m
> p
Cournot
> p
c
.
北京大学经济学院45
Firm 1’s
Reaction Curve
Firm 2’s
Reaction Curve
Duopoly Example
q
1
q
2
q
Cour.
q
Cour.
Cournot Equilibrium
q
C
q
C
Competitive Equilibrium (P = MC; Profit = 0)
Collusion
Curve
q
m
q
m
Collusive Equilibrium
For the firm, collusion is the best
outcome followed by the Cournot
Equilibrium and then the
competitive equilibrium
北京大学经济学院46
? Knowing p
Cournot
and q
i
, we can calculate each firm’s maximum
profit
? π
i
= (p
Cournot
–c)q
i
=[c + (a – c)/3 - c]* [(a – c)/3b]
? = (a – c)
2
/9b.
? Duopoly profit is
? π
Cournot
= 2π
i
= 2(a – c)
2
/9b.
? Monopoly profit would be
? π
m
= (p
m
–c)q
m
= [c + (a – c)/2 - c]* [(a – c)/2b]
? = (a – c)
2
/4b.
? Competitive industry profit would be
? π
c
= 0.
? Hence, π
m
> π
Cournot
> π
c
= 0.
北京大学经济学院47
Comparison of Outcomes
? Cournot equilibrium
? Price is < monopoly but > perfect competition
? Quantity is > monopoly but < perfect competition
? Total profit is < monopoly but > perfect competition
北京大学经济学院48
p
Cour.
q
Cour.
p
c
q
cq
i
p
Q
D
MC
CS
π
北京大学经济学院49
p
m
q
m
p
duo
q
duo
p
c
q
c
SW
m
< SW
Cour.
< SW
c
From PC to Duolopy,
?SW= - A
From Duolopy to Monopoly,
?SW= - (B+C)
q
i
price
quantity
D
MC
MR
B
A
C
北京大学经济学院50
Monopoly Duopoly PC
Output (a – c)/2b 2(a – c)/3b (a – c)/b
Price c + (a – c)/2 c + (a – c)/3 c
Profit (a – c)
2
/4b 2(a – c)
2
/9b 0
CS
0.5(a – p
m
)q
m
0.5(a – p
Cournot
)q
Cournot
0.5(a – p
c
)q
c
SW
CS
m
+ π
m
CS
Cournot
+ π
Cournot
CS (PS = 0)
北京大学经济学院51
Monopoly Duopoly PC
Output (a – c)/2b < 2(a – c)/3b < (a – c)/b
Price c + (a – c)/2 > c + (a – c)/3 > c
Profit (a – c)
2
/4b > 2(a – c)
2
/9b > 0
CS 0.5(a – p
m
)q
m
< 0.5(a – p
Cour.
)q
Cour.
< 0.5(a – p
c
)q
c
SW
CS
m
+ π
m
<
CS
Cour.
+ π
Cour.
< CS (PS = 0)
北京大学经济学院52
von Stackelberg Model: Introduction
? Assumption:
? One firm (larger firm) moves first
? Then “follower firms” react
? Both consider reactions of other
? Can compete in
? Quantity -- von Stackelberg Model
? Price -- Price leadership models
北京大学经济学院53
The von Stackelberg Model(斯泰克伯格模型)
? Outputs are strategic variables
? Firm 1 -- leader firm（领导者厂商）-- chooses
q
1
first
? Firm 2 – follower（跟随着厂商）-- then reacts
? Leader anticipates reaction of follower
? Issues
? What are prices, outputs, profits
? Is there a “first mover” advantage(先动优势)?
北京大学经济学院54
The von Stackelberg Model
? Follower firm will choose q
2
to maximize profit,
given leader firm q
1
(C-N assumption)
? Thus, follower reaction function: q
2
= R
2
(q
1
)
? Leader firm (1) anticipates follower firm’s (2)
reaction function, so chooses q
1
to max profit
Π
1
S
(q
1
) = p[q
1
+ R
2
(q
1
)] q
1
–c
1
(q
1
)
北京大学经济学院55
Von Stackelberg Model: Profits
? Return to duopoly example of different MC’s
? Leader firm 1 has lower costs
c
1
(q
1
) = q
1
2
? Follower firm 2 has higher costs
c
2
(q
2
) = 15q
2
+ q
2
2
北京大学经济学院56
Von Stackelberg Model: Example
? Market inverse demand function is
p = 60 - Q
? The firms’ cost functions are
c
1
(q
1
) = q
1
2
and c
2
(q
2
) = 15q
2
+ q
2
2
? Firm 2 is follower, with reaction function
1
221
45
()
4
q
qRq
?
==
北京大学经济学院57
2
11 1 21 1 1
2
1
111
2
11
() (60 ()
45
(60 )
4
195 7
44
s
qqRqqq
q
qqq
qq
Π=?? ?
?
= ?? ?
=?
Leader’s profit function is
For a profit-maximum, first order condition is
s
11
195 7
qq13.9
42
=?=
Von Stackelberg Model: Example
北京大学经济学院58
Follower firm’s response to q
1
=13.9 is
2
221
45 13.9
() 7.8
4
S
qRq
?
== =
Recall C-N outputs are (q
1
*,q
2
*) = (13,8)
So leader produces more than C-N output,
follower produces less than its C-N output
First mover advantage to leader (but modest
because leader also has cost advantage)
Von Stackelberg Model: Example
北京大学经济学院59
The End
北京大学经济学院60
Last Revised:
November. 22, 2005
北京大学经济学院1
Chapter 8
Oligopoly
(寡头市场)
? 2005 MOL
北京大学经济学院2
Chapter 8 includes:
? 8.1 What is Oligopoly?
? 8.2 Quantity Competition
? Cournot Duopoly（古诺双寡头模型）
? Stackelberg Duopoly（斯泰克伯格双寡头模型）
? Kinked Demand Curve Model（折弯的需求曲线模型）
? 8.3 Price Competition
? Bertrand Duopoly（伯特兰双寡头模型）
? Price Leaderships
北京大学经济学院3
Overview of Last Class
? What is Oligopoly?
? Quantity Competition
? Cournot Duopoly（古诺双寡头模型）
? Stackelberg Duopoly（斯泰克伯格双寡头
模型）
北京大学经济学院4
Outlines of Today’s Class
? Kinked Demand Curve Model（折弯
的需求曲线模型）
? Bertrand Duopoly（伯特兰双寡头
模型）
? Price Leaderships(价格领导者模型)
北京大学经济学院5
Readings about the part of this chapter
? Zhang: Chapter 10,P285-311
? Nicholson: Chapter 19, P528-537
北京大学经济学院6
Chapter 8 includes:
? 8.1 What is Oligopoly?
? 8.2 Quantity Competition
? Cournot Duopoly（古诺双寡头模型）
? Stackelberg Duopoly（斯泰克伯格双寡头模型）
? Kinked Demand Curve Model（折弯的需求曲线模型）
? 8.3 Price Competition
? Bertrand Duopoly（伯特兰双寡头模型）
? Price Leaderships
北京大学经济学院7
Sweezy (Kinked-Demand) Model斯威齐
（或折弯的需求曲线）模型
? Strategic interdependence: You aren’t in
complete control of your own destiny!
? The effect of a price reduction on the quantity
demanded of your product depends upon
whether your rivals(竞争对手) respond by cutting
their prices too!
? The effect of a price increase on the quantity
demanded of your product depends upon
whether your rivals respond by raising their
prices too!
北京大学经济学院8
Sweezy (Kinked-Demand) Model
? A Few firms in the market
? Each producing homogeneous products.
? Barriers to entry
? Each firm believes rivals will match (or follow)
price reductions, but won’t match (or follow)
price increases.
? Key feature of Sweezy Model
? Price-Rigidity(价格刚性)
北京大学经济学院9
Oligopoly - Kinked Demand Curve
? Underlying assumptions of firms
? competitors will lower prices if you do in
order to keep market share
? competitors will not raise prices if you do in
order to increase their market share
? Above leads to what, in reality, are two
demand curves; one for price increases
and another for decreases
北京大学经济学院10
Q
D
1
Q
0
(Rival holds its
price constant)
D
2
(Rival matches your price change)
P
P
H
P
0
P
L
北京大学经济学院11
P
Q
D
1
P
0
Q
0
D
2
(Rival matches your price change)
(Rival holds its
price constant)
D
Demand if Rivals Match Price
Reductions but not Price Increases
北京大学经济学院12
Oligopoly - Kinked Demand Curve
P
Q
Demand
北京大学经济学院13
Sweezy Marginal Revenue
D
1
Q
0
D
2
(Rival matches your price change)
(Rival holds its
price constant)
MR
1
MR
2
D
P
P
0
Q
MR
北京大学经济学院14
Oligopoly - Discontinuous MR Curve
P
Q
Demand
MR
北京大学经济学院15
Oligopoly - Kinked Demand Curve
? Occurs only in cutthroat competition
? Demand is kinked because of one firm’s
view of how other firms will react if they
raise or lower prices
? Kinked demand curve leads to
discontinuous(不连续) MR curve
? Result is that a firm is reluctant to change
prices either up or down
北京大学经济学院16
Oligopoly - Kinked Demand Curve
? Leads to sticky (administered) prices.
? Firms absorb many cost increases without
changing prices.
? Firms do not pass cost decreases on to the
consumer without changing prices.
北京大学经济学院17
Oligopoly - Sticky Prices
P
Q
D
MC
Q*
MR=MC
A
SAC
P*
MR
北京大学经济学院18
P
Q
D
MR
MC
1
MC
2
Q*
Price changes only
when MC shifts out
of the MR gap
Why are prices sticky under oligopoly?
If Costs increase
within the MR gap…
MC
3
MC
3
Q
3
P
3
P*
Price does not
change
北京大学经济学院19
Why are prices sticky under oligopoly?
Price does not
change if demand
curve shifts
Price does not
change if demand
curve shifts
北京大学经济学院20
The Kinked Demand Model
? Developed to explain why prices in
oligopoly markets tended to be
inflexible(僵硬的).
? Changes in costs were only rarely met by
changes in prices
? Price changes did occur when changes in
costs were large in magnitude.
