The comparative likelihood of an equivocal outcome under the plurality, Condorcet, and Borda voting procedures

Gillett, Raphael

doi:10.1007/bf00128125

Cited by 5 publications

(5 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…IC for arbitrary numbers of alternatives and agents. Gillett [Gil80] obtained a closed-form formula for Borda indecisiveness (two or more alternatives being tied) for m = 3 w.r.t. IC.…”

Section: Related Work and Discussionmentioning

confidence: 99%

How Likely Are Large Elections Tied?

Xia¹

2020

Preprint

View full text Add to dashboard Cite

Understanding the likelihood for an election to be tied is a classical topic in many disciplines including social choice, game theory, political science, and public choice. The problem is important not only as a fundamental problem in probability theory and statistics, but also because of its critical roles in many other important issues such as indecisiveness of voting, strategic voting, privacy of voting, voting power, voter turn out, etc. Despite a large body of literature and the common belief that ties are rare, little is known about how rare ties are in large elections except for a few simple positional scoring rules under the i.i.d. uniform distribution over the votes, known as the Impartial Culture (IC) in social choice. In particular, little progress was made after Marchant [Mar01] explicitly posed the likelihood of k-way ties under IC as an open question in 2001.We give an asymptotic answer to the open question for a wide range of commonly-studied voting rules under a model that is much more general and realistic than i.i.d. models including IC-the smoothed social choice framework [Xia20], which was inspired by the celebrated smoothed complexity analysis [ST09]. We prove dichotomy theorems on the smoothed likelihood of ties under a large class of voting rules called integer generalized irresolute scoring rules [Xia15], and refine the theorem for positional scoring rules, edge-order-based rules, and multi-round score-based elimination rules, which include commonly-studied voting rules such as plurality, Borda, veto, maximin, Copeland, ranked pairs, Schulze, STV, Coombs, and Baldwin as special cases. Our main technical tool is an improved dichotomous characterization on the smoothed likelihood for a Poisson multinomial variable to be in a polyhedron, which is proved by exploring the interplay between the V-representation and the matrix representation of polyhedra and might be of independent interest.

show abstract

“…IC for arbitrary numbers of alternatives and agents. Gillett [Gil80] obtained a closed-form formula for Borda indecisiveness (two or more alternatives being tied) for m = 3 w.r.t. IC.…”

Section: Related Work and Discussionmentioning

confidence: 99%

How Likely Are Large Elections Tied?

Xia¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In the paired-comparison method, a further type of indecisiveness can occur, namely, cyclical majorities in which A > B > C > A. An analysis of the likelihood of an indecisive outcome under each of the methods is provided in Gillett (1978Gillett ( , 1980b.…”

Section: Indecisive Outcomesmentioning

confidence: 99%

“…A substantial body of research into the validity of the procedure now exists (Loftus, 1979;Shepherd et al, 1982;Wells & Loftus, 1984;Wells & Turtle, 1986). The issue that has received most extensive investigation is the fairness of a lineup.…”

mentioning

confidence: 99%

See 1 more Smart Citation

An eyewitness paradox

Gillett

1992

Scandinavian J Psychology

Self Cite

View full text Add to dashboard Cite

Two methods of analysing data from an identification parade give rise to seemingly contradictory outcomes, both of which reach statistical significance. The traditional method is based on the first choices of eyewitnesses, whereas the paired‐comparison method takes account of each eyewitness's full rank ordering of the lineup participants. The person that most eyewitnesses pick out as the perpetrator of an offence (according to the first‐choice method) may in fact be the person that those same eyewitnesses as a group regard as least likely to have committed the act (according to the paired‐comparison method). Implications of the paradox for the analysis of lineup data are considered.

show abstract

“…Other calculated values for odd n up to49 and odd m up to 25 are shown as five digit entries in Table I. Gillett (1980b) used simulation in an attempt to find the p3 which minimize P(3, n, p3) for various n and conjectured that P(3, n, p3) is minimized by pl a = pa4 = ps 3 = 89 or p~ = p~ = p6 a = ~. Gillett (1977) calculated the probability of a unique Condorcet winner for three alternatives under impartial culture with an even number of voters for n up to 20.…”

Section: Impartial Anonymous Culture Condition (Ia C) -Under This Con-mentioning

confidence: 99%

Condorcet's paradox

Gehrlein

1983

Theor Decis

173

View full text Add to dashboard Cite

162WILLIAM V. GEHRLEIN ns a = 1. In this example A > sB, B > sC and C> sA where A > sB denotes group simple majority preference for A over B. As a result, we find that there is no alternative capable of defeating all other candidates on the basis of pairwise simple majority elections. If n~ = 1, n: 3 = 1, n~ = 1, we find that A is the Condorcet winner or simple majority winner since A > sC and A > sB.

show abstract

The comparative likelihood of an equivocal outcome under the plurality, Condorcet, and Borda voting procedures

Cited by 5 publications

References 8 publications

How Likely Are Large Elections Tied?

How Likely Are Large Elections Tied?

An eyewitness paradox

Condorcet's paradox

Contact Info

Product

Resources

About