國外博弈論課件lecture(5)-資料下載頁
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【正文】 FOC: a 2q1 q2* c = 0 q1 = (a q2* c)/2 May 22, 2021 73347 Game TheoryLecture 4 26 Cournot model of duopoly ? How to find a Nash equilibrium ? Solve Max u2(q1*, q2)=q2(a(q1*+q2)c) subject to 0 ? q2 ? +∞ FOC: a 2q2 – q1* – c = 0 q2 = (a – q1* – c)/2 May 22, 2021 73347 Game TheoryLecture 4 27 Cournot model of duopoly ? How to find a Nash equilibrium ? The quantity pair (q1*, q2*) is a Nash equilibrium if q1* = (a – q2* – c)/2 q2* = (a – q1* – c)/2 ? Solving these two equations gives us q1* = q2* = (a – c)/3 May 22, 2021 73347 Game TheoryLecture 4 28 Cournot model of duopoly ? Best response function ? Firm 1’s best function to firm 2’s quantity q2: R1(q2) = (a – q2 – c)/2 if q2 a– c。 0, othwise ? Firm 2’s best function to firm 1’s quantity q1: R2(q1) = (a – q1 – c)/2 if q1 a– c。 0, othwise q1 q2 (a – c)/2 (a – c)/2 a – c a – c Nash equilibrium May 22, 2021 73347 Game TheoryLecture 4 29 Cournot model of oligopoly ? A product is produced by only n firms: firm 1 to firm n. Firm i’s quantity is denoted by qi. Each firm chooses the quantity without knowing the other firms’ choices. ? The market priced is P(Q)=aQ, where a is a constant number and Q=q1+q2+...+qn. ? The cost to firm i of producing quantity qi is Ci(qi)=cqi. May 22, 2021 73347 Game TheoryLecture 4 30 Cournot model of oligopoly The normalform representation: ? Set of players: { Firm 1, ... Firm n} ? Sets of strategies: Si=[0, +∞), for i=1, 2, ..., n ? Payoff functions: ui(q1 ,..., qn)=qi(a(q1+q2 +...+qn)c) for i=1, 2, ..., n May 22, 2021 73347 Game TheoryLecture 4 31 Cournot model of oligopoly ? How to find a Nash equilibrium ? Find the quantities (q1*, ... qn*) such that qi* is firm i’s best response to other firms’ quantities ? That is, q1* solves Max u1(q1, q2*, ..., qn*)=q1(a(q1+q2* +...+qn*)c) subject to 0 ? q1 ? +∞ and q2* solves Max u2(q1*, q2 , q3*, ..., qn*)=q2(a(q1*+q2+q3*+ ...+ qn*)c) subject to 0 ? q2 ? +∞ ....... May 22, 2021 73347 Game TheoryLecture 4 32 Summary ? Nash equilibrium ? Concave function and maximization ? Cournot model of duopoly and oligopoly ? Next time ? Bertrand model of Duopoly ? Reading lists ? Sec of Gibbons and Sec of Osborne
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