Econ 521 - Week 10:
1. The 100 game - Consider a two player game, where players have to say a number, which will be added to the sum of all numbers said to that point. The sum start in 0 and alternatively each player add a number s ∈ {1, 2, ..., 10}. The player who reaches 100 wins.
(a) Play the game and solve it.
2. Dividing a cake fairly - Two players use the following procedure to divide a cake. Player 1 divides the cake into two pieces, and then player 2 chooses one of the pieces; player 1 obtains the remaining piece. The cake is continuously divisible (no lumps!), and each player likes all parts of it.
(a) Suppose that the cake is perfectly homogeneous, so that each player cares only about the size of the piece of cake she obtains. How is the cake divided in a subgame perfect equilibrium?
(b) Suppose that the cake is not homogeneous: the players evaluate different parts of it differently. Represent the cake by the set C, so that a piece of the cake is a subset P of C. Assume that if P is a subset of P' not equal to P' (smaller than P'), then each player prefers P' to P. Assume also that the palyers' preferences are continuous: if player i prefers P to P', then there is a subset of P not equal to P that player i also prefers to P'. Let (P1, P2) (where P1 and P2 together constitute the whole cake C) be the division chosen by player 1 in a subgame perfect equilibrium of the divide-and-choose game, P2 being the piece chosen by player 2. Show that player 2 is indifferent between P1 and P2, and player 1 likes P1 at least as much as P2. Give an example in which player 1 prefers P1 to P2.