博弈论 lecture5.ppt

博弈论 博弈论 * 博弈论 * Static (or Simultaneous-Move) Games of Complete Information Bertrand Model of Duopoly Contributing to a Public Good * 博弈论 * Outline of Static Games of Complete Information Introduction to games Normal-form (or strategic-form) representation Iterated elimination of strictly dominated strategies Nash equilibrium Review of concave functions, optimization Applications of Nash equilibrium Mixed strategy equilibrium * 博弈论 * Today’s Agenda Review of previous classes Bertrand model of duopoly (differentiated products) (sec 1.2.B of Gibbons) Bertrand model of duopoly (homogeneous products) (sec 3.2 of Osborne) Contributing to a public good (sec 2.8.4 of Osborne) * 博弈论 * Bertrand model of duopoly (differentiated products) Two firms: firm 1 and firm 2. Each firm chooses the price for its product without knowing the other firm has chosen. The prices are denoted by p1 and p2, respectively. The quantity that consumers demand from firm 1: q1(p1, p2) = a – p1 + bp2. The quantity that consumers demand from firm 2: q2(p1, p2) = a – p2 + bp1. The cost to firm i of producing quantity qi is Ci(qi)=cqi. * 博弈论 * Bertrand model of duopoly (differentiated products) The normal-form representation: Set of players: { Firm 1, Firm 2} Sets of strategies: S1=[0, +∞), S2=[0, +∞) Payoff functions: u1(p1, p2)=(a – p1 + bp2 )(p1 – c)u2(p1, p2)=(a – p2 + bp1 )(p2 – c) * 博弈论 * Bertrand model of duopoly (differentiated products) How to find a Nash equilibrium Find the price pair (p1*, p2*) such that p1* is firm 1’s best response to Firm 2’s price p2* and p2* is firm 2’s best response to Firm 1’s price p1* That is, p1* solves Max u1(p1, p2*) = (a – p1 + bp2* )(p1 – c)subject to 0 ? p1 ? +∞and p2* solvesMax u2(p1*, p2) = (a – p2 + bp1* )(p2 – c)subject to 0 ? p2 ? +∞ * 博弈论 * Bertrand model of duopoly (differentiated products) How to find a Nash equilibrium Solve firm 1’s maximization problemMax u1(p1, p2*) = (a – p1 + bp2* )(p1 – c)subject to 0 ? p1 ? +∞FOC: a