1. 1 Information, Control and Games Shi-Chung Chang EE-II 245, Tel: 2363-5251 ext. 245 scchang@cc.ee.ntu.edu.tw, http://recipe.ee.ntu.edu.tw/scc.htm Office Hours: Mon/Wed 1:00-2:00 pm or by appointment Yi-Nung Yang (03 ) 2655201 ext. 5205, yinung@cycu.edu.tw

2. 2 Extensive Games with Perfect Information

3. 3 Denotations in an Extensive Form Game Starting node (Empty history  ) Information set A simultaneous move info. set Left figure: payoff for DM1 Right figure: payoff for DM2 Terminal node

4. 4 Game Tree: Examples In last Lectures we analyzed games in normal form ~ All the dynamic aspects have been stripped Sometimes it is valuable to analyze games in extensive form with dynamics intact Example. Consider the following two-person non-zero sum game in extensive form to minimize costs Q. How to solve it? Two methods: –M1 ~ Convert to normal form –M2 ~ Deal directly in extensive form

5. 5 Normal form analysis –DM1: 3 strategies, L, M, and R –DM2: 2 3 = 8 strategies Game in normal form: DM1\2LLLLLRLRLLRRRLLRLRRRLRRR L (0, -1) (-2, 1) M (3, 2) (0, 3) (3, 2) (0, 3) R (2, 1)(-1, 0)(2, 1)(-1, 0)(2, 1)(-1, 0)(2, 1)(-1, 0) Q. Major difficulties? –Dimensionality can be very large (Recall the DP example) –Dynamic aspects are not appropriately considered Which of the four Nash solutions will actually happen?

6. 6 To overcome the difficulties, we shall analyze the extensive form directly. How? 2A2A 2B2B The solution process is backward induction –Starting from leaf nodes and work backward until the root node is reached, each time solve a simple problem –Then moving forward from the root to obtain the solution Q. If DM1 selected L, what should DM2 do? How about if DM1 selected M or R? –The solution is unique –(0, -1) is not a solution since DM1 who acts first will not select L –Extensive form is a reasonable approach for this problem

7. 7 Example. With a slight variation: 2A2A 2B2B If DM1 selected M or R, DM2 does not know how DM1 acted Again there are two methods: –M1 ~ Convert to a normal form –M2 ~ Deal directly in extensive form Normal form analysis: How? –DM1: 3 strategies, L, M, and R –DM2: 2 2 = 4 strategies Q. How to solve it?

8. 8 Game in normal form: 2A2A 2B2B DM1\DM2LLLRRLRR L(0, -1) (-2, 1) M(3, 2)(0, 3)(3, 2)(0, 3) R(2, 1)(-1, 0)(2, 1)(-1, 0) Q. Major difficulties? –Same as before Dimensionality can be very large Dynamic aspects are not appropriately considered

9. 9 Q. To analyze the extensive form directly. How? 2A2A 2B2B Q. At information set 2A, what should DM2 do? How about at 2B? What should DM1 select? –At 2A, DM2 should select L with costs (0, -1) DM1\DM2LR M(3, 2)(0, 3) R(2, 1)(-1, 0) Q. What problem does DM1 face? How should he select? –At 2B, DM2 faces the following normal game: The solution process is backward induction

10. 10 An exercise on backward induction

11. 11 Subgames and Subgame Perfection Subgames –for any non-terminal history h is the part of the game that remains after h occurred. Subgames –Subgame perfect equilibrium: No subgame can any player do better by choosing a different strategy

12. 12 Some examples that is not subgames 2A2A 2B2B

13. 13 Location Game Example: Dynamic Game of Perfect Information Grocery Shopping on Market Street Market Street is a one-way street. 1100 One-Way Market Street Two firms locate grocery stores on Market Street sequentially. That is, first firm 1 locates and then firm 2.

14. 14 Consumers live along streets 1-100. N W 1 23 i 99100 Consumers drive to market street, then drive west on market street (there are no left turns onto market street) until they reach a grocery store.

15. 15 Payoffs An example: Firm 1 locates at 15 and 2 locates at 47. 1547 Firm 1Firm 2 Consumers: 1 consumer uniformly distributed on each street. Since firm 1 gets all consumers who live on 15 th St, 16 th St, …45 th St and 46 th St.  1 (15, 47) = 47-15 = 32.  2 (15, 47) = 101-47 =54 100 1

16. 16 Now let’s use backward induction to find all subgame perfect Nash equilibrium. Recall that subgame perfection is an equilibrium refinement concept. If SGPNE then NE. 1 i 100 1 2 j 1 2 2 2 2 2 What are the pay-offs to Player 1? Player 2?

17. 17 Payoffs:  i (i,j) = 101-i if i>j (101-i)/2 if i=j j-i if i<j ji Firm jFirm i 100 1 101-i i-j i=j Firm jFirm i 100 1 (101-i)/2 Payoffs:  j (i,j) = i-j if i>j (101-i)/2 if i=j 101-j if i<j

18. 18 Backward Induction Fix a player 2 node i 0 (player 1 has located at i 0 ). What maximizes player 2’s payoff? First note that player 2 will always want to be at 1, i 0, or i 0 + 1. For example suppose player 1 has located at 4 (i.e. player 2 is at node 4). Where will 2 want to locate? Suppose player 1 has located at 75. Where will player 2 want to locate?

19. 19 Solving the game via backward induction At node i, firm 2 plays j= i+1 if 1  i  50 j= 1 if 51  i  100 Back at firm i’s node:  1 (i, j ) = i+1-i if 1  i  50 = 101-i if 51  i  100 Therefore unique subgame perfect equilibrium is: firm 1 plays 51 = i* firm 2 plays j=i+1 if 1  i  50 and j=1 if 51  i  100 = j* Note the way in which the strategies are stated.

20. 20 The end of market street Equilibrium path: firm 1 plays 51, firm 2 plays 1. Payoffs  1 (i*, j* ) = 50 and  2 (i*, j*) = 50 Also note that there is no first mover advantage.

21. 21 Location Game 2 What if Market Street is a two-way street. 1100 Two-Way Market Street Two firms locate grocery stores on Market Street sequentially. That is, first firm 1 locates and then firm 2.

22. 22 Payoffs of location game in a two-way street Payoffs:  i (i,j) = i + (j-i)/2  j (i,j) = (k-j) + (j-i)/2 i j Firm 1Firm 2 k 0 k/2

23. 23 General Distribution of Consumers’ Preference Single-peak distribution: –Principle of Minimum Differentiation Double-peak distribution ?

24. 24 Location Game 3 What if The Market Street is a Circle? Does one-way or two-way matter? (1) 0 x = ? (j’s location) Two-Way Market Street What if there are 3 firms? 0 = i’s location