This model explains why prices under monopoly tend to be “sticky”
北京大学经济学院21
3. Criticisms of the Kinked Demand Curve Theory
? This theory has two basic shortcomings:
? first, it ignores where the prevailing
(“kink”) price came from;
? second, empirical evidence points out a
number of oligopolies whose behavior
could not be explained by a kinked
demand curve.
北京大学经济学院22
Chapter 8 includes:
? 8.1 What is Oligopoly?
? 8.2 Quantity Competition
? Cournot Duopoly（古诺双寡头模型）
? Stackelberg Duopoly（斯泰克伯格双寡头模型）
? Kinked Demand Curve Model（折弯的需求曲线模型）
? 8.3 Price Competition
? Bertrand Duopoly（伯特兰双寡头模型）
? Price Leaderships(价格领导者模型)
? 8.4 Differentiated Product Market
? Cournot & Bertrand Duopoly Model
? Linear City Model(线形城市模型)
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Price Competition: Bertrand Model（伯特兰模
型）
? Alternative strategic behavior
? Firms compete using only price (not quantity)
? Bertrand Model
? Simultaneous game（同时博弈）
? Firms use price as strategic variable
? Get results dramatically different from quantity
competition
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Bertrand (1883) price competition.
? Both firms choose prices simultaneously and have
constant marginal cost：MC
1
=MC
2
=c.
? Firm 1 chooses p
1
. Firm 2 chooses p
2
.
? Consumers buy from the lowest price firm. (If p
1
=p
2
,
each firm gets half the consumers.)
? An equilibrium is a choice of prices p
1
and p
2
such
that
? firm 1 wouldn’t want to change his price given p
2
.
? firm 2 wouldn’t want to change her price given p
1
.
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Bertrand Model
? Example of Bertrand Model
? Each firm’s MC = c, constant
? All firms simultaneously set their prices
? Equilibrium: All firms set p=c
? All firms have same p, or high p loses all sales
? Any p>c, slight price reduction yields big profit
? Any p<c, lose money
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Bertrand Equilibrium
? Take firm 1’s decision if p
2
is strictly bigger than c:
? If he sets p
1
>p
2
, then he earns 0.
? If he sets p
1
=p
2
, then he earns 1/2*D(p
2
)*(p
2
-c).
? If he sets p
1
such that c<p
1
<p
2
he earns D(p
1
)*(p
1
-
c).
? For a large enough p
1
<p
2
, we have:
? D(p
1
)*(p
1
-c)>1/2*D(p
2
)*(p
2
-c).
? Each has incentive to slightly undercut the other.
? Equilibrium is that both firms charge p
1
=p
2
=c.
北京大学经济学院27
The Bertrand Paradox(伯特兰悖论)
? The conclusions of the Bertrand Model are the
following:
? (i) that firms price at marginal cost, and
? (ii) that firms do not make profit.
? These conclusion does not match the results of
cournot duopoly model. That means even duopoly
would suffice to restore competition, and we can get
the competitive equilibrium from oligopoly.
? We call this the Bertrand Paradox.
北京大学经济学院28
How to solve the Bertrand Paradox
? The Edgeworth Solution（埃奇沃思解法）
? Introducing capacity construction, by which firms cannot
sell more than they are capable of producing.
? The temporal Dimension(时间纬度)
? The “timing”(速度) of price reduction of each firm does
always not seem reflect economic reality.the more
collusive behavior than in the Bertrand equilibrium can be
sustained by the threat of future losses in a price war.
? Product Differentiation(产品差异化)
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The Edgeworth Model
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Price-leadership model
? Price-leadership
? Sequential game
? Price-leader firm sets its price
? Typically large, respected firm
? Dominant firm
? Barometric firm
? Follower firms – usually smaller – react to leader
? Note: Follower firms are price takers
? Analogous to competitive firms
北京大学经济学院31
Price Leadership
? Market demand function is D(p)
? Given leader price p, follower firms supply
q
f
(p), anticipated by leader
? So leader gets residual demand
L(p) = D(p) – q
f
(p)
? Leader’s chooses p to max profit
Π
L
(p) = p[D(p) - q
f
(p)] – c
L
[D(p) - q
f
(p)]
北京大学经济学院32
Price Leadership
? Results
? Followers act as competitors
? P=MC
? Economic profit of each is zero
? Leader acts as monopolist residual demand
? MR
L
=MC
L
? Only leader earns monopoly profits
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Co-operative Behavior: Collusion
? Collusion is illegal in US
? But not for international cartels
? OPEC
? Bauxite, copper, tin, coffee, tea, mercury, iodine
? Goal of cartel: Joint profit maximization
? Can achieve (joint) monopoly profits
? Must divide output, profits among cartel members
? If cartel fraction of market, like dominant firm model
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Co-operative Behavior: Collusion
? Fundamental tension for cartels
? Stability: Higher profits (share of joint max)
? Instability
? Successful cartel has p>>MC
? One member alone faces nearly fixed p
? Gets huge profits if lowers own price while
others hold price constant (cheat on agreement)
? Has led to downfall of most, but not all, cartels
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Co-operative Behavior: Collusion
? Factors that promote cartel cohesion
? Similar costs, expectations of demand, motives so
can agree on strategy
? Large share of market
? Small number of members
? Inelastic demand so potential profits large
(disincentive for cheating)
? Inelastic demand in LR so profits maintained
? Little expansion of supply by non-members in LR
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Comparison of Different Types of Markets
Perfect
Competition
Monopolistic
Competition Oligopoly Monopoly
Number of firms Very large
number
Many Few One
Type of product Standardized
(homogeneous)
Differentiated Standardized
or differentiated
Unique
Demand faced
by individual
firm
Price taker:
demand is
perfectly
elastic
Demand is price
elastic but not
perfectly elastic
Demand is less elastic
than demand facing
monopolistically
competitive firm
Firm faces market
demand curve
Entry conditions No barriers No barriers Large barriers from
government policies
or economies of scale
Large barriers from
economies of scale
or government
policies
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Comparison of Different Types of Markets
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The End
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Last Revised:
November. 22, 2005
北京大学经济学院1
Chapter 9
Game Theory
? 2005 MOL
北京大学经济学院2
Chapter 9 includes:
? 9.1 Introduction to Game Theory
? 9.2 Nash Equilibrium(纳什均衡)
? 9.3 Subgame Perfect Nash
Equilibrium（子博弈精炼纳什均衡）
? 9.4 Repeated Game（重复博弈）
北京大学经济学院3
Overview of Last Class
? What is Oligopoly?
? Quantity Competition
? Cournot Duopoly（古诺双寡头模型）
? Stackelberg Duopoly（斯泰克伯格双寡头模型）
? Kinked Demand Curve Model（折弯的需求曲线模
型）
? Price Competition
? Bertrand Duopoly（伯特兰双寡头模型）
? Price Leaderships
北京大学经济学院4
Outlines of This Week’s Class
? Game Theory: Introduction
? Elements of a game
? Classifications of Game
? Dominant strategy（占优策略）
? Dominant strategy equilibrium（占优策略均衡）
? Nash equilibrium（纳什均衡）
? Mixed Strategies
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Readings about the part of this chapter
? Zhang: Chapter 13,P375-408
? Nicholson: Chapter 10, P246-264
Chapter 20,P554-572
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Chapter 9 includes:
? 9.1 Introduction to Game Theory
? 9.2 Nash Equilibrium(纳什均衡)
? 9.3 Subgame Perfect Nash
Equilibrium（子博弈精炼纳什均衡）
? 9.4 Repeated Game（重复博弈）
北京大学经济学院7
Brief History of Game Theory
? 1913 - E. Zermelo provided the first theorem of game theory
asserts that chess is strictly determined
? 1928 - John von Neumann(冯.诺伊曼) proved the minimax
theorem(最小最大原理)
? 1944 - John von Neumann / Oskar Morgenstern’s wrote
"Theory of Games and Economic Behavior”
? 1950-1953, John Nash describes Nash equilibrium
? 1965-R. Selten(泽尔腾): Subgame perfect Nash
Equilibrium;Backward Induction.
? 1967-J. Harsanyi（海塞尼）: Game with incomplete
infroamtion; Bayesian Nash Equilibrium
北京大学经济学院8
Game Theory: Introduction
Evaluation of Game Theory（对博弈论的评价）
Rubinstein:
1950’s-- era of general equilibrium
1960’s-- era of growth
1970’s—era of economics of information
1980’s – era of game theory
("Introduction" in Game theory in Economics, eds by
A. Rubinstein, 1990, p.xi)
北京大学经济学院9
Game theory is everywhere - Economics & Business
? “Game theory is hot!” - Wall Street Journal, 13 February
1995,
? Auctioneer（拍卖人）and bidders（投标人）;
? Labor union negotiation with employer & repeated game;
? Negotiation between buyer and seller;
? Airlines’ price wars & tacit collusion;
? Timing of launch of new products in IT industry;
? Evaluation in merger and acquisitions & winner’s curse（赢者诅咒）;
? Insurance companies vs. the insured & asymmetric information;
? Others ...
北京大学经济学院10
Game theory is everywhere - Beyond
Economics & Business
? Presidential election
? International relations
? Office politics
? Dating strategies
? War
? Sports games
? Recruiting
? Everything...
北京大学经济学院11
Game theory is everywhere - Even hot in
Pop Culture these days
?John Nash &
Nash equilibr.
?2001
?Prisoner’s
dilemma
?2002
?Game of
chicken
北京大学经济学院12
How to describe a game?
? Players（博弈参与人）
? decision makers
? firms, governments, individuals
? Strategies(策略) (or choices or actions)
? output, prices, advertising budget
? Payoffs（得益）
? depend not only on own strategies, but on other’s
strategies as well
? games can be one-shot, repeated, or dynamic.
北京大学经济学院13
Examples of simple games
? Matching pennies（猜硬币博弈）
? 2 players: Player A, B
? actions: Heads（正面）or Tails（反面）
? payoffs: player A gets $1 (and player B loses $1)
if pennies match. Player A loses a dollar and
player B gains $1 if pennies don’t match.
? zero sum game（零和博弈）.
