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Puzzle: baffling raffling

Puzzle: baffling raffling Puzzle: Ba „ing Ra „ing MICHAEL L. LITTMAN Rutgers University and DANIEL REEVES Beeminder, LLC A good (a professional soft-serve ice cream maker, as it turns out) is being ra „ed o € by an auctioneer. Bidders can buy tickets in the ra „e for $1 apiece and at the end of the ra „e the good is assigned to one of the bidders. The probability of assigning the good to a given bidder is proportional to the number of tickets he or she bought. Five bidders are participating in the ra „e. They value the good at $480.50 (Max), $336.35 (Nora), $240.25 (Ozg¨r), $210.65 (Paolo) and $190.20 u (Qiao), and these values are common knowledge. If they were participating in a standard (English) auction, the good would go to the highest bidder for a price equal to the second highest bid (or bid plus epsilon, depending on the implementation). In this case, Max would win the good at Nora ™s price of $336.35 and would bring in a pro t of $480.50−$336.35=$144.15. None of the other bidders make a pro t and the auctioneer brings in $336.35. In the ra „e, how many tickets should each of the participants buy, assuming their goals are to maximize pro t knowing that the others are trying to do the same? Who, of the auctioneer, Max, Nora, Ozg¨r, Paolo, and Qiao, gained the most u expected pro t by the auctioneer ™s choice of using a ra „e instead of an English auction?1 Send solutions to the editor at dreeves@umich.edu with subject: SIGecom Exchanges Puzzle. The author(s) of the most elegant solution (as judged by the editor) will be allowed to publish it in the next issue of the Exchanges (ties broken in favor of earlier submissions). To make the solutions accessible to a wide audience, please try to minimize technical jargon. Until the winner is chosen the editor will not give any hints or feedback. 1 The authors gratefully acknowledge the input of Vincent Conitzer in the preparation of this puzzle. Authors ™ addresses: mlittman@cs.rutgers.edu, dreeves@umich.edu http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM SIGecom Exchanges Association for Computing Machinery

Puzzle: baffling raffling

ACM SIGecom Exchanges , Volume 10 (1) – Mar 1, 2011

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Publisher
Association for Computing Machinery
Copyright
Copyright © 2011 by ACM Inc.
ISSN
1551-9031
DOI
10.1145/1978721.1978731
Publisher site
See Article on Publisher Site

Abstract

Puzzle: Ba „ing Ra „ing MICHAEL L. LITTMAN Rutgers University and DANIEL REEVES Beeminder, LLC A good (a professional soft-serve ice cream maker, as it turns out) is being ra „ed o € by an auctioneer. Bidders can buy tickets in the ra „e for $1 apiece and at the end of the ra „e the good is assigned to one of the bidders. The probability of assigning the good to a given bidder is proportional to the number of tickets he or she bought. Five bidders are participating in the ra „e. They value the good at $480.50 (Max), $336.35 (Nora), $240.25 (Ozg¨r), $210.65 (Paolo) and $190.20 u (Qiao), and these values are common knowledge. If they were participating in a standard (English) auction, the good would go to the highest bidder for a price equal to the second highest bid (or bid plus epsilon, depending on the implementation). In this case, Max would win the good at Nora ™s price of $336.35 and would bring in a pro t of $480.50−$336.35=$144.15. None of the other bidders make a pro t and the auctioneer brings in $336.35. In the ra „e, how many tickets should each of the participants buy, assuming their goals are to maximize pro t knowing that the others are trying to do the same? Who, of the auctioneer, Max, Nora, Ozg¨r, Paolo, and Qiao, gained the most u expected pro t by the auctioneer ™s choice of using a ra „e instead of an English auction?1 Send solutions to the editor at dreeves@umich.edu with subject: SIGecom Exchanges Puzzle. The author(s) of the most elegant solution (as judged by the editor) will be allowed to publish it in the next issue of the Exchanges (ties broken in favor of earlier submissions). To make the solutions accessible to a wide audience, please try to minimize technical jargon. Until the winner is chosen the editor will not give any hints or feedback. 1 The authors gratefully acknowledge the input of Vincent Conitzer in the preparation of this puzzle. Authors ™ addresses: mlittman@cs.rutgers.edu, dreeves@umich.edu

Journal

ACM SIGecom ExchangesAssociation for Computing Machinery

Published: Mar 1, 2011

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