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1. Nim is a game that begins with several piles of matches. Two players make alternate moves. On a player’s move he or she selects one of the piles and removes at least one match from that pile. The last player to remove a match wins. Can Nim end in a draw? Construct the complete game tree of Nim when the game begins with three piles of matches, two containing only one match and the other containing two matches. Label each terminal nodes with either W or L to represent a win or loss, respectively, for player I (the player who chooses the first move). Identify a play that allows player I to win.

2. Five “Survivors”, Adam, Beatrice, Chloe, Donald and Eve, are marooned on a desert island when they find a treasure chest containing 100 gold coins. They agree to divide the treasure in the following way: The first person in alphabetical order (i.e. Adam) makes a proposal about the division, and everybody votes on it, including the proposer. If 50% or more are in favor, the proposal passes and is implemented straight away. Otherwise the proposer is banished from the Survivors community, and the procedure is repeated with the remaining Survivors, with the proposer being the next in line alphabetically. Assuming that all individuals are rational (i.e. they would each like to get as many gold coins as they can for themselves), and that the gold coins cannot be split or shared, what proposal should Adam make? (Hint: Work backward from the point when the game has got down to just two Survivors, Donald and Eve).

3. Consider the simplified game of Poker discussed in class. The dealer has four strategies, that we will denote by A, B, C and D, and the recipient has two strategies, that we will denote by alpha and beta
The payoff matrix given in class was:

    alpha    beta
A (0,0)    (2,-2)
B (1,-1)    (0,0)
C (-3,3)   (0,0)
D (-2,2)   (-2,2)
Show that the strategy pair (m,n ) is a Nash equilibrium, where m=1/3A+2/3B and n=2/3alpha+1/3beta remixed strategies for the dealer and recipient, respectively.

4. Consider the children’s game Rock-Scissors-Paper (R-S-P), with payoff matrix!
   R S P
R 0 1 -1
S -1 0 1
P 1 -1 0
Show that the mixed strategy B=1/3R+1/3S+1/3P is a symmetric Nash equilibrium of this game. Is it a strict Nash equilibrium?

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