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Matching Pennies （猜硬币博弈）
Player B
Heads Tails
Player A
Heads
Tails
1, -1 -1, 1
1, -1-1, 1
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Example: Rock paper scissors
Person B
rock paper
paper 1,-1
-1,1
0,0
0,0
rock
scissors
scissors
1,-1
-1,1
-1,1 1,-1 0,0
Person A
北京大学经济学院16
The Battle of the Sexes（性别之战或爱情博弈）
Football Opera
Maria
2,1 0,0
1,20,0
Football
Tom
Opera
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Prisoner’s Dilemma（囚徒困境）
探长
Prisoner A Prisoner B
北京大学经济学院18
Prisoner’s dilemma（囚徒的困境）
? Very important game
? 2 players: Prisoner A, B
? actions: deny（抵赖）or confess（坦白）
(cooperate or defect)
? payoffs: 0 if you confess while other denies, -1
if you both deny, -5 if you both confess, -8 if
you deny and the other guy confess.
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Prisoner’s Dilemma
北京大学经济学院20
Definition of Game Theory
? Game theory is a method of analysing
strategic behaviour – behaviour that takes
into account the expected behaviour of
others and the mutual recognition of
interdependence.
? Game theory seeks to understand oligopoly
by using games of all types, including games
of everyday life.
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Game Theory: Applications
? Game theory has been applied to analyze
? Oligopolies
? Cartels: OPEC
? Tax competition across jurisdictions, countries
? Military strategies
? Externalities: using common resources like fishery
北京大学经济学院22
What dose Game Theory Use for?
? Game theory examines oligopolistic behaviors as a
series of strategic moves and countermoves(对抗手段)
among rival firms.
? It analyses the behavior of decision-makers, or players,
whose choices affect one another.
? The focus is on the players’ incentives either to
cooperate or to compete.
? Game theory can be used to simulate oligopolistic
behavior in order to decide on optimal strategies.
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Elements of a game
Players
Players: decision making entities
Nature: A non-purposeful entity, it
chooses its actions
Strategic Players: rational decision
makers
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Elements of a game
Actions（行动）and
Strategies（策略）
Actions: The set of choices available each
decision node in a game is refered to as the
players’ actions.
Pure Strategy（纯策略）: A pure strategy for a
player is a rule that tells the player what actions
to take at each of his information set in the game.
It is a detailed plan that tells him what activities
take under every contingency.
北京大学经济学院25
Elements of a game
Mixed strategy（混合策略）: Mixed strategy
appears when player choose randomly（随机地）
between actions available to them. Formally, a
mixed strategy for a player consists of a probability
distribution on the set of pure strategies.
北京大学经济学院26
Elements of a game
Rationality and common knowledge
Rational behaviors: choices made according to
internally consistent criteria
Common knowledge: A “fact” in a game is said to
be “common knowledge” if every player knows it,
every player knows that every other player knows it,
every player knows that every other player knows
that every other player knows it, and so on.
北京大学经济学院27
Elements of a game
Some usual assumption
A1. The rationality of every player is common
knowledge.
A2. The complete description of the game ---
players, actions, strategies, order of play,
information, and payoff --- is also common
knowledge
北京大学经济学院28
Elements of a game
The order of play（博弈顺序）
Player make decisions at various points in a game. These
points are called decision nodes.
The sequence is which decisions are made as called order of
play.
If players in a game have to make their decision at the same
times, we call it a simultaneous move（同时行动）.
If players make their decisions in a particular sequence, one
after another, we call it a sequential move（序贯行动）.
北京大学经济学院29
Elements of a game
Information
A game is said to have PERFECT RECALL if no
player forgets any information he know, and all
player know the action they previously took.
A game of perfect information is one in which
every player at every decision node knows the
decision taken previously by every other player
(including Nature).
北京大学经济学院30
Classification of games:
These are several ways of classifying
games: by
(1) the number of players,
2 person game
n person game
北京大学经济学院31
Classification of games:
(2) the number of strategies available to
each of the players.
When the number of strategies is finite,
we have finite games.
When the number of strategies is infinite,
we have infinite games.
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Classification of games:
(3) the nature of payoffs.
zero-sum game
(the sum of the payoffs spends zero)
non-zero sum game
北京大学经济学院33
Classification of games:
? (4) the nature of preplay negotiation.
? cooperative games
? non-cooperative games
How to define cooperative vs. non-cooperative game
A game is
non-cooperative
if players can not make
binding commitments(or agreements)(约束性合
约), and cooperative otherwise, irrespective of the
possibility of communications.
北京大学经济学院34
Classification of games:
(5) the nature of the states:
A game is called a stochastic game（随机博弈）
if it contain random variables.
Otherwise it is a deterministic game（确定性博
弈）.
(6) Interactions over time
A game is called a Dynamic Game if actions of a
player at a particular time instance are affected by his
previous actions.
Otherwise it is a static games.
北京大学经济学院35
Classification of games:
(7) perfect information vs. imperfect information game
Perfect information（完美信息博弈）: at each move in the
game, the player with the move knows the full history of the
play of the game thus far.
ex) chess, football, English auction
Sequential and open moves -> [dynamic games] = games with
more than 2 stages
Imperfect information（非完美信息博弈）game = a player
does not know what others did.
ex) sealed bid auction, game of rock, paper, scissors.
Simultaneous and secret moves -> [static games] = games
with a single stage
北京大学经济学院36
(8) complete information（完全信息博弈）vs. incomplete
information（非完全信息博弈）game
Incomplete information = at least one player is uncertain about
another player’s payoff function. [Asymmetric or private
information]
Ex) Firm’s MC and workers’ ability may be private information.
北京大学经济学院37
Complete Incomplete
static
[Nash Equilibrium]
Nash, 1950-51
[Bayesian NE]
Harsanyi 1967-68
dynamic [Subgame Perfect NE]
Selten 1965, 1975
[Perfect Bayesian NE]
Harsanyi 1967-68
Nobel Prize in Economics in 1994
Three Game theorists, Nash, Harsanyi, and Selten won the Nobel
Memorial Prize in Economic Sciences in 1994 “for their pioneering
analysis of equilibria in the theory of non-cooperative
games”.
北京大学经济学院38
Ways to describe a game
? There are two ways to describe a
game:
? the “Normal” or matrix form
（标准型或距阵型）
? the “Extensive” form（扩展型）
北京大学经济学院39
Ways to describe a game
We can use the “Normal” or matrix form（标
准型或距阵型）if:
? There are only 2 (sometimes 3) players
? There are a finite number of strategies
? Actions approximately simultaneous
If actions are sequential, must use another
form, the “Extensive” form（扩展型）:
? Still only really feasible for 2 or 3 players,
although can accommodate “chance”
? Still must have finite number of strategies
北京大学经济学院40
Matching Pennies——the “Normal” or matrix form
Player B
Heads Tails
Player A
Heads
Tails
1, -1 -1, 1
1, -1-1, 1
北京大学经济学院41
Example of a game in normal form
Once again player 1 has two strategies, L and R.
Player 2 ha three strategies, L , M and R.
Player II
Player I
LMR
L (1,2) (3,1) (0,4)
R (6,1) (2,7) (7,8)
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Extensive Form Games
? Use a game “tree” to depict the order in which
players make decisions and the choices that
they have at each decision point.
? Decision points are called “nodes”.
? Players’ strategies or choices branch off from
each decision node.
? At the end of each branch on the game tree
are the payoffs the players would receive if
that branch were the path followed.
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Matching Pennies——Extensive Form
(-1,1)
(1,-1)
(-1,1)
(1,-1)
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Example of a Game is extensive form
Player 1 has two strategies L and R
Player 2 has 3 strategies L, M and R
L
M
R L
M
R
L
R
P1
P2
(1,2) (3,1)
(0,4) (6,1)
(2,7)
(7,8)
L
M
R L
M
R
L
R
P1
P2
(1,2) (3,1)
(0,4) (6,1)
(2,7)
(7,8)
The ordered pairs (1,2) gives the payoff of player 1 and the payoff
of the second player as 2. In the left hand side diagram, player 2
doesn’t know what Player 1 does. In the Right hand sides diagram,
P2 has access to P1’s decision.
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Chapter 9 includes:
? 9.1 Introduction to Game Theory
? 9.2 Nash Equilibrium(纳什均衡)
? 9.3 Subgame Perfect Nash
Equilibrium（子博弈精炼纳什均衡）
? 9.4 Repeated Game（重复博弈）
北京大学经济学院46
Concepts of Equilibrium for Games
? Dominant strategy（占优策略）is a strategy that
always gives a player a higher payoff than other
available strategies regardless of the actions of other
players
? Dominant strategy equilibrium（占优策略均衡）is the
predicted outcome where both players have dominant
strategies.
? Nash equilibrium（纳什均衡）is a situation in which
players interacting with one another each choose their
best strategy given the strategies that the other
players have chosen.
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Equilibrium for Games
? Dominant Strategies
? “I’m doing the best I can no matter what you do.”
? “You’re doing the best you can no matter what I
do.”
? Nash Equilibrium
? “I’m doing the best I can given what you are doing”
? “You’re doing the best you can given what I am
doing.”
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Nash’s Nash Equilibrium
( R.B. Myerson, "Nash Equilibrium and the History of Economic
Theory," Journal of Economic Literature, 37(3), September 1997,
1067-1082)
Nash's theory of noncooperative games should now be recognized as
one of the outstanding intellectual advances of the twentieth
century, comparable to the discovery of the DNA double helix in
the biological sciences.(p.1067)
Why? – It provides a general analytical framework (methodology) for
extending rational-choice analysis to non-market applications.
(p.1069)
So, economics could change from (marginalist era) social science
concerned with the production and allocation of material goods
to (today) the analysis of incentives in all social institutions.
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Two-Player Game: Example
? Players are A and B
? A has two strategies: “Up” and “Down”
? B has two strategies: “Left” and “Right”
? Payoff matrix - table showing payoffs to A and B
for each of four possible strategy combinations
? Payoff: (A,B)
北京大学经济学院50
Two-Player Game: Matrix
Payoff matrix for game
LR
U
D
(3,9)
(0,0)
(1,8)
(2,1)
Player B
Player A
Player A’s payoff is shown first
Player B’s payoff is shown second
北京大学经济学院51
Two-Player Game: Matrix
Player B
L R
(3,9)
(0,0)
(1,8)
(2,1)
U
Player A
D
If A plays Up and B plays Right then
A’s payoff is 1
B’s payoff is 8
北京大学经济学院52
Two-Player Game: Matrix
LR
U
D
(3,9)
(0,0)
(1,8)
(2,1)
Player B
Player A
A play is a pair such as (U,R) where
?1st element is strategy chosen by A
?2nd element is strategy chosen by B
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Two-Player Game: Matrix
Is (U,R) a likely play?
