Generating random weak orders and the probability of a Condorcet winner

Abstract

A b s tra c t. We present an algorithm for generating a random weak order of m objects in which all possible weak orders are equally likely. The form of the algorithm suggests analytic expressions for the probability of a Condorcet winner both for linear and for weak preference orders.

Responsibility and cross-subsidization in cost sharing

“…In the 3-agent case we have 13 such methods: the 6 ordered contributions methods, 3 methods based on preorderings of the type i ≺ j ∼ k, 3 based on preorderings of the type i ∼ j ≺ k, and the subsidy-free serial method. With 5 agents, there are 541 methods, and 47,293 with 7 agents: see Maassen and Bezembinder (2002) for a general formula.…”

Analysis of consumer preferences expressed by prioritization chains

“…The factors have the following levels: M = 10, 20, and 50 subjects and n = 5 and 10 alternatives, resulting in a total of six analyzed scenarios (ie, distinct pairs of M and n value). Under the null hypothesis of randomness, PCs on a set of n alternatives have been simulated using the algorithm proposed by Maassen and Bezembinder …”

Random preorders and alignments

“…Preorders are used in [14] to model the voting preferences of voters. (A different but equivalent definition of preorders is used in [14], where they are called weak orders.) We suppose that there are n candidates and m voters.…”

“…The assumption is made in [14] that each voter chooses his voting preference uniformly at random from all of the p(n) possibilities independently of the other voters. An algorithm for generating a random preorder is given in [14] and the ideas behind the algorithm are used to derive a formula for the probability of the occurrence of "Condorcet's paradox". See [11] for a survey of assumptions on voter preferences used in the study of Condorcet's paradox.…”