Simultaneous-move Games With Continuous Pure Strategies.

Simultaneous-move Games

With Continuous Pure Strategies

Pure strategies that are continuous 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 differentiatio

n, Hotelling Model), Choice of time to ..., and etc.

A model of price competition

Two firms selling substitutional (but not identical) products with demandsQx=44-2Px+PyQy=44-2Py+Px

Assuming MC=8 for each firm Profit for Firm X Bx=Qx (Px-8) =(44-2Px+Py)(Px-8)

Profit of Firm X at different Px when Py=0, 20 & 40

10 20 30 40

1000

500

500

Py=0

Py=20

Py=40

Px

Profit of Firm X

When Py=0, best Px=15

When Py=20, best Px=20

When Py=40, best Px=25

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

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.

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

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.)

Mathematically, NE is the solution to the simultaneous equations of best responsesPx=15+0.25PyPy=15+0.25Px

NE : (20, 20) →(288, 288)

Px

Py

0

15

20

40

15 20 25

X’s best response to Py

Y’s best response to Px

NE

•NE is where two best response curves intersects.

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 isPx=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

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

Players: X & Y (candidates) Strategies: x & y (advertising expense

s) from 0 to ∞. Payoffs:

Ux=a•x/(a•x+c•y)-b•xUy=c•y/(a•x+c•y)-d•y

First assume a=b=c=d=1

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= yy

Best Responses and N.E.

X’s best response

Y’s best response

x

y

N.E. (1/4, 1/4)

Critical Discussion on N.E.

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*

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

Another prisoner’s dilemma Asymmetric cases

If b<d, X is more cost-savingex:a=c=1,b=1/2,d=1,→x*=4/9,y*=2/9 If a>c, X is more effective gaining shareex:a=2,c=1,b=d=1, →x*=y*=2/9

ex:a=c=1,b=1/2,d=1,→x*=4/9,y*=2/9

X’s best response

Y’s best response

x

y

N.E. (4/9, 2/9)

ex:a=2,c=1,b=d=1, →x*=y*=2/9

X’s best response

Y’s best response

x

y

N.E. (2/9, 2/9)

Critiques on Nash equilibrium 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

Example 2

Left Right

Up 9, 10 8, 9.9

Down 10, 10 -1000, 9.9

Rationality leading to N.E 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

∂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

24

12

30

15

NE=(12, 6)

X’s best response

Y’s best response

9

7.5

Homework Question 3 on page 152 (Cournot model) Consider an industry with 3 identical firms e

ach 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?