More on Extensive Form Games

1. More on Extensive Form Games

2. Histories and subhistories • A terminal history is a listing of every play in a possible course of the game, all the way to the end. • A proper subhistoryis a listing of every play in the course of the game up to some point before the end. • Every proper subhistory induces a game, called a subgamewhich is defined by the remaining possibilities for play and resulting payoffs.

3. Proper Subgames • For any proper subhistory, there is a well-defined extensive form game that follows this subhistory. • A subgame following any non-empty proper subhistory is called a proper subgame.

4. Subgame perfect Nash Equilibrium • A strategy specifies what each person will do at any possible point in the game where it is his turn. • A strategy profile (i.e. list of strategies chosen by each player) then determines the course of play in every possible subgame. • A subgame perfect Nash equilibrium (SPNE) is a strategy profile such that each person’s play in each subgame is a best response to the other players’ actions in that subgame.

5. Berlin or Havana? Prob 156.2 c

6. Histories and subgames • Terminal histories: • Proper subhistories: • Player functions: • Proper subgames:

7. How many proper subgames does the game on the the blackboard have? • 6 • 10 • 4 • 3 • 5+

8. In the game on the blackboard, what is the payoff to Player 2 in a subgame perfect Nash equilibrium? • 0 • 1 • 2 • 3 • There are two subgame perfect equilibria. In one of them he gets 2 and in one of them he gets 1.

9. Choosing Sides Game Ex 174.1 Two choosers, 3 players, a, b and c. Chooser 1 gets to choose first, then 2 chooses, then 1 gets a second choice.

10. Game Tree: Choosing Sides

11. Analysis • Player 2 never gets his last choice. • Therefore it never makes sense for Chooser 1 to choose Chooser 2’s last choice first. Chooser 1 is always going to get that player anyway. • Chooser 1’s first choice should be the one that he likes better of the two who are not Chooser 2’s last choice.

12. Variant of All-Pay Auction: 175.2 • Two bidders compete for an object that is worth $2.50 to each of them. • They bid sequentially. They must bid an integer number of dollars. When it is your turn you must either raise the bid by $1 or pass. Nobody can afford to bid more than $3. • If you pass, other bidder gets object. Both must pay the amount they bid.

13. Game tree

14. What if nobody can bid more than $4?

15. Repeated Prisoners’ Dilemma • Backwards induction solution? • Does this solution seem reasonable if game is repeated 100 times?