## Abstract

In Kalai (2002) [10], Kalai investigated the probability of a rational outcome for a generalized social welfare function (GSWF) on three alternatives, when the individual preferences are uniform and independent. In this paper we generalize Kalai's results to a broader class of distributions of the individual preferences, and obtain new lower bounds on the probability of a rational outcome in several classes of GSWFs. In particular, we show that if the GSWF is monotone and balanced and the distribution of the preferences is uniform, then the probability of a rational outcome is at least 3/4, proving a conjecture raised by Kalai. The tools used in the paper are analytic: the Fourier-Walsh expansion of Boolean functions on the discrete cube, properties of the Bonamie-Beckner noise operator, and the FKG inequality.

Original language | English |
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Pages (from-to) | 389-410 |

Number of pages | 22 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 117 |

Issue number | 4 |

DOIs | |

State | Published - May 2010 |

Externally published | Yes |

### Bibliographical note

Funding Information:1 This research is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

### Funding

1 This research is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

Funders | Funder number |
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Israel Academy of Sciences and Humanities |

## Keywords

- Arrow's theorem
- Discrete harmonic analysis
- Fourier-Walsh expansion