Related papers: What majority decisions are possible

Suppose we are given a family of choice functions on pairs from a given finite set (with at least three elements) closed under permutations of the given set. The set is considered the set of alternatives (say candidates for an office). The…

Suppose we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential "voters". The question is, what choice…

Let $B$ be a fixed rational function of one complex variable of degree at least two. In this paper, we study solutions of the functional equation $A\circ X=X\circ B$ in rational functions $A$ and $X$. Our main result states that, unless $B$…

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a…

In a recent paper it was shown that No Free Lunch results hold for any subset F of the set of all possible functions from a finite set X to a finite set Y iff F is closed under permutation of X. In this article, we prove that the number of…

Given information about which options a decision-maker definitely rejects from given finite sets of options, we study the implications for decision-making with E-admissibility. This means that from any finite set of options, we reject those…

Let m>2 be an integer. We show that ZF + "For every integer n, Every countable family of non-empty sets of cardinality at most n has an infinite partial choice function" is not strong enough to prove that every countable set of m-element…

Methods for choosing from a set of options are often based on a strict partial order on these options, or on a set of such partial orders. I here provide a very general axiomatic characterisation for choice functions of this form. It…

We consider sets of operations on a set A that are closed under permutation of variables, addition of dummy variables and composition. We describe these closed sets in terms of a Galois connection between operations and systems of pointed…

If uncertainty is modelled by a probability measure, decisions are typically made by choosing the option with the highest expected utility. If an imprecise probability model is used instead, this decision rule can be generalised in several…

The following system of equations {x_1 \cdot x_1=x_2, x_2 \cdot x_2=x_3, 2^{2^{x_1}}=x_3, x_4 \cdot x_5=x_2, x_6 \cdot x_7=x_2} has exactly one solution in ({\mathbb N}\{0,1})^7, namely (2,4,16,2,2,2,2). Hypothesis 1 states that if a system…

Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets being enumerated by…

May's classical theorem states that in a single-winner choose-one voting system with just two candidates, majority rule is the only social choice function satisfying anonimity, neutrality and positive responsiveness axiom. Anonimity and…

A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that…

We prove that the following problem is decidable: given a finite set of relations, decide whether this set admits a near-unanimity function.

Many mathematical statements have the following form. If something is true for all finite subsets of an infinite set $I$, then it is true for all of $I$. This paper describes some old and new results on infinite sets of linear and…

We describe a model that explains possibly indecisive choice behavior, that is, quasi-choices (choice correspondences that may be empty on some menus). The justification is here provided by a proportion of ballots, which are quasi-choices…

Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…

Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision…

We prove that it is decidable if a finitely based permutation class contains infinitely many simple permutations, and establish an unavoidable substructure result for simple permutations: every sufficiently long simple permutation contains…