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1、Game Theory (Microeconomic Theory (IV),Instructor: Yongqin Wang Email: yongqin_ School of Economics, Fudan University December, 2004 Main Reference: Robert Gibbons,1992: Game Theory for Applied Economists, Princeton University Press Fudenberg and Tirole,1991: Game Theory, MIT Press,1.Static Game of
2、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 ar
3、e 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 is best response to the strategies specified for the n-1 other players,Contd,Proposition B In the -player normal form game if the strategies
4、 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
5、) 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.,Contd,Firm As problem:,Contd,By symmetry, firm Bs problem. Figure Illustration: Response Function, Tatonneme
6、nt 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. Hardi
7、n(1968) : The Tragedy of Commons,Contd,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 grazing each goat is .,Cont
8、d,A maximum number of goats : , for but for Also The villagers problem is simultaneously choosing how many goats to own (to choose ).,Contd,His payoff is (1) In NE , for each , must maximize (1), given that other farmers choose,Contd,First order condition (FOC): (2) (where ) Summing up all farmers F
9、OC and then dividing by yields (3),Contd,In contrast, the social optimum should resolve FOC: (4) Comparing (3) and (4), we can see that Implications for social and economic systems (Coase Theorem),2. Dynamic Games of Complete Information,2.1 Dynamic Games of Complete and Perfect Information 2.1.A Th
10、eory: Backward Induction Example: The Trust Game General features: (1) Player 1 chooses an action from the feasible set . (2) Player 2 observes and then chooses an action from the feasible set . (3) Payoffs are and .,Contd,Backward Induction: Then “People think backwards”,2.1.B An example: Stackelbe
11、rg Model of Duopoly,Two firms quantity compete sequentially. Timing: (1) Firm 1 chooses a quantity ; (2) Firm 2 observes and then chooses a quantity ； (3) The payoff to firm is given by the profit function is the inverse demand function, , and is the constant marginal cost of production (fixed cost
12、being zero).,Contd,We solve this game with backward induction (provided that ).,Contd,Now, firm 1s problem so, .,Contd,Compare with the Cournot model. Having more information may be a bad thing Exercise: Extend the analysis to firm case.,2.2 Two stage games of complete but imperfect information2.2.A
13、 Theory: Sub-Game Perfection,Here the information set is not a singleton. Consider following games (1)Players 1 and 2 simultaneously choose actions and from feasible sets and , respectively. (2) Players 3 and 4 observe the outcome of the first stage ( , ) and then simultaneously choose actions and f
14、rom feasible sets and , respectively. (3) Payoffs are ,An approach similar to Backward Induction,1 and 2 anticipate the second behavior of 3 and 4 will be given by then the first stage interaction between 1 and 2 amounts to the following simultaneous-move game: (1)Players 1 and 2 simultaneously choo
15、se actions and from feasible sets and respectively. (2) Payoffs are Sub-game perfect Nash Equilibrium is,2.2B An Example: Banks Runs,Two depositors: each deposits D in a bank, which invest these deposits in a long-term project. Early liquidation before the project matures, 2r can be recovered, where DrD/2. If the bank allows the investment to reach maturity, the project will pay out a total of 2R, where RD. Assume there is no discounting. Insert Matrixes Interpretation of The model, good versus bad equilibrium.,Contd,Date 1 Date 2,Contd,In Equilibrium Interpretation o
