# Game Theory and Personal Choices

Problem 1

1. Suppose you and one of your two roommates have just finished
cleaning your dorm suite and found 13 quarters which you put on a
table in the middle of the room. The third roommate who did none of
the cleaning comes in from an afternoon of fun and relaxation and
proposes that you divide the coins up the following way: The two who
cleaned and collected the quarters will take turns after flipping a coin
to see who goes first. At each turn the player has a choice to take 1 or
2 quarters. If the player takes 1 coin, then the next player gets a turn. If
the player takes 2 coins, then the game ends and the third roommate
gets the rest of the money on the table.

a. The roommate who helped you clean says, "Okay that sounds fair.
what the equilibrium outcome of the game will be. (i.e., indicate
how much money each roommate will get and why). State the
assumptions you make. (Hint: what would you do if it was your
turn and there were two coins left?)

b. If you and the roommate who helped you clean could write a
binding contract describing how each would play the game, what
might the contract say? How would the outcome of the game
played according to the contract differ from the game as played in
part a.?
c. The game in part a. is a repeated game in the sense that several
turns are possible. In a repeated game there is an opportunity to
build or erode trust. Can you image getting an outcome similar to
that in b without a contract? Explain.

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#### Solution Preview

...ould you do if it was your turn and there were two coins left?)
b. If you and the roommate who helped you clean could write a binding contract describing how each would play the game, what might the contract say? How would the outcome of the game played according to the contract differ from the game as played in
part a.?
c. The game in part a. is a repeated game in the sense that several turns are possible. In a repeated game there is an opportunity to build or erode trust. Can you image getting an outcome similar to that in b without a contract? Explain.

First, list all possible outcomes (see table). If there are two coins left and the second player takes both, the outcome will be as shown in yellow (second line from the bottom). If the second player takes only one and the first takes the remaining one the outcome will be as highlighted in blue. However, the first player may stop the game earlier, as shown in pink (third line from the bottom) and take the ...