博弈论(复旦大学 王永钦).pptVIP

Game Theory Microeconomic Theory IV Instructor: Yongqin Wang Email: yongqin_wang@ School of Economics and CCES, Fudan University December, 2004 1.Static Game of Complete Information 1.3 Further Discussion on Nash Equilibrium NE 1.3.1 NE versus Iterated Elimination of Strict Dominance Strategies Proposition A In the -player normal form game if iterated elimination of strictly dominated strategies eliminates all but the strategies , then these strategies are the unique NE of the game. A Formal Definition of NE In the n-player normal form the strategies are a NE, if for each player i, is at least tied for player i’s best response to the strategies specified for the n-1 other players, Cont’d Proposition B In the -player normal form game if the strategies are a NE, then they survive iterated elimination of strictly dominated strategies. 1.3.2 Existence of NE Theorem Nash, 1950 : In the -player normal form game if is finite and is finite for every , then there exist at least one NE, possibly involving mixed strategies. See Fudenberg and Tirole 1991 for a rigorous proof. 1.4 Applications 1.4.1 Cournot Model Two firms A and B quantity compete. Inverse demand function They have the same constant marginal cost, and there is no fixed cost. Cont’d Firm A’s problem: Cont’d By symmetry, firm B’s problem. Figure Illustration: Response Function, Tatonnement Process Exercise: what will happens if there are n identical Cournot competing firms? Convergence to Competitive Equilibrium 1.4.2 The problem of Commons David Hume 1739 : if people respond only to private incentives, public goods will be underprovided and public resources over-utilized. Hardin 1968 : The Tragedy of Commons Cont’d There are farmers in a village. They all graze their goat on the village green. Denote the number of goats the farmer owns by , and the total number of goats in the village by Buying and caring each goat cost and value to a farmer of