This is about game theory.

Players 1,2, and 3 play the following game. Each player starts with a pile containing one coin in front of him. In turn, each player says either 'stop' si or 'continue' ci. If a player says 'stop,' then the game ends and each player receives his pile of coins. If a player says 'continue,' then his pile looses one coin, while the other two piles gain one coin each. The players play in the order that they are numbered. (Of course, a player gets to play only if all players before him say 'continue.') If all the players say 'continue,' then after the coins are adjusted following player 3's statement, the game ends and each receives all coins in his pile.

(a) Draw the extensive form of this game. (Be sure to label each branch with the associated action, and remember that actions at different information sets cannot have the same label.) (b) Find all Subgame Perfect Nash Equilibria.

(c) Convert the game into the normal form. Be sure to make clear which player chooses row, column and matrix. Be sure to make clear which payoff goes to each player. (If you are using the diagonal method of listing payoffs that I use, then I recommend that player 1 chooses rows, player 2 to chooses matrices, and player 3 to chooses columns.)

(d) Use the circling method to find all pure strategy Nash equilibria. Clearly write down these Nash equilibria as well as showing the circles on the matrix.

(e) Select a Nash equilibrium which is not a Subgame Perfect Nash Equilibrium. Select a player who is not demonstrating sequential rationality (in this equilibrium.) Explain the violation of sequential rationality.

(f) Argue that a strategy in a pure strategy Nash equilibrium must be (normal form) Rational- izable.

(g) Given your answer to (f) what strategies must be (normal form) rationalizable in this game? (Hint: this is not a trick question. It should be very easy, and is really just an intermediate step towards getting (h).)

(h) Show that the remaining strategies (not indicated in part g) must also be (normal form) rationalizable.