Session 4: Topics

1. Session 4: Topics • Strategy space, pure and mixed strategies • Solution concepts • Backward induction • Dominance • Nash equilibrium

2. 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)

3. 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

4. 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.

5. Behavioral Strategies

6. Kuhn’s Theorem Every game of perfect information with finite number of Nodes has a solution of backward induction. Comment: If pay-offs to players at all terminal nodes are Unequal then the solution is unique.

7. Backward Induction The concept of backwards induction corresponds to the assumption that it is common knowledge that each player will act rationally at each node where he moves — even if his rationality would imply that such a node will not be reached

8. Example: Stackelberg • Firm 1 moves first , firm 2 reacts • Firm 2 finds the optimum level of production q2 given q1 • Firm 1 takes into account the reaction hence calculates • its production • There is a “first mover” advantage for the firm 1

9. Example : Christine and Lion • A group of hungry lions • What happens if number of lions is odd / even?

10. The centipede game Go on Go on Go on Jack Jill Jack Jill Go on stop stop stop stop (5, 3) (4, 7) (2, 0) (1, 4) Go on Go on Jill Jill Jack (99, 99) stop stop (94, 97) (98, 96) (97, 100)

11. Solution Concepts • Backward induction • Dominance / Rationalizibility • Nash Equilibrium

12. Dominance Definition: A strategy S dominates a strategy T if every outcome in S is at least as good as the corresponding outcome in T, and at least one outcome in S is strictly better than the corresponding outcome in T. Dominance Principle: A rational player would never play a dominated strategy.

13. Dominance • Weakly Dominant Strategy: • Strategy si dominates strategy si’ if for all strategies s-i, si yields higher payoffs than si’ for player i. • Strictly Dominant Strategy: • Strategy si strictly dominates strategy si’ if for all strategies s-i, si yields strictly higher payoffs than si’ for player i. • Note: Strictly dominated strategy enables the elimination of certain strategies. In equilibrium, a player will never choose a strictly dominated strategy.

14. Left Middle Right Top 4, 3 5, 1 6, 2 Middle 2, 1 8, 4 3, 6 Bottom 3, 0 9, 6 2, 8 Example

15. Example : Second Price Auction

16. Example : Guess Average

17. Example : Bertrand Competition

18. Left Middle Right Top 5, 2 5, 1 6, 4 Middle 2, 3 8, 6 3, 1 Bottom 3, 4 6, 2 2, 8 Mixed Strategy Dominance

19. Pure and Mixed Dominated Strategies

20. Dominance Solvability • To use dominance to solve a game: • Delete dominated strategies for each of the players • Look at the smaller game with these strategies eliminated • Now delete dominated strategies for each side from the smaller game • Continue this process until no further deletion is possible • If only single strategies remain, the game is dominance solvable

21. Best Respond Strategy

22. Dominance and Best Responce Theorem: A dominated strategy si is never a best reply

23. Extraction of Best Respond Strategy

24. 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,

25. N.E as the fixed point of BR correspondence Not a fixed point A fixed point

26. 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.

27. Finding N.E in Normal Forms Example:

28. Example: Dominant Shareholder

29. Example: Durable Goods Monopoly

30. Example: Rational Expectation of R&D , Human Capital

31. Example: Search Games • Two firms 1 , 2 • Many consumers • Each consumers knows firm i with probability a • Informed consumers choose lowest offer • Semi-informed consumers choose the firm they know • Uninformed consumers choose each firm with equal chance