Player B
L R
(3,9)
(0,0)
(1,8)
(2,1)
U
Player A
D
If B plays Right, A will play Down (2 v. 1)
So (U,R) is not a likely play
北京大学经济学院54
Two-Player Game: Matrix
Is (D,R) a likely play?
Player B
L R
(3,9)
(0,0)
(1,8)
(2,1)
U
Player A
D
If B plays Right then A’s best reply is Down
If A plays Down then B’s best reply is Right
So (D,R) is a likely play
北京大学经济学院55
Two-Player Game: Matrix
Is (D,L) a likely play?
Player B
L R
(3,9)
(0,0)
(1,8)
(2,1)
U
Player A
D
If A plays Down then B’s best reply is Right
So (D,L) is not a likely play
北京大学经济学院56
Two-Player Game: Matrix
Player B
Is (U,L) a likely play?
L R
(3,9)
(0,0)
(1,8)
(2,1)
U
Player A
D
If A plays Up then B’s best reply is Left
If B plays Left then A’s best reply is Up
So (U,L) is a likely play
北京大学经济学院57
Nash Equilibrium: Introduction
? Nash equilibrium
? a play where each strategy is a best response to
the other
? Can be multiple Nash equilibria
? Example has two Nash equilibria
? (U,L) and
? (D,R)
北京大学经济学院58
Nash Equilibrium: Matrix
Player B
L R
(3,9)
(0,0)
(1,8)
(2,1)
Player A
U
D
(U,L) and (D,R) are both Nash equilibria
But (U,L) is preferred to (D,R) by both A & B
(U,L) is Pareto preferred to (D,R)
Is (U,L) the only (likely) equilibrium? NO
北京大学经济学院59
Nash Equilibrium and Simultaneous Move Games
Example 1: Prisoners’ dilemma
? Two prisoners – A & B are accused of collaborating
in a crime.
? Separate jail cells.
? Each asked to confess.
? If both confess, each receive 5 years.
? If neither confesses, each receive only 2 years.
? If one confesses & other doesn’t, the one that
confesses receives one year & other 10 years.
北京大学经济学院60
Simultaneous Move Games
Example 1: Prisoners’ dilemma
? Each faces dilemma, if both don’t confess only
receive 2 years. But can they trust each other?
? If A (B) doesn’t confess risks being taken
advantage of by B (A).
? No matter what A (B) does, B (A) comes out
ahead by confessing.
? Both have incentive to confess.
北京大学经济学院61
Simultaneous Move Games
Example 1: Prisoners’ dilemma
? Players – Two prisoners – A & B
? Each player has two strategies - Confess or Not
Confess
? Four possible outcomes – { (C,C), (C,NC),
(NC,C), (NC,NC) }; player A strategy listed first
& player B second.
北京大学经济学院62
Prisoners’ Dilemma
Prisoners’ Dilemma outcome:
? Player A – dominant strategy to confess
? Player B - dominant strategy to confess
? Dominant strategy equilibrium – both confess
(special case of Nash equilibrium).
北京大学经济学院63
The Prisoner’s Dilemma Game
NC=Not Confess, C=Confess, Payoffs are years in jail
Prisoner B
NC C
(-2,-2) (-10,-1)
(-1,-10)
(-5,-5)
NC
Prisoner A
C
北京大学经济学院64
The Prisoner’s Dilemma: Matrix
Prisoner B
NC C
Prisoner A
(-2,-2) (-10,-1)
(-1,-10)
(-5,-5)
NC
C
If Prisoner A plays No Confess,
Then Prisoner B’s best reply is Confess
北京大学经济学院65
The Prisoner’s Dilemma: Matrix
Prisoner B
NC C
Prisoner A
(-2,-2) (-10,-1)
(-1,-10)
(-5,-5)
NC
C
If Prisoner A plays No Confess then Prisoner
B’s best reply is Confess
If Prisoner A plays Confess then Prisoner B’s
best reply is Confess
北京大学经济学院66
The Prisoner’s Dilemma: Matrix
Prisoner B
NC C
(-2,-2) (-10,-1)
(-1,-10)
(-5,-5)
NC
Prisoner A
C
So no matter what Prisoner A plays,
Prisoner B’s best reply is always Confess
Confess is a dominant strategy for Prisoner B
北京大学经济学院67
The Prisoner’s Dilemma: Matrix
Prisoner B
NC C
Similarly, no matter what Prisoner B plays,
Prisoner A best reply is always Confess
Confess is a dominant strategy for Prisoner A also
Prisoner A
(-2,-2) (-10,-1)
(-1,-10)
(-5,-5)
NC
C
北京大学经济学院68
The Prisoner’s Dilemma: Matrix
Prisoner B
NC C
(-2,-2) (-10,-1)
(-1,-10)
(-5,-5)
NC
Prisoner A
C
Only Nash equilibrium for this game is (C,C), even
though (NC,NC) gives both players better payoffs
So, the only Nash equilibrium is inefficient
北京大学经济学院69
Example of Prisoner’s Dilemma Game
? Cheating on cartel
? Both firms benefit if both maintain agreement
? But if one cooperates, big incentive (much higher
profits) for other to cheat
? If other cheats, may as well cheat also
? Nash equilibrium is both cheat, but both lose
relative to maintaining agreement
北京大学经济学院70
Example of Prisoner’s Dilemma Game
? Tax Competition
? If all countries agree to tax mobile capital（资
本流动）, can extract rents (if capital fixed
internationally)
? If all others tax, can attract much capital by
lowering tax rate
? If others lower rate, best to follow
? If all lower tax rates, all have less revenue, no
capital reallocation, and lower public services
北京大学经济学院71
Nash Equilibrium and Simultaneous Move
Games
Example 2: Coordination Game（协调博弈）
Arms Race（军备竞赛）
? Players – USA & USSR
? Each player has two strategies – {Build,
Refrain}
? Four possible outcomes{ (B,B), (R,R), (B,R),
(R,B) }
北京大学经济学院72
Example of Prisoner’s Dilemma Game
? Arms Race
? Best for both countries to stop arms race
? But if one cooperates, strategic gain to other for
“cheating” on arms agreement
? If other does not cooperate, should expand arms
as well
? But, if both don’t cooperate, spend money on
arms build-up with no additional security
北京大学经济学院73
Coordination Game
USSR
REFRAIN BUILD
REFRAIN 4,4 1,3
BUILD 3,1 2,2
USA
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Coordination Game
USSR
REFRAIN BUILD
REFRAIN 4,4 1,3
BUILD 3,1 2,2
USA
北京大学经济学院75
Coordination Game
USSR
REFRAIN BUILD
REFRAIN 4,4 1,3
BUILD 3,1 2,2
USA
北京大学经济学院76
Coordination Game
USSR
REFRAIN BUILD
REFRAIN 4,4 1,3
BUILD 3,1 2,2
USA
北京大学经济学院77
Coordination Game
USSR
REFRAIN BUILD
REFRAIN 4,4 1,3
BUILD 3,1 2,2
USA
北京大学经济学院78
Coordination Game
Coordination game outcome:
? Two Nash Equilibrium – (B,B) & (R,R).
? Preferred equilibrium is (R,R).
? Need to coordinate choices – can be tacit
? Requires mutual assurance to refrain.
? To achieve assurance and coordination – use
strategic moves or creation of focal point
(convergence of expectations).
北京大学经济学院79
Nash Equilibrium: Battle of the Sexes
Woman
Football Ballet
Football 2, 1 0, 0
Ballet 0, 0 1, 2
Man
? (Football, Football) is a NE: Best responses to each
other
? (Ballet, Ballet) is a NE: Best responses to each other
北京大学经济学院80
N-Person Non-cooperative Games
? N players
? Non-cooperative vs. Cooperative: :
? Players cannot make binding commitments
? Players join and split the gains out of cooperation
? Solution concept: Nash Equilibrium
北京大学经济学院81
N-Person Non-cooperative Games
? Normal Form Games
? N players
? S
i
=Strategy set of player i (Pure Strategy)
? Single simultaneous move: each player i chooses a
strategy s
i
∈S
i
? Nobody observes others’ move
? The strategy combination (s
1
, s
2
, …, s
N
) gives payoff (u
1
,
u
2
, …, u
N
) to the N players
? All the above information is known to all the players and it
is common knowledge
北京大学经济学院82
Nash Equilibrium
? Nash Equilibrium is a strategy combination s
*
= (s
1
*
,
s
2
*
, …, s
N
*
), such that s
i
*
is a best response to
(s
1
*
, …,s
i-1
*
,s
i+1
*
,…, s
N
*
), for each i
? (s
1
*
, s
2
*
, s
3
*
) is a Nash Equilibrium (3 player game) iff
? s
1
*
is the best response of 1, if 2 chooses s
2
*
and 3 chooses
s
3
*
? s
2
*
is the best response of 2, if 1 chooses s
1
*
and 3 chooses
s
3
*
? s
3
*
is the best response of 3, if 1 chooses s
1
*
and 2 chooses
s
2
*
? Note: It is a simultaneous game and nobody knows what
exactly the choice of other agents
? Nash Equilibrium assumes correct and consistent beliefs
北京大学经济学院83
Nash Equilibrium (NE)
? Formally, a set of strategies forms a NE
if, for every player i,
π
i
(s
i
*
, s
-i
) ≥π
i
(s
i
, s
-i
).
? Note that the equilibrium is defined in
terms of strategies, not payoffs.
? Why is this a solution? Because it’s a
rest point - no incentive for one player
to change unilaterally.
北京大学经济学院84
The Approaches for The
Solution to Nash Equilibrium
北京大学经济学院85
How Do We Find NE?
? Elimination of Dominated Strategies （重复剔
除劣策略）.
