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Unformatted text preview: Problem Set 6 Solution Key 1 Consider the following game. Player 1 chooses the row (U or D). Her payoffis the ﬁrst number in the cel].
Player 2 chooses the column (L or R). Her payoff is the second number in the cell.
Player 3 chooses the box (1, 2, or 3). Her payoffis the third number in the cell.
Choices are made simultaneously. Box I L R U (5, g, 2) (0, 0, 0)
D (2, 2, 0) (2,3, 2)
Box 2 L R U (0,1, 0) (a, 1, i)
D (3,1,0) (1, 0, 2)
Box 3 L R U (3,1, 3) (1,1, 1
D (1,1,1) (1,2,2) Forp/ayer 1: BR1 (L, Boxl) : U, BR] (R, Boxl) : D, BR] (L, 80x2) 2 D, BR; (R, 80x2) = U, BR] (L, 30x3) 2
U, BR1 (R, 30x3) 6 {U, D}. Forp/ayer2: BR; (U, Box 1) = L, BR; (D, Box 1) = R, BR; (U, 30x2) 6 {L, R}, BR; (D, 30x2) = L, BR; (U, 30x3) 6
{L, R}, BR; (D, 30x3) 2 R. Forp/ayer 3.’ BR; (U, L) = 30x3, BR; (D, L) : 30x3, BR; (U, R): 30x2, BR; (D, R) = 30x2. Therefore, NE={(U,R,B0x2), (U,L,Box3)}.  5 g F v .gin this game there are two players and two boxes. One ol‘ the boxes is marked “player 1" and the other is marked “player
2." At the beginning ot‘tho game, each box contains three dollars. Player l is given the choice between stopping the game and
continuing. If he chooses to stop then each player receives the money in his own box and the game ends. Il‘ player 1 chooses to
continue, then two dollars are removed from his box and three dollars are added to player 2’s box. Then player 2 must choose
between stopping the game and continuing. If he stops, then the game ends and each player keeps the money in his own box.
If player 2 elects to continue, then two dollars are removed from his box and three dollars are added to player l's box. Play
continues like this, alternating between players, until either one of them decides to stop or k rounds of play have elapsed. At
each round both players play unless player I chooses to stop. If neither player chooses to stop by the end of the kth round, then
both players obtain 9 dollars. Assume each player wants to maximize the amount of money he earns. a. Draw this game’s tree for k=2.
b. Find the subgame perfect equilibrium of this game. (9.9) ...
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 Fall '07
 Codrin

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