Extensive and Strategic Form Games

1. Extensive and Strategic Form Games Econ 171

2. Reminder: Course requirements • Class website Go to economics department home page. Under Links, find Class pages, then click on Econ 171 • Textbook: Games, Strategies, and Decision Making  by Joseph E. Harrington, Jr. • Clicker: Available at campus bookstore i>clicker Register your clicker at www.i>clicker.com/registration

3. Rock, Paper, Scissors

4. Rock-Paper-Scissors First let’s draw the game tree on the blackboard for the game played with full information, where Player A moves first. How would we draw the game tree for this game when players move simultaneously?

5. Vocabulary for Extensive form games • Decision Tree • Decision Node-Specifies whose turn • Branches-Options • Terminal Node—End of play • Payoffs—For each person at each terminal node. • Strategy—What will you do at each decision node where it is your turn

6. Clicker Question How many strategies are possible for Player B in the perfect information version of Rock-Paper-Scissors • 3 • 6 • 9 • 12 • 27

7. What is a strategy? • “A strategy is not a sequence of actions, but rather a catalog of contingency plans, what to do in every situation. ‘’ Harrington, page 34. (Read this section with extra care.) • A strategy is a list stating what you would do at each possible decision node where it is your turn.

8. Strategies for B in perfect information rock, paper, scissors game • A strategy for B in perfect rock, paper, scissors answers 3 questions: • what will I do if I see rock?, • what will I do if I see paper? • What will I do if I see scissors? • There are 3 possible answers to each question. Hence there are 3x3x3=27 possible strategies. • Examples: • Paper if rock, rock if paper, rock if scissors • Or Rock if rock, scissors if paper, paper if scissors And so on… 27 possibilities

9. Details of strategic form game • Set of Players • For each player a strategy set—list of all the strategies that the player could choose. Remember that a strategy tells everything you would do on any occasion when its your turn. • Strategy profile: List of strategies chosen by every player. • Payoff to each player depends on the strategy profile that was chosen.

10. Two player game matrix in strategic form Make a two-by-two table with one row for each strategy that player 1 could choose and one column for every strategy that player 2 could choose. Enter payoffs to players 1 and 2 in appropriate spots.

11. Example: Simultaneous Move Matching Pennies • In this case each player has only two possible strategies. Choose Heads, Choose tails. • Payoff to Player 1 (row chooser) is written first, then payoff to Player 2.

12. Matching PenniesStrategic Form of Game Player 2 Tails Heads Heads Player 1 Tails

13. Rock, Paper, Scissors—Simultaneous Move Rock Paper Scissors Rock Paper Scissors

14. More complicated game Player 1 C D Player 2 2 0 E F Player 1 3 1 G H 0 0 1 2 4 Possible Strategies for Player 1 : What are they? 2 Possible Strategies for Player 2: What are they?

15. Strategic Form Player 2 E F C,G C,H Player 1 D,G D,H

16. WMDs: What are the strategies?

17. Clicker Question 2 • How many possible strategies are there for the U.S. in this game? • 2 • 4 • 6 • 8 • 16

18. Prisoners’ Dilemma Game Player 2 Cooperate Defect P LAyER 1 Cooperate Defect

19. Clicker Question 3 Players A and B play two rounds of simultaneous move prisoners’ dilemma. They don’t get to see how the other player played until both rounds are over. How many strategies are possible for each player? • 2 • 4 • 8 • 16 • 32

20. Clicker Question 4 Players A and B play two rounds of simultaneous move prisoners’ dilemma. Each gets to see the other’s move in round 1 before choosing an action for round 2. How many strategies are possible for each player? • 2 • 3 • 4 • 16 • 32

21. The game of Chicken James Dean story. Alternatively—Two animals both want a resource. Each has two possible strategies. Fight or give up. A fight is very bad for both of them. How do we make an interesting game of this?

22. Swerve Don’t Swerve Swerve Don’t Swerve

23. Common Knowledge of a fact • Three ladies in a railway car. All have dirty faces. • They can see each other’s faces, but not their own. Each would blush visibly if she knew her own face was dirty. • All are brilliant logicians and they all know this. • The conductor comes into the car and announces for all to hear. “Someone in this car has a dirty face.”

24. Common Knowledge Why should this news matter? All three can see two dirty faces. In fact, all three know that the others can see at least one dirty face. Lady 1 says, Suppose that my face is clean. Then Lady 2 will see exactly one dirty face—that of Lady 3. Lady 2 will reason, if my face is clean, then Lady 3 will see 2 clean faces. If Lady 3 saw 2 clean faces, she would know her face was dirty and would blush. If Lady 3 doesn’t blush, lady 2 would conclude that her own face is dirty and would blush. Therefore if Ladies 2 and 3 don’t blush, Lady 1 must conclude that her own face if dirty.

25. So long…at least for now.