? A player has a dominated strategy（劣策略）
if there is one action/strategy which always
provides a lower payoff than another strategy,
no matter what other players do.
? If you cross off all dominated strategies,
sometimes you are left with only NE.
北京大学经济学院86
Dominated Strategy(劣策略)
? If Strategy A’s payoffs are always greater than
Strategy B’s, then we say Strategy B is
dominated by Strategy A, and Strategy B is a
(strictly) dominated strategy.
北京大学经济学院87
Iterated Elimination of Dominated
Strategies（重复剔除劣策略）
? 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.
北京大学经济学院88
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 ...
北京大学经济学院89
Game A
Column Player →
Row Player ↓
Red
Black
Red
2,2
5,0
Black
0,5
3,3
北京大学经济学院90
Game B
LR
1, 4
T 3, 5
C 2, 6 4, 5
B 1, 2 0, 3
北京大学经济学院91
Solution: Iterated Elimination
4, 5
1, 4
2nd
LR
T 3, 5
3rd
C 2, 6
B 1, 2 0, 3
1st
北京大学经济学院92
Repeated elimination can find the NE
Left Center Right
Top 3,2 5,4 4,3
Middle 1,6 4,2 2,5
Bottom 1,1 6,3 5,4
1st
2nd
3rd
4th
北京大学经济学院93
? Elimination of dominated strategies only
works if the strategies are strictly
dominated
? Always worse, not just equal to or worse
Left Center Right
Top 3,2 5,4 4,3
Middle 3,6 4,2 2,5
Bottom 1,1 6,3 5,4
1st
2nd
3rd
4th
北京大学经济学院94
Sometimes there aren’t dominated
strategies so you have to check for NE cell
by cell
“Battle of the
Sexes” Scream The Beach
Scream 2,1 0,0
The
Beach 0,0 1,2
北京大学经济学院95
Sometimes there aren’t any NE
“Matching
Fingers”
1 finger
2 fingers
1 finger
1,-1
-1,1
2 fingers
-1,1
1,-1
北京大学经济学院96
Applications to Oligopoly
Some examples of the application of game
theory to Oligopoly are:
? Pricing decisions
? Advertising games
? Product choice games
? Production decisions
北京大学经济学院97
Game Theory and the Prisoners’
Dilemma
? The prisoners’ dilemma illustrates the difficulty
in maintaining cooperation.
? Often people (firms) fail to cooperate with one
another even when cooperation would make
them better off.
北京大学经济学院98
The Prisoners’ Dilemma
? Cooperation is difficult to maintain, because
cooperation is not in the best interest of the
individual player.
北京大学经济学院99
Oligopolies as a Prisoners’ Dilemma
? Self-interest makes it difficult for the oligopoly
to maintain a cooperative outcome with low
production, high prices, and monopoly profits.
北京大学经济学院100
Oligopolies as a Prisoners’ Dilemma
? The monopoly outcome is jointly rational for
the oligopoly, but each oligopolist has an
incentive to cheat.
北京大学经济学院101
Pricing Game
FIRM B
LOW PRICE HIGH PRICE
LOW PRICE 10,10 100,-50
HIGH PRICE -50,100 50,50
FIRM A
北京大学经济学院102
Pricing Game
FIRM B
LOW PRICE HIGH PRICE
LOW PRICE 10,10 100,-50
HIGH PRICE -50,100 50,50
FIRM A
北京大学经济学院103
Pricing Game
FIRM B
LOW PRICE HIGH PRICE
LOW PRICE 10,10 100,-50
HIGH PRICE -50,100 50,50
FIRM A
北京大学经济学院104
Pricing Game
FIRM B
LOW PRICE HIGH PRICE
LOW PRICE 10,10 100,-50
HIGH PRICE -50,100 50,50
FIRM A
北京大学经济学院105
Pricing Game
FIRM B
LOW PRICE HIGH PRICE
LOW PRICE 10,10 100,-50
HIGH PRICE -50,100 50,50
FIRM A
北京大学经济学院106
Pricing Game
? Collusion: joint profit maximisation is for both
firms to set high prices - outcome (50,50).
? Dominant strategies for both firms to set low
prices.
? Dominant strategy equilibrium (Nash
equilibrium) is both to set low prices –
outcome (10,10).
北京大学经济学院107
Advertising Game
FIRM B
ADVERTISE NOT ADVERT
ADVERTISE 10,5 15,0
NOT ADVERT 6,8 20,2
FIRM A
北京大学经济学院108
Advertising Game
FIRM B
ADVERTISE NOT ADVERT
ADVERTISE 10,5 15,0
NOT ADVERT 6,8 20,2
FIRM A
北京大学经济学院109
Advertising Game
FIRM B
ADVERTISE NOT ADVERT
ADVERTISE 10,5 15,0
NOT ADVERT 6,8 20,2
FIRM A
北京大学经济学院110
Advertising Game
FIRM B
ADVERTISE NOT ADVERT
ADVERTISE 10,5 15,0
NOT ADVERT 6,8 20,2
FIRM A
北京大学经济学院111
Advertising Game
FIRM B
ADVERTISE NOT ADVERT
ADVERTISE 10,5 15,0
NOT ADVERT 6,8 20,2
FIRM A
北京大学经济学院112
Advertising Game
? Dominant strategy for Firm B to advertise.
? No dominant strategy for Firm A.
? Nash equilibrium is for both firms to advertise.
北京大学经济学院113
Product Choice Game: Simultaneous Moves
FIRM B
CRISPY SWEET
CRISPY -5,-5 10,10
SWEET 10,10 -5,-5
FIRM A
北京大学经济学院114
Product Choice Game:
Simultaneous Moves
FIRM B
CRISPY SWEET
CRISPY -5,-5 10,10
SWEET 10,10 -5,-5
FIRM A
北京大学经济学院115
Product Choice Game:
Simultaneous Moves
FIRM B
CRISPY SWEET
CRISPY -5,-5 10,10
SWEET 10,10 -5,-5
FIRM A
北京大学经济学院116
Product Choice Game: Simultaneous Moves
FIRM B
CRISPY SWEET
CRISPY -5,-5 10,10
SWEET 10,10 -5,-5
FIRM A
北京大学经济学院117
Product Choice Game: Simultaneous Moves
FIRM B
CRISPY SWEET
CRISPY -5,-5 10,10
SWEET 10,10 -5,-5
FIRM A
北京大学经济学院118
Product Choice Game: Simultaneous Moves
? Coordination: firms divide market
? Non-cooperation: incentive to move to one of
the Nash equilibrium
? Without more information can say which
equilibrium.
? Firms can ‘signal’ each other about intentions.
北京大学经济学院119
Product Choice Game: Simultaneous Moves
FIRM B
CRISPY SWEET
CRISPY -5,-5 10,20
SWEET 20,10 -5,-5
FIRM A
北京大学经济学院120
Product Choice Problem: Simultaneous Moves
FIRM B
CRISPY SWEET
CRISPY -5,-5 10,20
SWEET 20,10 -5,-5
FIRM A
北京大学经济学院121
Why People Sometimes Cooperate
? Firms in oligopolies have a strong incentive to
collude in order to reduce production, raise
prices, and increase profits.
? Firms that care about future profits will
cooperate in repeated games rather than
cheating in a single game to achieve a one-
time gain.
北京大学经济学院122
Water Production Problem
FIRM 2
PROD 30 PROD 40
PROD 30 1800,1800 1500,2000
PROD 40 2000,1500 1600,1600
FIRM 1
北京大学经济学院123
Water Production Problem
FIRM 2
PROD 30 PROD 40
PROD 30 1800,1800 1500,2000
PROD 40 2000,1500 1600,1600
FIRM 1
北京大学经济学院124
Water Production Problem
FIRM 2
PROD 30 PROD 40
PROD 30 1800,1800 1500,2000
PROD 40 2000,1500 1600,1600
FIRM 1
北京大学经济学院125
Water Production Problem
FIRM 2
PROD 30 PROD 40
PROD 30 1800,1800 1500,2000
PROD 40 2000,1500 1600,1600
FIRM 1
北京大学经济学院126
Water Production Problem
FIRM 2
PROD 30 PROD 40
PROD 30 1800,1800 1500,2000
PROD 40 2000,1500 1600,1600
FIRM 1
北京大学经济学院127
Water Production Problem
? Non collusive strategy is the Nash equilibrium if
firms interact once only - each firm produces 40.
? Collusive strategy can be Nash equilibrium if firms
interact repeatedly – switching from collusion
produces gain of 200 in first period, but losses of
200 for every future period.
? Firms respond to breaking collusion by use of Tit-
for-tat and trigger strategies.
? Provided that they expect to interact for at least two
future periods then the Nash equilibrium is to set
output equal to 30 for each firm.
北京大学经济学院128
Mixed Strategies and Nash
Equilibrium
北京大学经济学院129
Mixed Strategies（混合策略）
? Thus far, all “pure” strategies
? Players choose a single strategy
? E.g., player A plays only U or only D
? Alternative: Mixed strategies
? Choose combination of stratgies
? Example: A chooses
? U with probability 0.25 and
? D with probability 0.75
北京大学经济学院130
Pure Strategies: Original Example Matrix
Player B
L R
(3,9)
(0,0)
(1,8)
(2,1)
U
Player A
D
Original example
Simultaneous game with two
Pure Strategies Nash Equilibria
(U,L) and (D,R)
北京大学经济学院131
Pure Strategies: New Matrix
Player B
LR
(1,2) (0,4)
(0,5) (3,2)
U
Player A
D
Does pure strategy Nash equilibria exist?
北京大学经济学院132
Pure Strategies: Matrix
Player B
LR
(1,2) (0,4)
(0,5) (3,2)
U
Player A
D
(U,L), (U,R), (D,L), (D,R) are not Nash equilibrium.
So, no Pure Strategy Nash Eq’m. in this game. BUT
北京大学经济学院133
Mixed Strategies: Player A
? Suppose Player A chooses mixed strategy
? with probability π
U
Player A plays Up, and
? with probability 1-π
U
Player A plays Down
? I.e., mixing pure strategies
? Mixed strategy has probability distribution
(π
U
,1-π
U
)
北京大学经济学院134
Mixed Strategies: Player B
? Similarly, Player B has mixed strategy with
probability distribution (π
L
,1-π
L
)
? with probability π
L
Player B plays Left and
? with probability 1-π
L
Player B plays Right
? Nash Equilibrium in Mixed Strategies
? Each player chooses optimal probabilities, given
opponent’s probabilities
? Each set of expectations satisfied in eq’m.
