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Solution to exchanges 7.3 puzzle: product adoption in a social network

Solution to exchanges 7.3 puzzle: product adoption in a social network Solution to Exchanges 7.3 Puzzle: Product Adoption in a Social Network Aneesh Sharma and Sicco Verwer Two correct solutions were submitted to the puzzle in SIGecom exchanges given at http://www.sigecom.org/exchanges/volume_7/3/PUZZLE.pdf. Both of these solutions are listed below. The rst is by Aneesh Sharma, the second by Sicco Verwer. Solution 1 Let Bi denote the expected number of B adoptors after i agents have made their adoption decisions. We are interested in computing Bn/n. First, we observe that B1 = p0 as the rst agent can only adopt B if she is a B fanatic. Further, we observe that for any i > 1: Bi+1 = p0 + Bi Bi p1 (Bi + 1) + 1 ’ p0 ’ p1 Bi n n This is because the expected number of agents go up by 1 only if either agent i + 1 is a B fanatic or if the agent that i + 1 has chosen to admire has already chosen to adopt B (with probability Bi/n). In the remaining cases, the expected number of agents remain the same. Now, we can simplify the above equation to get: p1 Bi+1 = p0 + 1 + Bi n We can http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM SIGecom Exchanges Association for Computing Machinery

Solution to exchanges 7.3 puzzle: product adoption in a social network

Sharma, Aneesh; Verwer, Sicco
ACM SIGecom Exchanges , Volume 8 (1) – Jul 1, 2009
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Publisher
Association for Computing Machinery
Copyright
The ACM Portal is published by the Association for Computing Machinery. Copyright © 2010 ACM, Inc.
ISSN
1551-9031
DOI
10.1145/1598780.1598793
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Abstract

Solution to Exchanges 7.3 Puzzle: Product Adoption in a Social Network Aneesh Sharma and Sicco Verwer Two correct solutions were submitted to the puzzle in SIGecom exchanges given at http://www.sigecom.org/exchanges/volume_7/3/PUZZLE.pdf. Both of these solutions are listed below. The rst is by Aneesh Sharma, the second by Sicco Verwer. Solution 1 Let Bi denote the expected number of B adoptors after i agents have made their adoption decisions. We are interested in computing Bn/n. First, we observe that B1 = p0 as the rst agent can only adopt B if she is a B fanatic. Further, we observe that for any i > 1: Bi+1 = p0 + Bi Bi p1 (Bi + 1) + 1 ’ p0 ’ p1 Bi n n This is because the expected number of agents go up by 1 only if either agent i + 1 is a B fanatic or if the agent that i + 1 has chosen to admire has already chosen to adopt B (with probability Bi/n). In the remaining cases, the expected number of agents remain the same. Now, we can simplify the above equation to get: p1 Bi+1 = p0 + 1 + Bi n We can

Journal

ACM SIGecom Exchanges – Association for Computing Machinery

Published: Jul 1, 2009

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