Editor's puzzle: product adoption in a social network

Vincent Conitzer

doi:10.1145/1486877.1486889

Conitzer, Vincent

2008-11-01 00:00:00

Editor ™s Puzzle: Product Adoption in a Social Network VINCENT CONITZER Duke University Solutions should be sent to the editor at conitzer@cs.duke.edu with subject header SIGecom Exchanges Puzzle. The author(s) of the most elegant solution (as judged by the editor) will be allowed to publish his or her or their proof in the next issue of the Exchanges (ties broken towards earlier submissions). To make the solution accessible to a wide audience, try to minimize technical jargon in the proof. The editor will not give any feedback on submitted solutions and ignore any requests for hints, etc. We have n agents who, in sequence, have to decide whether to adopt technology A or technology B. The decision is irreversible. Each agent has a single other agent that she admires, drawn uniformly at random from the other agents. (Note that if x admires y, it is not necessarily the case that y admires x.) With probability p0 , an agent is a œB fanatic who will certainly adopt B. With probability p1 , an agent is a œB copycat who will adopt B if the agent that she admires has already adopted B, and A otherwise. With probability 1 ’ p0 ’ p1 , an agent is an œA fanatic who will certainly adopt A. All random draws are independent. Assuming that n is large, give a simple formula for the nal fraction of agents who adopt B. Hint: In a simulation with n = 1000000, p0 = .5, and p1 = .5, repeated 1000 š times, the average nal fraction of adopters was 0.6487, approximately e ’ 1. In a simulation with n = 1000000, p0 = .1, and p1 = .1, repeated 1000 times, the average nal fraction of adopters was 0.1052. In a simulation with n = 1000000, p0 = .1, and p1 = .2, repeated 1000 times, the average nal fraction of adopters was 0.1107. Authors ™ addresses: conitzer@cs.duke.edu Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for pro t or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior speci c permission and/or a fee. c 2008 ACM 1529-3785/2008/0700-0001 $5.00

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ACM SIGecom Exchanges Association for Computing Machinery

http://www.deepdyve.com/lp/association-for-computing-machinery/editor-s-puzzle-product-adoption-in-a-social-network-mwkbhbhpDq

$Editor ™s Puzzle: Product Adoption in a Social Network VINCENT CONITZER Duke University Solutions should be sent to the editor at conitzer@cs.duke.edu with subject header SIGecom Exchanges Puzzle. The author(s) of the most elegant solution (as judged by the editor) will be allowed to publish his or her or their proof in the next issue of the Exchanges (ties broken towards earlier submissions). To make the solution accessible to a wide audience, try to minimize technical jargon in the proof. The editor will not give any feedback on submitted solutions and ignore any requests for hints, etc. We have n agents who, in sequence, have to decide whether to adopt technology A or technology B. The decision is irreversible. Each agent has a single other agent that she admires, drawn uniformly at random from the other agents. (Note that if x admires y, it is not necessarily the case that y admires x.) With probability p0 , an agent is a œB fanatic who will certainly adopt B. With probability p1 , an agent is a œB copycat who will adopt B if the agent that she admires has already adopted B, and A otherwise. With probability 1 ’ p0 ’ p1 , an agent is an œA fanatic who will certainly adopt A. All random draws are independent. Assuming that n is large, give a simple formula for the nal fraction of agents who adopt B. Hint: In a simulation with n = 1000000, p0 = .5, and p1 = .5, repeated 1000 š times, the average nal fraction of adopters was 0.6487, approximately e ’ 1. In a simulation with n = 1000000, p0 = .1, and p1 = .1, repeated 1000 times, the average nal fraction of adopters was 0.1052. In a simulation with n = 1000000, p0 = .1, and p1 = .2, repeated 1000 times, the average nal fraction of adopters was 0.1107. Authors ™ addresses: conitzer@cs.duke.edu Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for pro t or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior speci c permission and/or a fee. c 2008 ACM 1529-3785/2008/0700-0001 $5.00$

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Editor's puzzle: product adoption in a social network

Editor's puzzle: product adoption in a social network

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Editor's puzzle: product adoption in a social network

Editor's puzzle: product adoption in a social network

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