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PartIISec5_9

# PartIISec5_9 - Game Theory Solutions to Exercises The...

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Game Theory Solutions to Exercises: The Extensive Form of a Game Jan-Jaap Oosterwijk Fall 2007

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5 The Extensive Form of a Game 5.9.1 The Silver Dollar Player II chooses one of two rooms in which to hide a silver dollar. Then, Player I, not knowing which room contains the dollar, selects one of the rooms to search. However, the search is not always successful. In fact, if the dollar is in room #1 and I searches there, then (by a chance move) he has only probability 1 / 2 of finding it, and if the dollar is in room #2 and I searches there, then he has only probability 1 / 3 of finding it. Of course, if he searches the wrong room, he certainly won’t find it. If he does find the coin, he keeps it; otherwise the dollar is returned to Player II. Draw the game tree. Solution: (See graphical appendix at the end of this document for the tree.) The fact that the game is started by Player II and not I may seem to be meant to trick you, but it’s actually very much in line with another convention: that we work with the winnings for Player I. In this game, the beginning player, Player II, either loses his coin or ends up keeping it, so Player I is the one with positive winnings. Hence, Ferguson has kept your numerical calculations positive. Remember that “Player I, not knowing which room contains the dollar” implies that the two positions at which I moves in the tree are contained in one information set. Finally, don’t make the ‘mistake’ of replacing Player I’s move and the following chance move (performed by Nature) to one single move, representing the expected winnings. From a ‘strategic form’ point of view, the solution of both trees might be the same, but the Kuhn tree is part of analysis in ‘extensive form’ in which each and every move, also those by Nature, should be visible and winnings should be actual winnings, not expected ones. To see why it’s sometimes even crucial to draw these, also have a look at the next exercise ‘Two Guesses for the Silver Dollar’. 5.9.2 Two Guesses for the Silver Dollar Draw the game tree for problem 1, if when I is unsuccessful in his first attempt to find the dollar, he is given a second chance to choose a room and search for it with the same probabilities of success, independent of his previous search. (Player II does not get to hide the dollar again.)
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PartIISec5_9 - Game Theory Solutions to Exercises The...

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