北京大学经济学院135
Mixed Strategies: Matrix
Player B
LR
(1,2) (0,4)
(0,5) (3,2)
U
Player A
D
Game has no pure strategy Nash equilibria, but
has Nash equilibrium in mixed strategies
北京大学经济学院136
Mixed Strategies: Matrix
Player B
(1,2) (0,4)
(0,5) (3,2)
L,π
L
R,1-π
L
U,π
U
Player A
D,1-π
U
If B plays Left, expected payoff is
2π
U
+ 5(1 - π
U
)
北京大学经济学院137
Mixed Strategies: Matrix
Player B
(1,2) (0,4)
(0,5) (3,2)
L,π
L
R,1-π
L
U,π
U
Player A
D,1-π
U
If B plays Left, expected payoff is
2π
U
+ 5(1 - π
U
)
If B plays Right, expected payoff is
4π
U
+ 2(1 - π
U
)
北京大学经济学院138
Mixed Strategies: Matrix
Player B
L,π
L
R,1-π
L
Player A
(1,2) (0,4)
(0,5) (3,2)
U,π
U
D,1-π
U
For a Nash equilibrium to exist, B must
be indifferent between playing Left or Right
i.e. 2π
U
+ 5(1 - π
U
) = 4π
U
+ 2(1 - π
U
)
Dπ
U
= 3/5
北京大学经济学院139
Mixed Strategies: Matrix
Player B
L,π
L
R,1-π
L
5
3
(1,2) (0,4)
(0,5) (3,2)
U,
Player A
5
2
D,
Dπ
U
= 3/5, (1 - π
U
) =2/5
北京大学经济学院140
Mixed Strategies: Matrix
Player B
(1,2) (0,4)
(0,5) (3,2)
L,π
L
R,1-π
L
5
3
U,
Player A
5
2
D,
If A plays Up, expected payoff is
1 x π
L
+ 0(1 - π
L
) = π
L
北京大学经济学院141
Mixed Strategies: Matrix
Player B
L,π
L
R,1-π
L
5
3
(1,2) (0,4)
(0,5) (3,2)
U,
Player A
5
2
D,
If A plays Up, expected payoff is
1 x π
L
+ 0(1 - π
L
) = π
L
If A plays Down, expected payoff is
0 x π
L
+ 3(1 - π
L
) = 3(1 - π
L
)
北京大学经济学院142
Mixed Strategies: Matrix
Player B
L,π
L
R,1-π
L
5
3
(1,2) (0,4)
(0,5) (3,2)
U,
Player A
5
2
D,
For a Nash equilibrium to exist, A must be
indifferent between playing Up or Down
i.e. π
L
= 3(1 – π
L
) D π
L
= 3/4
北京大学经济学院143
Mixed Strategies: Matrix
(1,2) (0,4)
(0,5) (3,2)
L, R,
4
3
4
1
Player B
5
3
U,
Player A
5
2
D,
π
L
= 3/4, (1 – π
L
) =1/4
北京大学经济学院144
Mixed Strategies: Matrix
Player B
(1,2) (0,4)
(0,5) (3,2)
L, R,
4
3
4
1
5
3
U,
Player A
5
2
D,
Only Nash equilibrium:
A plays mixed strategy (3/5, 2/5)
B plays mixed strategy (3/4, 1/4)
北京大学经济学院145
Mixed Strategies: Matrix
Player B
(1,2)
(0,4)
(0,5) (3,2)
L, R,
4
3
4
1
9/205
3
U,
Player A
5
2
D,
The payoffs will be (1,2) with probability
20
9
4
3
5
3
=×
北京大学经济学院146
Mixed Strategies: Matrix
Player B
(0,4)
(0,5) (3,2)
L, R,
4
3
4
1
(1,2)
9/20 3/205
3
U,
Player A
5
2
D,
The payoffs will be (0,4) with probability
20
3
4
1
5
3
=×
北京大学经济学院147
Mixed Strategies: Matrix
Player B
(0,4)
(0,5)
L, R,
4
3
4
1
(1,2)
9/20 3/20
6/20
(3,2)
5
3
U,
Player A
5
2
D,
The payoffs will be (0,5) with probability
20
6
4
3
5
2
=×
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Mixed Strategies: Matrix
Player B
(0,4)
L, R,
4
3
4
1
(1,2)
9/20 3/20
(0,5) (3,2)
6/20 2/20
5
3
U,
Player A
5
2
D,
The payoffs will be (3,2) with probability
20
2
4
1
5
2
=×
北京大学经济学院149
Player B
Player A
A’s expected Nash equilibrium payoff is
4
3
20
2
3
20
6
0
20
3
0
20
9
1 =×+×+×+×
B’s expected Nash equilibrium payoff is
5
16
20
2
2
20
6
5
20
3
4
20
9
2 =×+×+×+×
(0,4)
U,
D,
L, R,
4
3
4
1
5
3
5
2
(1,2)
9/20 3/20
(0,5) (3,2)
6/20 2/20
Mixed Strategies: Matrix
北京大学经济学院150
Penalty
? Ronaldo vs. Oliver Kahn (goalie)
? Oliver Kahn
? LR
? Ronaldo L -1,1 1,-1
? R1,-1-1,1
北京大学经济学院151
Penalty, Heads and Tails
? No Nash equilbrium in pure strategies.
? But a Nash equilibrium in mixed strategies
? If Ronaldo shoots left with prob 1/2 it really
doesnt matter for Kahn what he does; if he
jumps left with prob 1/2, Ronaldo is indifferent
between left and right
北京大学经济学院152
Mixed strategies
? A mixed strategy for player i is a probability
distribution which assigns a probability to each of
player i’s strategies.
? (p
i1
,…,p
iK
) such that 0≤ p
ik
≤1 and Σ
k
p
ik
=1.
? Theorem: (Nash 1950). Consider a finite game G, then
there exist a Nash equilibrium in mixed strategies.
Gave a Nobel prize!
? Finite: Finitely many players and each player has only
finitely many strategies
北京大学经济学院153
Mixed strategies, discussion
? Sounds weird, but works!
? Mixed strategies can be thought of as the uncertainty
the other player assigns to my choice. Think of Oliver
Kahn.
? The penalty (Matching pennies) is a situation where
players would like to outguess each other. Many such
situations.
? Such games possess no NE in pure strat. But
important nevertheless!
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Mixed strategies, cont.
? A pure strategy can be thought of as a mixed
strategy, which assigns probability one to one
strategy and zero to the other strategies.
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Existence of Nash equilibrium
? Kahn
? L R
? Rona L-1,1 1,-1
? ldo R 1,-1 -1,1
? q 1-q
? Look at Ronaldo: L yields
? (-1)q + 1(1-q)=1-2q
? R yields:
? 1q+(-1)(1-q)=-1+2q
? L best reply if
? 1-2q>-1+2q?q<1/2
? R best reply if
? q>1/2
北京大学经济学院156
Existence
? Kahn
? L R
? Rona L-1,1 1,-1 r
? ldo R 1,-1 -1,1 1-r
? q 1-q
? Look at Kahn:
? Jump L: 1r + (-1)(1-r)
? = 2r-1
? Jump R: (-1)r+1(1-r)=
? 1-2r
? L best if 2r-1>1-2r
? r>1/2
? Indifferent if r=1/2
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Best replies, equilibrium
1/2
1/2
1
? r
Ronaldo’s best reply
Kahn’s
best reply
q
1
北京大学经济学院158
Mixed strategy equilibrium
? Equilibrium r=q=1/2
? Notice, r=1/2 makes Peter indifferent,
? q=1/2 makes Louis indifferent
? In this case both are willing to choose r=1/2
and q=1/2 respectively
北京大学经济学院159
Battle of the sexes again
? OB
? q 1-q
? O2,10,0r
? B0,01,21-r
? Mixed strategy equilibrium,
? 2q+0=0+1*1-q ?q=1/3
? r + 0 = 0 + 2(1-r) ? r= 2/3
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Existence of Nash Equilibrium
? Consider game with
? finite number of players
? each with a finite number of pure strategies
? Such a game has
? at least one (pure or mixed strategy) Nash
equilibrium
? If no pure strategy Nash equilibrium, then must
have at least one mixed strategy Nash equilibrium
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John Nash
北京大学经济学院162
? Was born on June 13
th
, 1928
in Bluefield, West Virginia,
USA
? “a singular little boy, solitary
and introverted”
? Started experiments in
science at age 12, and
mathematics at age 14
? His father gave him science
books when most parents
were giving their children
coloring books
His Life….
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Hardships…
o Throughout his life, he was
never recognized as a genius,
he was always looked upon
as a social outcast
o In 1955, Nash had an affair
with a student Alicia Larde
and they were later married
and had a son
o Was diagnosed with
schizophrenia and entered
into McLean Hospital
o In 1961, his family
committed him to Trenton
State hospital where he
endured insulin-coma
therapy
o Over years, he recovered
and wrote a paper for the
World Congress of Psychiatry
where he described his
illness
o He cured himself of
schizophrenia over time
北京大学经济学院164
Education and achievements….
? In 1941, he entered
Bluefield college
? Was accepted into the
Carnegie Institute of
Technology
? In 1948, received a BC
and MA in mathematics
? Went to Princeton
? In 1949, he wrote a
paper which would win
the Nobel Prize 45 years
later
? In 1952, taught at the
Massachusetts Institute
of Technology
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Mathematic Theories and Achievements
? In 1954, his paper C1 was published and it talked
about the Riemannian manifold
? Nash’s equilibrium n-tuple won the Nobel Prize
for economic game theory on election strategies,
cause of war, and agenda manipulation in the
legislature
? Published his paper of solutions of parabolic,
elliptic equations in the American Journal of
Mathematics
北京大学经济学院166
? Worked with RAND, a government organization and
became an expert on the cold war
? Published a paper on Real Algebraic Manifolds in the
Annals of Mathematics
? In 1999, John Nash was also awarded the Leroy P.
