# Solution to exchanges 10.1 puzzle: Baffling Raffling debaffled

Solution to exchanges 10.1 puzzle: Baffling Raffling debaffled Solution to Exchanges 10.1 Puzzle: Ba „ing Ra „ing Deba „ed PRESTON MCAFEE Yahoo! Research [Puzzle Editor ™s Note: This is the winning solution to Ba „ing Ra „ing from issue 10.1. The mechanism described there is sometimes known as a Chinese Auction. It is also equivalent, as McAfee points out, to a special case of a Cournot problem. An alternative formulation is: I decide a bid x, pay it in full, and then win the good with probability x/X where X is the sum of all the bids. Generalizing the question in the original puzzle, this solves the game for an arbitrary vector of common-knowledge valuations, i.e., the complete-information case with n agents.] Notation: xi is i ™s bid (the number of tickets bought by i) and vi is the value of i, indexed so that v1 ¥ v2 ¥ . . .. Let X ’i = j=i xj and X = j xj . xi vi ’xi . The xi + X ’i choice of xi is restricted to xi ¥ 0, and probably should be restricted to integers. I will ignore this constraint. [This turns out to be moot for the speci c (carefully constructed) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM SIGecom Exchanges Association for Computing Machinery

# Solution to exchanges 10.1 puzzle: Baffling Raffling debaffled

, Volume 10 (3) – Dec 1, 2011
3 pages

/lp/association-for-computing-machinery/solution-to-exchanges-10-1-puzzle-baffling-raffling-debaffled-cGyejDA0wm
Publisher
Association for Computing Machinery
ISSN
1551-9031
DOI
10.1145/2325702.2325712
Publisher site
See Article on Publisher Site

### Abstract

Solution to Exchanges 10.1 Puzzle: Ba „ing Ra „ing Deba „ed PRESTON MCAFEE Yahoo! Research [Puzzle Editor ™s Note: This is the winning solution to Ba „ing Ra „ing from issue 10.1. The mechanism described there is sometimes known as a Chinese Auction. It is also equivalent, as McAfee points out, to a special case of a Cournot problem. An alternative formulation is: I decide a bid x, pay it in full, and then win the good with probability x/X where X is the sum of all the bids. Generalizing the question in the original puzzle, this solves the game for an arbitrary vector of common-knowledge valuations, i.e., the complete-information case with n agents.] Notation: xi is i ™s bid (the number of tickets bought by i) and vi is the value of i, indexed so that v1 ¥ v2 ¥ . . .. Let X ’i = j=i xj and X = j xj . xi vi ’xi . The xi + X ’i choice of xi is restricted to xi ¥ 0, and probably should be restricted to integers. I will ignore this constraint. [This turns out to be moot for the speci c (carefully constructed)

### Journal

ACM SIGecom ExchangesAssociation for Computing Machinery

Published: Dec 1, 2011