arXiv subjects

Gabriel Gendler

Publications and source records attributed to Gabriel Gendler.

4 recordsLinked to original sources

arXiv2025

Partial results for union-closed conjectures on the weighted cube

The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can pose more general conjectures by choosing a different probability distribution on the cube. In particular, for any sequence of probabilities $(p_i)_{i=1}^d$ we can consider the product of $d$ independent Bernoulli random variables, with success probabilities $p_i$. In this short note, we find a generalised form of Karpas' special case of the union-closed conjecture for families $\mathcal{F}$ with density at least half. We also generalise Knill's logarithmic lower bound.

math.CO↗

arXiv2024

Good election rules with more than three candidates are Borda

Arrow proved that for three or more candidates, the IIA condition is enough to forbid all non-dictatorial election rules (or Social Welfare Functions). Maskin introduced the weaker MIIA condition, which permits the ``Borda'' election rules where each voter assigns points linearly to each candidate according to their order of preference. However, in previous work we demonstrated that there exist Social Welfare Functions between three candidates and satisfying the MIIA condition which are far from being Borda rules. We demonstrate that this phenomenon is unique to the case of three candidates. As soon as a fourth candidate is introduced, and indeed for any larger number of candidates, the only good election rules are the unweighted Borda rules.

math.CO↗

arXiv2024

Condorcet cycle elections with influential voting blocs

A Condorcet cycle election is an election (often called a Social Welfare Function, or SWF) between three candidates, where each voter ranks the three candidates according to a fixed cyclic order. Maskin showed that if such a SWF obeys the MIIA condition, and respects the complete anonymity of each voter, then it must be a Borda election, where each voter assigns two points to their preferred candidate, one to their second preference and none to their least preferred candidate. We introduce a relaxed anonymity condition called ``transitive anonymity'', whereby a group $G$ acting transitively on the set of voters $V$ maintains the outcome of the SWF. Elections across multiple constituencies of equal size are common examples of elections with transitive anonymity but without full anonymity. First, we demonstrate that under this relaxed anonymity condition, non-Borda elections do exist. On the other hand, by modifying Kalai's proof of Arrow's Impossibility Theorem, which employs methods from the analysis of Boolean functions, we show that this can only occur when the number of voters is not a multiple of three, and we demonstrate that even these non-Borda elections are very close to being Borda.

math.CO↗

arXiv2024

Non-Borda elections under relaxed IIA conditions

Arrow's celebrated Impossibility Theorem asserts that an election rule, or Social Welfare Function (SWF), between three or more candidates meeting a set of strict criteria cannot exist. Maskin suggests that Arrow's conditions for SWFs are too strict. In particular he weakens the "Independence of Irrelevant Alternatives" condition (IIA), which states that if in two elections, each voter's binary preference between candidates $c_i$ and $c_j$ is the same, then the two results must agree on their preference between $c_i$ and $c_j$. Instead, he proposes a modified IIA condition (MIIA). Under this condition, the result between $c_i$ and $c_j$ can be affected not just by the order of $c_i$ and $c_j$ in each voter's ranking, but also the number of candidates between them. More candidates between $c_i$ and $c_j$ communicates some information about the strength of a voter's preference between the two candidates, and Maskin argues that it should be admissible evidence in deciding on a final ranking. We construct SWFs for three-party elections which meet the MIIA criterion along with other sensibility criteria, but are far from being Borda elections (where each voter assigns a score to each candidate linearly according to their ranking). On the other hand, we give cases in which any SWF must be the Borda rule.

cs.GT↗