## Abstract

For f a weighted voting scheme used by n voters to choose between two candidates, the n Shapley-Shubik Indices (or Shapley values) of f provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 [SS54] and are widely studied in social choice theory as a measure of the "influence" of voters. The Inverse Shapley Value Problem is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant ε > 0 our algorithm runs in fixed poly(n) time (the degree of the polynomial is independent of ε) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error ε. If there is a "reasonable" voting scheme in which all voting weights are integers at most poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error ε = n^{-1/8}.

Original language | English (US) |
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Title of host publication | Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings |

Pages | 266-277 |

Number of pages | 12 |

Edition | PART 1 |

DOIs | |

State | Published - Dec 1 2012 |

Event | 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012 - Warwick, United Kingdom Duration: Jul 9 2012 → Jul 13 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 1 |

Volume | 7391 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012 |
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Country | United Kingdom |

City | Warwick |

Period | 7/9/12 → 7/13/12 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)