Transcript Simultaneous-move Games - National Cheng Kung University

Simultaneous-move Games
With Continuous Pure Strategies
Pure strategies that are
continuous
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Price Competition
Pi is any number from 0 to ∞
Quantity Competition (Cournot Model)
Qi is any quantity from 0 to ∞
Political Campaign Advertising
Location to sell (Product differentiation,
Hotelling Model), Choice of time to ...,
and etc.
A model of price competition
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Two firms selling substitutional (but not
identical) products with demands
Qx=44-2Px+Py
Qy=44-2Py+Px
Assuming MC=8 for each firm
Profit for Firm X
Bx=Qx (Px-8) =(44-2Px+Py)(Px-8)
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Profit of Firm X at different Px when Py=0, 20 & 40
Profit of Firm X
500
Py=40
Px
10
20
30
When Py=0, best Px=15
When Py=20, best Px=20
500
Py=20
Py=0
When Py=40, best Px=25
1000
40
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At every level of Py, Firm X finds a Px to
maximize its profit (regarding Py as
fixed)
Bx=Qx (Px-8) =(44-2Px+Py)(Px-8)
∂ Bx/ ∂ Px=-2(Px-8)+(44-2Px+Py)(1)
=60-4Px+Py
∂ Bx/ ∂ Px=0 when Px=15+0.25Py
Best response of Px to Py
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For instance,
When Py=0,
best response Px=15+0.25x0=15.
When Py=20,
best response Px=15+0.25x20=20.
When Py=40,
best response Px=15+0.25x40=25.
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Similarly, at every level of Px, Firm Y
finds a Py to maximizes its profit.
By=Qy (Py-8) =(44-2Py+Px)(Py-8)
∂ By/ ∂ Py=-2(Py-8)+(44-2Py+Px)(1)
=60-4Py+Px
∂ By/ ∂ Py=0 when Py=15+0.25Px
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Nash Equilibrium is where best
response coincides.
X’s equilibrium strategy is his best
response to Y’s equilibrium strategy
which is also her best response to X’s
equilibrium strategy. (Best response to
each other, such that no incentive for
each one to deviate.)
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Mathematically, NE is the solution to the
simultaneous equations of best
responses
Px=15+0.25Py
Py=15+0.25Px
NE : (20, 20) →(288, 288)
Py
X’s best response to Py
40
Y’s best response to Px
20
15
0
NE
15 20 25
Px
•NE is where two best response curves intersects.
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Note that the joint profits are maximized
($324 each) if the two cooperate and both
charge $26.
However, when Py=26, X’s best response is
Px=15+0.25x26=21.5 (earning $364.5).
Similar to the prisoner’s dilemma, each has
an incentive to deviate from the best
outcome, such that to undercut the price.
Bertrand Competition
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Firms selling identical products and
engaging in price competing.
Dx=a-Px if Px<Py
=(a-Px)/2 if Px=Py
=0 if Px>Py, similar for Firm Y
Assuming (constant) MCx<MCy
At equilibrium, Px slightly below MCy.
Political Campaign Advertising
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Players: X & Y (candidates)
Strategies: x & y (advertising expenses)
from 0 to ∞.
Payoffs:
Ux=a•x/(a•x+c•y)-b•x
Uy=c•y/(a•x+c•y)-d•y
First assume a=b=c=d=1
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To find the best response of x for every
level of y, find partial derivative of Ux,
with respect to x, (regarding y as given)
and set it to 0.
∂Ux/ ∂x=0
→y/(x+y)2-1=0
→x= y  y
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Best Responses and N.E.
y
X’s best response
N.E. (1/4, 1/4)
Y’s best response
x
Critical Discussion on N.E.
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Similarly Y’s best response is y=x1/2-x
N.E. (x*, y*) must satisfy the following
x* is the best response to y*, while y*
is the best response to x*.
(x*, y*) solves the simultaneous eqs.
x*=y*1/2-y*
y= x*1/2-x*
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x*=(x*1/2-x*)1/2-(x*1/2-x*)
x*1/2= (x*1/2-x*)1/2
x*= x*1/2-x*
4x*2=x*
x*=0 or 1/4
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Another prisoner’s dilemma
Asymmetric cases
If b<d, X is more cost-saving
ex:a=c=1,b=1/2,d=1,→x*=4/9,y*=2/9
If a>c, X is more effective gaining share
ex:a=2,c=1,b=d=1, →x*=y*=2/9
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ex:a=c=1,b=1/2,d=1,→x*=4/9,y*=2/9
y
X’s best response
N.E. (4/9, 2/9)
Y’s best response
x
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ex:a=2,c=1,b=d=1, →x*=y*=2/9
y
X’s best response
N.E. (2/9, 2/9)
Y’s best response
x
Critiques on Nash equilibrium
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Example 1
A
B
C
A
2, 2
3, 1
0, 2
B
1, 3
2, 2
3, 2
C
2, 0
2, 3
2, 2
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Example 2
Left
Right
Up
9, 10
8, 9.9
Down
10, 10
-1000, 9.9
Rationality leading to N.E
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A costal town with two competitive
boats, each decide to fish x and y
barrels of fish per night.
P=60-(x+y)
Costs are $30 and $36 per barrel
U=[60-(x+y)-30]x
V=[60-(x+y)-36]y
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∂U/∂x=0
→60-x-y-30-x=0
→x=15-y/2
∂V/∂y=0
→60-x-y-36-y=0
→y=12-x/2
30
X’s best response
NE=(12, 6)
12
7.5
Y’s best response
9
15
24
Homework
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Question 3 on page 152
(Cournot model) Consider an industry with 3 identical firms each
producing with a constant cost $c per unit. The inverse demand
function is P=a-Q where P is the market price and
Q=q1+q2+q3, is the total industry output. Each firm is assumed
choosing a quantity (qi) to maximizes its own profit.
(A) Describe firm 1’s profit function as a function of q1, q2 & q3.
(B) Find the best response of q1 when other firms are producing
q2 and q3.
(C) The game has a unique NE where every firm produces the
same quantity. Find the equilibrium output for every firm and its
profit. Also find the market price and industry’s total output.
(D) As the number of firms goes to infinity, how will the market
price change? And how will each firm’s profit change?