More on Extensive Form Games

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 subhistory is 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 subgame which is defined by the remaining possibilities for play and resulting payoffs.

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.

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.

Berlin or Havana? Prob c

Histories and subgames Terminal histories: Proper subhistories: Player functions: Proper subgames:

How many proper subgames does the game on the the blackboard have? A)6 B)10 C)4 D)3 E)5+

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

Choosing Sides Game Ex Two choosers, 3 players, a, b and c. Chooser 1 gets to choose first, then 2 chooses, then 1 gets a second choice. PlayerValue to Chooser 1Value to Chooser 2 a31 b23 c12

Game Tree: Choosing Sides

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.

Variant of All-Pay Auction: 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.

Game tree

What if nobody can bid more than $4?

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