## Abstract

Finitely many agents have preferences on a finite set of alternatives, single-peaked with respect to a connected graph with these alternatives as vertices. A probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles at which all peaks are on leaves of the tree and, thus, extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leaves. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation.

Original language | English |
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Pages (from-to) | 811-833 |

Number of pages | 23 |

Journal | Mathematics of Operations Research |

Volume | 46 |

Issue number | 2 |

Early online date | 11 Mar 2021 |

DOIs | |

Publication status | Published - May 2021 |

## JEL classifications

- d71 - "Social Choice; Clubs; Committees; Associations"

## Keywords

- Probabilistic rules
- SCHEMES
- SOCIAL CHOICE
- Single-peaked preferences.
- block trees
- graphs
- probabilistic rules
- single-peaked preferences
- strategy-proofness
- unanimity
- Block trees
- Unanimity
- Strategy-proofness
- Single-peaked preferences
- Graphs