هوالحق

1. هوالحق Short Course on Game Theory Hamed Ghoddusi Von Neumann Aumann Selton Harsanyi

2. “In war the will is directed at an animate object that reacts.” - Karl Von Clausewitz, On War

3. Course Plan • Part 1: Definition of games, Typical Examples of Games, Common Knowledge, Expected Utility, States and Strategies • Part 2: Extensive Forms with Perfect Information, Backward Induction, Imperfect Information, Incomplete Information, Normal forms • Part 3: Solution Concepts: Rationalizibility, Strong and Weak Dominance, Nash equilibrium, Bayesian-Nash equilibrium, Correlated equilibrium , Sub-game perfection, signalling games • Part 4 (If time permits): Trembling hand, repeated games, supermodular games

4. Game Theory • Study of strategic behavior • Strategic behavior : Taking into account the behavior of other • players

5. Examples of Strategic Behavior • Industrial Competition • Firm / Capital market relationship • Voting decisions • Auctions/Biddings • …

6. Game to be discussed in the class: • Shareholders voting game • Search models • Durable goods monopolist • R&D • Bargaining • Job market signaling • …

7. Games We Study • Played once / more than once • Finite number of players • Finite/infinite strategy space

8. Prisoner’s Dilemma Silent Cooperate 1 , 1 10 , 0 Silent 0 , 10 8 , 8 Cooperate

9. Chicken Game Drive Stop -100 , -100 1 , 0 Drive 0 , 1 Stop 0 , 0

10. Hawk and Dove Hawk Dove (V-C)/2 V , 0 Hawk 0 , V Dove V/2

11. Battle of Sexes Theater Restaurant Theater 10 , 5 4 , 4 Restaurant 5 , 10 4 , 4

12. Game Theory Decision Theory Representation Theory Solution Theory

13. ٍExtensive Form • Players • Rules of Game • Strategies

14. Normal Form • Players • Space of Pure Strategies • Payoffs

15. Differences between Extensive and Normal Form?

16. Games • Competitive : zero-sum games , non zero-sum games • Non Competitive : supermodular (complementary) games • search games for instance

17. Games • One shot games • Repeated games

18. Information • Perfect Information : Chess • Imperfect Information : Cards • Incomplete Information

19. Characteristics of Players • Self-interested, utility maximizer • Rational : Friedman’s “as if” paradigm • Remark: One person and several identity situation

20. Tree of Ulysses‘s problem

21. Decision Under Uncertainty • Decision over lotteries • Von Neumann Morgenstern expected utility • Risk Aversion

22. States • Mutually exclusive • Collectively exhaustive • Independent of players decision

23. Information Sets

24. Bayesian Decision Making • Updating of prior beliefs • P(E|E’)= P(E∩E’) / P(E’)

25. Strategy Detailed plan of actions, contingent to any possible occurrence in the game : a plan that you could leave to somebody else playing for you. Strategy as a function which maps states of the world to the actions A W

26. Strategy • Strategy in the simultaneous move games • Strategy in the Bayesian games • Finite and infinite strategy space

27. Example 1 : How many strategies? L 2 L H 1 L H 2 M H

28. Example 2 : Equivalent Normal Form? 1 L H 2 2 H L H L

29. What do you think if strategies always assign the same value to some elements of the domain?

30. Strategy in Extensive Form A strategy for player X is a sub-tree of a game tree which satisfies the following conditions: • It is rooted at the root of the game tree • whenever it is player X's turn at a node that belongs to the subtree, exactly • one of the available moves belongs to the subtree; • whenever it is not player X's turn at a node that belongs to the subtree, all • of the available moves belong to the subtree

31. Bargaining Games • Gains from trade, the problem of distribution of benefits • In the absence of market • Disagreement value • Rubenstein smart solution

32. Rubenstein’s Bargaining Game

33. Real Life Example • Company-Union Negotiations

34. Solution to Bargaining Game

35. Extensive Form Games • Set Theoretic Definition • Graph Theoretic Definition

36. Set Theory Representation • (W,N) • W set of Plays • N collection of non-empty sub-sets of W (Nodes) • {w} є N • Predecessor function

37. Set Theory Representation • Plays : States • Nodes : Events • Moves • Terminal Moves

38. Example 1 2 2 2 w5 w6 w3 w4 w2 w1

39. Example : Modeling of Bargaining • W = ? • N = ?

40. Graph Theoretic Representation • Graph (V,E) • Nodes (States) • Branches (Actions) • History • Immediate Predecessor

41. Example 1 H A N 2 2 2 H A A N H N

42. Games with Perfect Information • An Extensive Form • Assignment of Decision Points • Pay off function ** Simultaneous move is ruled out.

43. Games with imperfect Information • An Extensive Form • Collection of Information set • Pay off function

44. Example 1 2 2

45. Sub Games • Sub-tree • Contains the whole information set

46. Example: Noisy Stackelberg L 2 1 H QL 2 1 QH 2

47. Strategy Space The product structure of strategies of all players S = S1 * S2 * S3 * … * Sk = ∏ Si S(-i) = Strategies of all player but player (i)

48. Pure Strategy vs Mixes Strategy The subset of pure strategies entering the mix with a strictly positive weight is the support of the mixed strategy. Point: A pure strategy may be strictly dominated by a mixed strategy even if it does not strictly dominated by any pure strategy

49. Behavioral Strategy vs Mixes Strategy When a player implements a mixed strategy, she spins the roulette wheel a single time; the outcome of this spin determines which pure strategy (set of deterministic choices at each information set) she will play. When she implements a behavior strategy, she independently spins the roulette wheel every time she reaches a new information set.

50. Behavioral Strategies