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1、Game Theory (Microeconomic Theory (IV)Instructor: Yongqin Wang Email: yongqin_School of Economics, Fudan UniversityDecember, 2004Main Reference: Robert Gibbons,1992: Game Theory for Applied Economists, Princeton University Press Fudenberg and Tirole,1991: Game Theory, MIT Press1.Static Game of Compl
2、ete Informationn1.3 Further Discussion on Nash Equilibrium (NE)n1.3.1 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
3、 unique NE of the game.11,.,;,.,nnGSSuun*1(,.,)nssA Formal Definition of NEnIn 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, 11,.,;,.,nnGSSuu*1(,.,)nss*is*111111(,.,.,)(,.,
4、.,)iiiniiiinsssssusssssContd Proposition B In the -player normal form game if the strategies are a NE, then they survive iterated elimination of strictly dominated strategies.11 ,.,;,.,nnGSS uu*1(,.,)nssn1.3.2 Existence of NETheorem (Nash, 1950): In the -player normal form game if is finite and is f
5、inite for every , then there exist at least one NE, possibly involving mixed strategies.See Fudenberg and Tirole (1991) for a rigorous proof.n11 ,.,;,.,nnGSS uuniSi1.4 Applications 1.4.1 Cournot ModelTwo firms A and B quantity compete. Inverse demand function They have the same constant marginal cos
6、t, and there is no fixed cost. ,0PaQ aContd Firm As problem:22()20220AAAABAAAABABAAAPqcqaqq qcqdaqqcdqaqcqddq Contd By symmetry, firm Bs problem.Figure Illustration: Response Function, Tatonnement Process Exercise: what will happens if there are n identical Cournot competing firms? (Convergence to C
7、ompetitive 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 CommonsContdThere are farmers in a village. They all graze their goat on the vill
8、age 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 . nthiig1.nG gg c()v GContdA maximum number of goats : , for but forAlso The villagers problem is simultaneously c
9、hoosing how many goats to own (to choose ). max: ( )0Gv G maxGG( )0v G maxGG( )0, ( )0v Gv GigContdHis payoff is (1) In NE , for each , must maximize (1), given that other farmers choose 111(.)iiiinigv gggggcg *1(,.,)ngg*igi*111(,.,)iinggggContdFirst order condition (FOC): (2)(where )Summing up all
10、farmers FOC and then dividing by yields (3) *()()0iiiiiv ggg v ggc*111.iiingggggnn*1(*)(*)0v GG vGcnContdIn contrast, the social optimum should resolveFOC: (4)Comparing (3) and (4), we can see that Implications for social and economic systems (Coase Theorem)*Gmax()Gv GGc( *)* ( *)0v GGv Gc *GG2. Dyn
11、amic Games of Complete Informationn2.1 Dynamic Games of Complete and Perfect Informationn2.1.A Theory: 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
12、 are and .1a1A1a2a2A112(,)u a a212(,)u a aContdBackward Induction:Then“People think backwards” 2212argmax(,)au a a11121argmax( ,()au a R a2.1.B An example: Stackelberg Model of DuopolyTwo firms quantity compete sequentially. Timing: (1) Firm 1 chooses a quantity ; (2) Firm 2 observes and then choose
13、s 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 being zero). 10q 1q20q ( ,) ( )iijiq qq P Qci( )P Qa Q 12QqqcContd We solve this game with backward induction (provided that ). 2212212*
14、1221argmax(,)()()2qq qq aqqcaqcqR q1qacContdNow, firm 1s problemso, . 111211121*1argmax(,()()2qq R qq aqR qcacq*24acqContdCompare with the Cournot model.Having more information may be a bad thingExercise: Extend the analysis to firm case.n2.2 Two stage games of complete but imperfect information2.2.
15、A Theory: Sub-Game PerfectionnHere the information set is not a singleton. nConsider 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
16、 from feasible sets and , respectively. (3) Payoffs are , 2a1a1A2A1a2a3A4A1234 (,)iu a a a a1,2,3,4i An approach similar to Backward Induction1 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
17、 game:(1)Players 1 and 2 simultaneously choose actions and from feasible sets and respectively.(2) Payoffs are Sub-game perfect Nash Equilibrium is 1a*312412(,),(,)a a aa a a2a*12312412(,(,),(,)iu a a a a aa a a*1234(,)a a a a1A2A2.2B An Example: Banks RunsnTwo depositors: each deposits D in a bank,
18、 which invest these deposits in a long-term project. nEarly 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.nAssume there is no discounting.nInsert Matrixes nInterpretation
19、 of The model, good versus bad equilibrium.ContdnDate 1nDate 2r, rD,2r-D2r-D, DNext stageR, R2R-D, DD, 2R-DR, RContdnIn EquilibriumnInterpretation of the Model and the Role of law and other institutionsr, rD, 2r-D2r-D, DR, R2.3 Repeated Gamen2.3A Theory: Two-Stage Repeated GameRepeated Prisoners Dil
20、emma Stage Game1,15,00,54,42,26,11,65,5ContdnDefinition Given a stage game G, let the finitely repeated game in which G is played T times, with the outcomes of all preceding plays observed before the next play begins. The payoff for G(T) are simply the sum of the payoffs from the stage games.nPropos
21、ition If the stage game G has a unique NE, then for any finite T , the repeated game G(T) has a unique sub-game perfect outcome: the Nash equilibrium of G is played in every stage. (The paradox of backward induction)Some Ways out of the ParadoxnBounded Rationality (Trembles may matter)nMultiple Nash
22、 Equilibrium( An Two-Period Example)nUncertainty about other playersnUncertainty about the futures2.3B Theory: Infinitely Repeated GamesnDefinition 1 Given the discount factor , the present value of the infinitely repeated sequence of payoffs isnDefinition 2 (Selten, 1965) A Nash Equilibrium is subg
23、ame perfect if the players strategies constitute a Nash equilibrium in every subgame. 123,. 211231.ttt ContdnDefinition3: Given the discounted factor , the average payoff of the infinite sequence of payoffs isFolk Theorem (Friedman,1971): See Gibbons (p97). Discuss Reputation Model 123,. 11(1)ttt2.4 Dynamic Games with Complete but Imperfect InformationnInformation set is not a singleton.nJustification for Sub-Game Perfect Argument.nCommitment, Reputation, Sunk Cost and Cheap talk.