Steele Prize by the American Mathematical Society
? He cured himself of Schizophrenia by establishing
which objects were in existance and which ones
weren’t in his life
Work cont….
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Time period in which John Nash Lived….
9 Lived during a time when the
cold war raged through the
world leaving everyone in
paranoia
9 On Feb.9
th
1950, Joseph
McCarthy, a senator from
Wisconsin declared that any
communist supporters which
belonged to the ACP would
be persecuted and would
have to face the
consequences
北京大学经济学院168
American era cont…
? During the 1950’s and 60’s,
scientists did not know very
much about mental illness,
therefore “shock” therapy
and a misunderstanding for
those who were ill were
apparent
? Discrimination against
homosexuals was present,
and so it was hard for those
who showed signs of this
My beginning as a legally
北京大学经济学院169
His Life Today
? Presently, John Nash is a
professor at Princeton and
continues to work on his
mathematical theories
? He still struggles with his
illness on occasion and has
learned to live with his
disease
? The movie “A Beautiful Mind”
was made last year to
recognize his amazing story
of existance and recovery
北京大学经济学院170
http://www.math.princeton.edu/jfnj/
Bibliography
http://namiscc.org/newsletters/February02/John
NashDrugFreeRecovery.htm
http://www-history.mcs.st-
and.ac.uk/history/Mathematicians/Nash.htm
http://www.mathforum.com/
http://www.nobel.se/economics/laureates/1994/
nash-autobio.html
北京大学经济学院171
The End
北京大学经济学院172
Last Revised:
Dec. 2, 2005
北京大学经济学院1
Chapter 9
Game Theory
? 2005 MOL
北京大学经济学院2
Chapter 9 includes:
? 9.1 Introduction to Game Theory
? 9.2 Nash Equilibrium(纳什均衡)
? 9.3 Subgame Perfect Nash
Equilibrium（子博弈精炼纳什均衡）
? 9.4 Repeated Game（重复博弈）
北京大学经济学院3
Overview of Last Class
? Game Theory: Introduction
? Elements of a game
? Classifications of Game
? Dominant strategy（占优策略）
? Dominant strategy equilibrium（占优策略均衡）
? Nash equilibrium（纳什均衡）
? Mixed Strategies
北京大学经济学院4
Readings about the part of this chapter
? Zhang: Chapter 13,P375-408
? Nicholson: Chapter 10, P246-264
Chapter 20,P554-572
北京大学经济学院5
Outlines of Today’s Class
? Definition of Subgame
? Perfect Equilibrium (SPE)(子博弈精
炼均衡)
? Repeated Games
? Finitely repeated games(有限重复博
弈)
? Infinitely repeated games（无限重复
博弈）
北京大学经济学院6
Chapter 9 includes:
? 9.1 Introduction to Game Theory
? 9.2 Nash Equilibrium(纳什均衡)
? 9.3 Subgame Perfect Nash
Equilibrium（子博弈精炼纳什均衡）
? 9.4 Repeated Game（重复博弈）
北京大学经济学院7
Sequential-move games: Introduction
There is a strict order of play
Players know what the ones that moved before have done
A player must consider how others will react
It will be convenient to describe sequential-move games
using the extensive-form representation (tree)
Important note: Actually, each type of game is capable of
each type of representation!
北京大学经济学院8
Example: 仿冒与反仿冒博弈
A企业是仿冒企业，B企
业是被仿冒企业。如果
被仿冒企业采取措施制
止，仿冒企业就会停止
仿冒，如果被仿冒企业
不采取措施制止，那么
仿冒企业就会继续仿冒
下去。
假设仿冒最多进行两次
北京大学经济学院9
The extensive form (game tree)
1. Decision nodes(决策结): corresponding to players
2. Branches（决策枝）: corresponding to actions
3. Terminal nodes（终点结）:
corresponding to
payoff
A
N
2
2
A
N
A
N
-1,-1
3,0
0,3
1
2,2
北京大学经济学院10
The Solution to Sequential-move Game
——Backward induction(逆向归纳法)
A
N
1
2
2
A
N
A
N
-1,-1
3,0
0,3
2,2
北京大学经济学院11
How did we reason?
First: Player 2 chooses N after A and A after N
Second: Realising this, player 1 will choose A
Notice: If no player is ever indifferent between two
actions, backward induction produces a unique
outcome
北京大学经济学院12
Definition of Subgame（子博弈）
? A subgame in an extensive-form game has the
following properties:
? It begins at a node of the tree corresponding to
an information set reduced to a singleton (the set
contains only one set)(单结)
? It encompasses all parts of the tree following the
starting node.
? It never divides an information set.
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Matching Pennies——Extensive Form
(-1,1)
(1,-1)
(-1,1)
(1,-1)
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Matching Pennies——Extensive Form
(-1,1)
(1,-1)
(-1,1)
(1,-1)
北京大学经济学院15
Example of a Game is extensive form
Player 1 has two strategies L and R
Player 2 has 3 strategies L, M and R
L
M
R L
M
R
L
R
P1
P2
(1,2) (3,1)
(0,4) (6,1)
(2,7)
(7,8)
L
M
R L
M
R
L
R
P1
P2
(1,2) (3,1)
(0,4) (6,1)
(2,7)
(7,8)
The ordered pairs (1,2) gives the payoff of player 1 and the payoff
of the second player as 2. In the left hand side diagram, player 2
doesn’t know what Player 1 does. In the Right hand sides diagram,
P2 has access to P1’s decision.
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Example: 仿冒与反仿冒博弈
北京大学经济学院17
Subgame Perfect Equilibrium (SPNE)
(子博弈精炼均衡)
? A NE is a SPNE if the strategies of
the players yield a NE in every
subgame, whether these subgames
are reached with a positive
probability at the equilibrium or not.
北京大学经济学院18
Further Understanding SPNE
?如果在一个完美信息的动态博弈中，各博弈方的策略构
成的一个策略组合满足：在整个动态博弈中及它的所有
子博弈中都构成纳什均衡，那么，这么策略组合称为该
动态博弈的一个SPNE。
?子博弈是原博弈的一个分支，它本身可以作为一个独立
的博弈来进行分析。
? SPNE实现的条件：
? 1.策略组合(s
1
*
,s
2
*
,… s
n
*
)是原博弈的纳什均衡；
? 2.策略组合(s
1
*
,s
2
*
,… s
n
*
)在每一个子博弈上给出纳什均衡。
北京大学经济学院19
Sequential Game: Extensive Form
A plays first
B plays second
UD
LLRR
(3,9)
(1,8)
(0,0) (2,1)
A
BB
北京大学经济学院20
Sequential Game: Extensive Form
UD
LLRR
(3,9)
(1,8)
(0,0) (2,1)
A
BB
(U,L) and (D,R) are both Nash equilibria
Which is more likely to occur?
北京大学经济学院21
Sequential Game: Extensive Form
UD
LLRR
(3,9)
(1,8)
(0,0) (2,1)
A
BB
If A plays U then B plays L; A gets 3
北京大学经济学院22
Sequential Game: Extensive Form
UD
LLRR
(3,9)
(1,8)
(0,0) (2,1)
A
BB
If A plays U then B plays L; A gets 3
If A plays D then B plays R; A gets 2
北京大学经济学院23
Sequential Game: Extensive Form
UD
LLRR
(3,9)
(1,8)
(0,0) (2,1)
A
BB
If A plays U then B plays L; A gets 3
If A plays D then B plays R; A gets 2
A anticipates B, so (U,L) is likely Nash equilibrium
北京大学经济学院24
Example:
Example:
? Scenario
? Two new (sweet, crispy) cereals
? Successful only if each firm produces one
cereal
? Sweet will sell better
? Both still profitable with only one producer
北京大学经济学院25
Modified Product Choice Problem
Crispy Sweet
Firm 2
Crispy
Sweet
-5, -5 10, 20
-5, -520, 10
Firm 1
北京大学经济学院26
Modified Product Choice Problem
? Question
? What is the likely
outcome if both
make their
decisions
independently,
simultaneously,
and without
knowledge of the
other’s intentions?
Firm 1
Crispy Sweet
Crispy
Sweet
Firm 2
-5, -5 10, 20
-5, -520, 10
北京大学经济学院27
Modified Product Choice Problem
The Extensive Form of a Game
The Extensive Form of a Game
? Assume that Firm 1 will introduce its new
cereal first (a sequential game).
? Question
? What will be the outcome of this game?
北京大学经济学院28
Product Choice Game in Extensive Form
The Advantage of Moving First
In this product-choice game, there is a clear
advantage to moving first.
Firm 1 decides produce sweet cereals at first
Crispy -5, -5
Sweet
Crispy
Sweet
10, 20
20, 10
-5, -5
Firm 1
Crispy
Sweet
Firm 2
Firm 2
北京大学经济学院29
Threats（威胁）, Commitments（承诺）,
and Credibility（可信度）
? Strategic Moves
? What actions can a firm take to gain
advantage in the marketplace?
? Deter entry（阻止进入）
? Induce（引诱）competitors to reduce output,
leave, raise price
? Implicit agreements that benefit other firm
北京大学经济学院30
Threats, Commitments, and Credibility
? How To Make the First Move
? Demonstrate Commitment（宣布承诺）
? Firm 1 must constrain his behavior to the
extent Firm 2 is convinced that he is
committed
北京大学经济学院31
Threats, Commitments, and Credibility
? Empty（Incredibility）Threats（虚假的威
胁）(不可置信的承诺)
? If a firm will be worse off if it charges a low
price, the threat of a low price is not credible
in the eyes of the competitors.
北京大学经济学院32
Threats, Commitments, and Credibility
? Scenario（假定如下情况）
? Race Car Motors, Inc. (RCM) produces cars
? Far Out Engines (FOE) produces specialty car
engines and sells most of them to RCM
? Sequential game with RCM as the leader
? FOE has no power to threaten to build big
since RCM controls output.
北京大学经济学院33
Production Choice Problem
Race Car Motors
Small cars Big cars
Far Out Engines
Small engines
Big engines
3, 6 3, 0
8, 31, 1
北京大学经济学院34
Threats, Commitments, and Credibility
? Question
? How could FOE force RCM to shift to big cars?
