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Simultaneous-move Games

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

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Simultaneous-move Games

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  1. Simultaneous-move Games With Continuous Pure Strategies

  2. 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.

  3. 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)

  4. Profit of Firm X at different Px when Py=0, 20 & 40 Profit of Firm X Py=40 Px Py=20 When Py=0, best Px=15 When Py=20, best Px=20 Py=0 When Py=40, best Px=25

  5. 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

  6. 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.

  7. 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

  8. 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.)

  9. 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)

  10. Py X’s best response to Py 40 Y’s best response to Px 20 15 NE 0 Px 15 25 20 • NE is where two best response curves intersects.

  11. 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.

  12. 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.

  13. Political Campaign Advertising • 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

  14. 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=

  15. Best Responses and N.E. y X’s best response N.E. (1/4, 1/4) Y’s best response x

  16. 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*

  17. 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

  18. 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

  19. 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

  20. 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

  21. Critiques on Nash equilibrium • Example 1

  22. Example 2

  23. 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

  24. ∂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

  25. 30 X’s best response NE=(12, 6) 12 7.5 Y’s best response 9 24 15

  26. 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?

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