2. Game Theory (Microeconomic Theory (IV)) Instructor: Yongqin Wang Email: yongqin_wang@yahoo.com.cn School of Economics and CCES, Fudan University December, 2004

3. 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.

4. 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,

5. 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.

6. 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.

7. 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.

8. Cont ’ d Firm A ’ s problem:

9. 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)

10. 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

11. 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.

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

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

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

15. 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)

16. 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.

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

18. 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).

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

20. Cont ’ d Now, firm 1 ’ s problem so,.

21. Cont ’ d Compare with the Cournot model. Having more information may be a bad thing Exercise: Extend the analysis to firm case.

22. 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,

23. 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

24. 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.

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

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

27. 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

28. 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)

29. 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

30. 2.3B Theory: Infinitely Repeated Games Definition 1 Given the discount factor, the present value of the infinitely repeated sequence of payoffs is Interpretation of the discount factor. Definition 2 (Selten, 1965) A Nash Equilibrium is subgame perfect if the players ’ strategies constitute a Nash equilibrium in every subgame.

31. Existence of SPE

32. Cont ’ d Definition3: Given the discounted factor, the average payoff of the infinite sequence of payoffs is

33. Cont ’ d

34. Implications of Repeated Games Reputation-building Collusion Social mobility and social capital Organization theory (Kreps) Exit and Voice

35. 2.4 Dynamic Games with Complete but Imperfect Information At least some information set is not a singleton Sub-game Perfection

36. Static (or Simultaneous-Move) Games of Incomplete Information Introduction to Static Bayesian Games

37. Static (or simultaneous-move) games of complete information A set of players (at least two players) For each player, a set of strategies/actions Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies All these are common knowledge among all the players.

38. Static (or simultaneous-move) games of INCOMPLETE information Payoffs are no longer common knowledge Incomplete information means that  At least one player is uncertain about some other player ’ s payoff function. Static games of incomplete information are also called static Bayesian games

39. Cournot duopoly model of complete information The normal-form representation:  Set of players: { Firm 1, Firm 2}  Sets of strategies: S 1 =[0, +∞), S 2 =[0, +∞)  Payoff functions: u 1 (q 1, q 2 )=q 1 (a-(q 1 +q 2 )-c), u 2 (q 1, q 2 )=q 2 (a-(q 1 +q 2 )-c) All these information is common knowledge

40. Cont ’ d A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q 1 and q 2, respectively. They choose their quantities simultaneously. The market price: P(Q)=a-Q, where a is a constant number and Q=q 1 +q 2. Firm 1 ’ s cost function: C 1 (q 1 )=cq 1. All the above are common knowledge

41. Cont ’ d Firm 2 ’ s marginal cost depends on some factor (e.g. technology) that only firm 2 knows. Its marginal cost can be  HIGH: cost function: C 2 (q 2 )=c H q 2.  LOW: cost function: C 2 (q 2 )=c L q 2. Before production, firm 2 can observe the factor and know exactly which level of marginal cost is in. However, firm 1 cannot know exactly firm 2 ’ s cost. Equivalently, it is uncertain about firm 2 ’ s payoff. Firm 1 believes that firm 2 ’ s cost function is  C 2 (q 2 )=c H q 2 with probability , and  C 2 (q 2 )=c L q 2 with probability 1 – . All the above are common knowledge

42. Cont ’ d

48. 3. Static Games of Incomplete Information 3.1 Theory: Static Bayesian Games and Bayesian NE 3.1.A: An Example: Cournot Competition under Asymmetric Information The basic Set-up:

49. Cont ’ d Introduction of asymmetric information: with probability and with probability

50. Cont ’ d Firm 2 knows its cost functions and firm 1 ’ s, but firm 1 only knows its own function and that firm 2 ’ s Marginal cost is with Probability,and with probability All of this is common knowledge.

51. Cont ’ d

52. The FOC:

53. cont ’ d Solutions: Comparison with the Complete-Information version

54. 3.1.B Normal Form Representation of Static Bayesian Games Definition: The normal form representation of an n-player static Bayesian game specifies the player ’ s action spaces, their type space, their beliefs, and their payoff functions. Player ’ s type, privately known by player, determines player ’ s payoff function, and is a member of the set of possible types.

55. Cont ’ d Player ’ s belief describes ’ s uncertainty about the other players ’ possible types, given ’ s own type. We denote this game by

56. cont ’ d Harsanyi Transformation Time of a static Bayesian game Nature draws a type vector, ; Nature reveals to player, but not to any other player; The players simultaneously choose actions, player choosing ; Payoffs are received. Some remarks

57. 3.1C Definition of Bayesian Nash Equilibrium (BNE) Definition 1 In the game of static Bayesian game, a strategy for player is a function, where for each type in, specifies the action from the feasible set that type would choose if drawn by nature.

58. 3.2A Mixed Strategies Revisited

59. Cont ’ d

64. First-price sealed-bid auction

65. cont ’ d

66. Cont ’ d

67. cont ’ d

68. Cont ’ d

69. Relation with Information Economics Bayesian Game and Mechanism Design(Adverse Selection) Dynamic Bayesian Games and Signaling Moral Hazard Dynamics

70. Equilibrium Concepts in Game Theory NE, SPE, BNE, PBNE Embarrassment of richness(merits and demerits) and Refinements Evolutionary Game Theory Behavior Game Theory

71. Concluding Remarks Taking Stock Further Reading Gibbons (1992)