Game Theory (Microeconomic Theory (IV)) Instructor: Yongqin Wang 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 Complete Information 1.3 Further Discussion on Nash Equilibrium (NE) 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 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 grazing each goat is.

Cont ’ d A maximum number of goats :, for but for Also The villagers ’ problem is simultaneously choosing how many goats to own (to choose ).

Cont ’ d His payoff is (1) In NE, for each, must maximize (1), given that other farmers choose

Cont ’ d First order condition (FOC): (2) (where ) Summing up all farmers ’ FOC and then dividing by yields (3)

Cont ’ d 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 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 are and.

Cont ’ d Backward Induction: Then “ People think backwards ”

2.1.B An example: Stackelberg 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 being zero).

Cont ’ d We solve this game with backward induction (provided that ).

Cont ’ d Now, firm 1 ’ s problem so,.

Cont ’ d 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 information 2.2.A 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 from 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 choose 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 D>r>D/2. If the bank allows the investment to reach maturity, the project will pay out a total of 2R, where R>D. Assume there is no discounting. Insert Matrixes Interpretation of The model, good versus bad equilibrium.

Cont ’ d Date 1 Date 2 r, r D,2r-D 2r-D, DNext stage R, R2R-D, D D, 2R-DR, R

Cont ’ d In Equilibrium Interpretation of the Model and the Role of law and other institutions r, rD, 2r-D 2r-D, DR, R

2.3 Repeated Game 2.3A Theory: Two-Stage Repeated Game Repeated Prisoners ’ Dilemma Stage Game 1,15,0 0,54,4 2,26,1 1,65,5

Cont ’ d Definition 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. Proposition 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 Paradox Bounded Rationality (Trembles may matter) Multiple Nash Equilibrium( An Two-Period Example) Uncertainty about other players Uncertainty about the futures

2.3B Theory: Infinitely Repeated Games Definition 1 Given the discount factor, the present value of the infinitely repeated sequence of payoffs is Definition 2 (Selten, 1965) A Nash Equilibrium is subgame perfect if the players ’ strategies constitute a Nash equilibrium in every subgame.

Cont ’ d Definition3: Given the discounted factor, the average payoff of the infinite sequence of payoffs is Folk Theorem (Friedman,1971): See Gibbons (p97). Discuss Reputation Model

2.4 Dynamic Games with Complete but Imperfect Information Information set is not a singleton. Justification for Sub-Game Perfect Argument. Commitment, Reputation, Sunk Cost and Cheap talk.