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solutions-167

# solutions-167 - Selected Solutions from Thomas S...

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Unformatted text preview: Selected Solutions from Thomas S. Ferguson’s Game Theory Jeffrey Lee Hellrung, Jr. April 03, 2009 1.1.3 (a) The strategy of the previous exercise will not work. You would begin by choosing a 3, and each time your opponent chooses a 4, your response is to choose another 3. We thus get the sequence of moves (I)3, (II)4, (I)3, (II)4, (I)3, (II)4, (I)3, (II)4, bringing the total to 28 with your move next. The previous exercise’s strategy now is, again, to choose a 3, bringing the total to exactly 31, but this is not possible since four 3’s have already been used. It is easy to see that, at this point, your opponent will win. (b) The key observation is that 7 n + 3 is a P-position if there are at least 4 − n of each card left. Thus, 3, 10, 17, and 24 are P-positions if, respectively, at least 4, 3, 2, and 1 of each card remain. Using this observation, we discover that an optimal first move is to choose a 5: I II I II I II I II I +--A-- 6 --4-- 10* |--2-- 7 --3-- 10* 0 --5-- 5 --+--3-- 8 --2-- 10* +--A-- 13 --4-- 17* |--4-- 9 --A-- 10* |--2-- 14 --3-- 17* |--5-- 10 --2-- 12 --+--3-- 15 --2-- 17* +--A-- 20 --4-- 24* +--6-- 11 --6-- 17* |--4-- 16 --A-- 17* |--2-- 21 --3-- 24* |--5-- 17 --2-- 19 --+--3-- 22 --2-- 24* +--A-- 27 --4-- 31** +--6-- 18 --6-- 24* |--4-- 23 --A-- 24* |--2-- 28 --3-- 31** |--5-- 24 --2-- 26 --+--3-- 29 --2-- 31** +--6-- 25 --6-- 31** |--4-- 30 --A-- 31** +--6-- 32** Those positions marked with a single asterisk ( * ) are P-positions by the above observation; those marked with double asterisk ( ** ) are trivially P-positions or N-positions. Thus, no matter what player II plays, player I wins. It was easy to see that choosing a 4 or 6 is not an optimal first move for player I, and will result in a win for player II. In both cases, player II may bring the total to 10 on his or her turn with at least 3 of each card left, which is a P-position. Choosing a 3 is also not an optimal first move for player I. The analysis is similar to the case of choosing a 5 initially, except we start from player II’s optimial response, which is to choose a 4: I II I II I II I II I II +--A-- 8 --2-- 10* +--A-- 15 --2-- 17* +--A-- 22 --2-- 24* +--A-- 29 --2-- 31** |--2-- 9 --A-- 10* |--2-- 16 --A-- 17* |--2-- 23 --A-- 24* |--2-- 30 --A-- 31** 0 --3-- 3 --4-- 7 --+--3-- 10 --4-- 14 --+--3-- 17 --4-- 21 --+--3-- 24 --4-- 28 --+--4-- 32** |--4-- 11 --6-- 17* |--4-- 18 --6-- 24* |--4-- 25 --6-- 31** |--5-- 33** |--5-- 12 --5-- 17* |--5-- 19 --5-- 24* |--5-- 26 --5-- 31** +--6-- 34** +--6-- 13 --4-- 17* +--6-- 20 --4-- 24* +--6-- 27 --4-- 31** The analysis for choosing an A or 2 initially is more complex. If someone can show that these are optimal or non-optimal, let me know....
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solutions-167 - Selected Solutions from Thomas S...

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