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

A model of price competition 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)

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

For instance, When Py=0, best response Px= x0=15. When Py=20, best response Px= x20=20. When Py=40, best response Px= x40=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= Px

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 responses Px= Py Py= Px NE : (20, 20) → (288, 288)

Px Py 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 is Px= x26=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 expenses) from 0 to ∞. Payoffs: Ux=ax/(ax+cy)-bx Uy=cy/(ax+cy)-dy 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=

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=x 1/2 -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-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

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 ABC A2, 23, 10, 2 B1, 32, 23, 2 C2, 02, 32, 2

Example 2 LeftRight Up9, 108, 9.9 Down10, , 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

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 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?