Principles of Game Theory

1. Principles of Game Theory Lecture 7: Subgame perfection

2. Administrative • Quiz 2 results this week (again, hopefully Tuesday) • Homework results on blackboard this afternoon • Exam 1 Sept 17th. Next Tuesday. • Part 1 of Dixit et al: chapters 1-8 • Chapters 1 and 2 of Gibbons

3. Subgame perfection What are the strategies for each player? Congress: • Balance • Deficit Fed • L if B; H if D • H if B; L if D • H if B; H if D • L if B; L if D 3, 4

4. Subgame perfection • Recall how we constructed normal form games: • A matrix of strategies We now have 2 equilibria!? • Why? • When players can condition behavior, all players know this and it opens up the possibility for more equilibria

5. Nash Equilibria • Remember what a Nash Eq is: • Mutual best response. • Equilibrium in beliefs; mutually supporting. • But (Deficit, High Always) seems odd…

6. Subgame perfection • There are 2 equilibria to this game • But only 1 of them is what we call a subgame perfect eq

7. Subgames • How many subgames? 3, 4

8. Subgame Perfection • An equilibrium is called a subgame-perfect equilibrium if it is an equilibrium in everysubgame.

9. Subgame Perfection • 2 Nash equilibria: • But only 1 subgame perfect eq

10. Subgame Perfect Eq • The set of subgame perfect equilibria is always a subset of the set of Nash equilibria • Not necessarily a proper subset.

11. Credible threats • Subgame perfection is often used to describe credible and non-credible threats. • Bob: “I’m going to go no matter what Alice!!” • Alice: “Sure you are, Bob…”

12. Credible threats • A non-credible threat is a threat that a rational player would not actually carry out because it would not be in their interest to do so. • A credible threat needs to be • Publicly observable • Irreversible.

13. Imperfect Information What happens when player 2 knows that player 1 moved first but doesn’t know what he did? • Before we said it would be “as if” it were simultaneous. Now we’ll see why. • What makes a sequential game different? • Players can condition their behavior on the history of play – or their information up to that node.

14. Imperfect information Tennis example from the book (fig 6.9) • N. doesn’t know what E. did:

15. Imperfect information • Back to the Congress/Fed example: how many subgames? How many equilibria? 3, 4

16. Congress/Fed • When the 2nd player can’t condition behavior on any history, we’re back to the original normal form game:

17. Observable actions • Questions on subgame perfection? • Credible threats? • Why does it need to be publicly* observable • Irreversible? • Imperfect information?