北京大学经济学院35
Modified Production Choice Problem
Race Car Motors
Small cars Big cars
0, 6 0, 0
8, 31, 1
Far Out Engines
Small engines
Big engines
北京大学经济学院36
Modified Production Choice Problem
? Questions
1) What is the risk of this strategy?
2) How could irrational behavior give
FOE some power to control output?
北京大学经济学院37
Classification of Strategic Moves
Classification of Strategic Moves:
1. Unconditional Moves: Move First
2. Conditional Moves:
(a) Threat（威胁）: punish others who fail to cooperate
with you.
(1)Compellent Threat（强制性威胁）:threat to
induce someone to action.
(2)Deterrent Threat（威慑性威胁）: threat to prevent
someone from taking an action
(b) Promises（承诺）: offer a reward who cooperative
with you.
Compellent vs. deterrent promise.
北京大学经济学院38
Application:金矿开采博弈
策略：
乙的最佳策略是不借，
甲的最佳策略是不分；其
分的策略是不可信的的承
诺
北京大学经济学院39
Application:金矿开采博弈
?甲乙双方策略：
?乙的完整策略是在
第一阶段选择“借”，
如第二阶段甲选择
“不分”，第三阶段选
择“打”官司。
?甲的完整策略是第
二阶段选择“分”
北京大学经济学院40
Application:金矿开采博弈
?乙在第三阶段选择“打”官司
的威胁就不是一个“可信的
威胁”，而是一个Empty
Threats。
?策略：
?乙的最优选择是第一阶段不
借；
?甲的最优策略是第二阶段不
分。
北京大学经济学院41
Chapter 9 includes:
? 9.1 Introduction to Game Theory
? 9.2 Nash Equilibrium(纳什均衡)
? 9.3 Subgame Perfect Nash
Equilibrium（子博弈精炼纳什均衡）
? 9.4 Repeated Game（重复博弈）
北京大学经济学院42
Repeated games（重复博弈）
So far we have considered games that are played only once
However, in real life the same games are played by
the same players over and over again
Two kinds of repeated games:
1. Finitely repeated (played a fixed number of times)
2. Infinitely repeated (played an indefinite number of times)
北京大学经济学院43
Example 1: prisoner’s dilemma repeated twice
At the first period, players choose simultaneously between
confess (defect) or not confess (cooperate)
After observing what happened in the first period, they
again choose simultaneously between cooperate and defect
北京大学经济学院44
One could expect that if the game is repeated players
can achieve better outcomes
Players might be able to build trust and punish others
for defection
Will this be the case?
To find the subgame perfect Nash equilibrium, we solve
the game starting from the end
北京大学经济学院45
Prisoner’s Dilemma
? Each player has a dominant strategy
? Equilibrium that arises from using dominant
strategies(占优策略) is worse for every player
than the outcome that would arise if every
player used her dominated strategy（劣势策
略）instead
? Private rationality→collective irrationality
? Goal:
? To sustain mutually beneficial cooperative
outcome overcoming incentives to cheat
北京大学经济学院46
Duopoly Competition
? Two firms: Firm 1 and Firm 2
? Two prices: low ($4) or high ($5 )
北京大学经济学院47
Prisoner’s Dilemma
Equilibrium: $24 K
Firm 2
Low High
Low
24 , 24 40 , 10
High
10 , 40 30 , 30
Firm 1
Cooperation: $30 K
北京大学经济学院48
Repeated Interaction
? Repeated Interaction
? Ongoing relationship between players
? Current action affects future interactions
? History-Dependent Strategies
? Choose an action today dependent on the
history of interaction
? Can history-dependent strategies help
enforce mutual cooperation?
北京大学经济学院49
Finite Repetition
? Suppose the market relationship lasts for only T
periods
? Use backward induction (逆向归纳法)
? T
th
period: no incentive to cooperate
? No future loss to worry about in last period
? T-1
th
period: no incentive to cooperate
? No cooperation in T
th
period in any case
? No opportunity cost to cheating in period T-1
? Unraveling: logic goes back to period 1
北京大学经济学院50
Finite Repetition
? Cooperation is impossible if the relationship
between players is for a fixed and known
length of time.
? Why do people cooperate even though they
don’t live forever?
北京大学经济学院51
Infinite Repetition（无限重复博弈）
? No last period, so no rollback
? Use history-dependent strategies
? Trigger strategies（触发策略）
? Begin by cooperating
? Cooperate as long as the rivals do
? Upon observing a defection（背叛）:
immediately revert to a period of punishment of
specified length in which everyone plays non-
cooperatively
北京大学经济学院52
Two Trigger Strategies（两种触发策略）
? Grim Trigger Strategy（冷酷触发策略）
? Cooperate until a rival deviates
? Once a deviation occurs, play non-cooperatively
for the rest of the game
? Tit-for-Tat Strategy（针锋相对策略）
? Cooperate if your rival cooperated in the most
recent period
? Cheat if your rival cheated in the most recent
period
北京大学经济学院53
Grim Trigger Strategy
? In any period t, a firm faces one of two histories of
play:
? Zero deviations up to that point（不偏离合作点）
? Charge the high price in the next period
? One or more deviations up to that point
? Charge the low price from that point on in every
period
? Since { low, low } is the Nash equilibrium, each
firm is doing the best it can
北京大学经济学院54
Equilibrium in GTS: Discounting（贴现）
Definition: Given a discount factor δ, the present value of an
infinite sequence of payoffs ......., , , ,
4321
ππππ is
∑
∞
=
?
=++++
1
1
4
3
3
2
21
.......
t
t
t
πδπδπδδππ
Example 1: The present value of an infinite sequence of payoffs 1
1, 1, ....... ( 1=
t
π , for all t) is
δ?1
1
.
北京大学经济学院55
Equilibrium in GTS: Discounting（贴现）
? Discounting:
present value of future profits is less than value
of current profits
? Discounting rate is
1
1 r
δ =
+
Note: r is interest rate
北京大学经济学院56
Equilibrium in GTS
? For GTS to be an equilibrium, the present value of
colluding must be greater than the present value of
cheating
? PV(collude)
= 30 + 30δ + 30 δ
2
+ …= 30/1- δ
? PV(cheat)
= 40 + 24 δ + 24 δ
2
+ …= 40 + 24 δ /1- δ
北京大学经济学院57
Payoff Stream
profit
40
collude
30
cheat
24
t t+1 t+2 t+3 time
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Equilibrium in GTS
? Equilibrium if: PV(collude) > PV(cheat)
30/1- δ > 40 + 24 δ /1- δ
δ>5/8 or r<60%
? Cooperation is sustainable using grim trigger
strategies as long as r < 60% or δ>5/8
? Or… as long as $1 invested today returns less than
$1.60 next period
? As long as firms value the future enough
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Sustainability（持续性）
? The minimum discount rate required to sustain
the collusive outcome depends on the payoff
structure
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Tit-for-Tat（以牙还牙策略）
? Tit-for-Tat is nicer than GTS
? a) If rival uses GTS, cooperate if:
Colluding is better than cheating
30、30、30 … >40、24、24、24、24…
? b) If rival uses tit-for-tat, cooperate if:
Colluding is better than cheating once
30、30、30 … > 40、10、30、30、30…
北京大学经济学院61
Axelrod’s Simulation埃克斯罗德的重复博
弈模拟实验
? R. Axelrod, The Evolution of Cooperation
? Prisoner’s Dilemma repeated 200 times
? Economists submitted strategies
? Pairs of strategies competed
? Winner: Tit-for-Tat
? Reasons:
? Forgiving, Nice, Provocable, Clear
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Main Ideas
? Not necessarily tit-for-tat
? Doesn’t always work
? Don’t be envious
? Don’t be the first to cheat
? Reciprocate opponent’s behavior
? cooperation and defection
? Don’t be too clever
北京大学经济学院63
Trigger Strategies
? GTS and Tit-for-Tat are extremes
? Two goals:
Deterrence
? GTS is adequate punishment
? Tit-for-tat might be too little
Credibility
? GTS hurts the punisher too much
? Tit-for-tat is credible
北京大学经济学院64
重复博弈下实现合作的条件
?（1）博弈要重复无数次，或至少在博弈有限的存续期
间看不到或预测不到博弈的结束的时间。
?（2）都采取一种“触发式”策略：
?冷酷式触发策略
?以牙还牙触发策略
?（3）贴现系数要足够的大，即人们在进行重复博弈时
要有足够的耐心，不要只顾眼前的利益，要有长远的打
算。
北京大学经济学院65
Application:无穷次重复博弈下的假冒伪劣产品
?一个市场有n家相同企业，生产同一种产品。
?企业可以选择优质，也可能选择假冒伪劣。
?如企业选择优质，则会承受边际成本c.
?如企业进行生产假冒伪劣产品，则不用承受该成本，即
c=0。
?消费者可以选择购买或不买，若购买后发现产品质量是
优质的，则会继续购买，如发现是劣质的，则从此以后
不再购买。
?问：要使企业提供优质产品的条件是什么？
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Application:无穷次重复博弈下的假冒伪劣产品
?如企业提供劣质产品，其好处是在一时期省下c，但其
后则会永远失去消费者信任，以后不能再卖出去产品。
这样该企业得利为pq
i
?如果企业已开始就生产优质产品就会获利(p-c)q
i
以后该企业会永远获利，则该企业的总得利为
（p-c）q
i
(1+δ+ δ
2
+ δ
3
+…..)
?要使企业提供优质产品，就要求
pq
i
≤（p-c）q
i
(1+δ+ δ
2
+ δ
3
+…..)
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Application:无穷次重复博弈下的假冒伪劣产品
?整理可得：
?企业无动力提供劣质品的充要条件：
p≥(1+r)c
?含义：只要价格p足够的高，企业才无动力去生产假冒伪
品。
反过来，p<(1+r)c,则一定有次品，即“便衣没好货”
r
cpq
r
pq
ii
)(
1
?
≤
+
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The End
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Last Revised:
Dec. 2, 2005
| 课件名称： | 北京大学：中级微观经济学（下） |
| 课件分类： | 经济 |
| 课件类型： | 电子教案 |
| 文件大小： | 4.04MB |
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| 用户评分： | 6.5 |
