国外博弈论课件lecture6.ppt

May 27, 2003 73-347 Game Theory--Lecture 6 May 27, 2003 Lecture 6 Static (or Simultaneous-Move) Games of Complete Information The Problems of Commons Mixed Strategy Equilibrium 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 Today’s Agenda The problems of commons (sec 1.2.D of Gibbons) Mixed strategies Solving matching pennies The problems of commons n farmers in a village. Each summer, all the farmers graze their goats on the village green. Let gi denote the number of goats owned by farmer i. The cost of buying and caring for a goat is c, independent of how many goats a farmer owns. The value of a goat is v(G) per goat, where G = g1 + g2 + ... + gn There is a maximum number of goats that can be grazed on the green. That is, v(G)0 if G Gmax, and v(G)=0 if G ? Gmax. Assumptions on v(G): v’(G) 0 and v”(G) 0. Each spring, all the farmers simultaneously choose how many goats to own. The problems of commons The normal-form representation: Set of players: { Farmer 1, ... Farmer n} Sets of strategies: Si=[0, Gmax), for i=1, 2,..., n Payoff functions: ui(g1, ..., gn)=gi v(g1 + ...+ gn) – c gi for i = 1, 2, ..., n. The problems of commons How to find a Nash equilibrium Find (g1*, g2*, ..., gn*) such that gi* is farmer i’s best response to other farmers’ choices. That is, g1* solves Max u1(g1, g2*, ..., gn*)= g1 v(g1 + g2* ...+ gn*) – c g1 subject to 0 ? g1 Gmaxand g2* solvesMax u2(g1*, g2 , g3*, ..., gn*)= g2v(g1*+g2+g3*+ ...+ gn*)–cg2 subject to 0 ? g2 Gmax ....... The problems of commons How to find a Nash equilibrium and gn* solvesMax un(g1*, ..., gn-1*, gn)= gnv(g1*+...+ gn-1*+ gn)–cgn subject to 0 ? gn Gmax ....... The problems of commons FOCs: The problems of commons How to find a Nash equilibrium