Mathematical Social Sciences 37 1999 45–57
Beta distributions in a simplex and impartial anonymous cultures
Eivind Stensholt
Norwegian School of Economics and Business Administration , Helleveien 30, 5035 Bergen-Sandviken,
Norway Received 30 June 1997; received in revised form 6 January 1998; accepted 11 February 1998
Abstract
Variations of IAC are introduced and simulated. A uniformly distributed point P 5 X , X , . . . ,
1 2
X in a simplex S is generated by a map
´ , ´ , . . . , ´ →
P from the unit cube to S surjective
n 11 1
2 n
with bijective restriction to interiors with the ´ ’s rectangular and i.i.d on [0,1]. The fraction xyz
i
of the electorate with preference x . y . z is a sum of X ’s. The variations allow different
i
correlations e.g. rxyz, xzy ± rxyz, zyx while they all are 2 0.2 under IAC. Simulation of two
such variations give smaller Condorcet paradox propability than IAC. This is explained heuristically with a graphic ‘‘pictogram’’ representation of the profile.
1999 Elsevier Science
B.V. All rights reserved. Keywords
: Beta distributions; preference structures; Condorcet’s paradox; impartial anonymous cultures
JEL classification : D71
1. Introduction
Let a random point P 5 X , X , . . . , X be uniformly distributed in a simplex S.
1 2
n11
The sums of barycentric coordinates have beta-distributions, which are known as distributions for order statistics. In a recent paper, Tovey 1997 relates the order
statistics directly to S and uses them to generate P. Section 2 generates P by a transformation Eq. 4 which also allows an elementary proof of the density formulas.
The ‘‘cultures’’ IC and IAC, which are compared in Section 3, are common assumptions about the stochastic distribution of profiles. They are often used in
Tel.: 147-5595-9298; fax: 147-5595-9234; e-mail: eivindhamilton.nhh.no 0165-4896 99 – see front matter
1999 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 9 8 0 0 0 1 6 - X
46 E
. Stensholt Mathematical Social Sciences 37 1999 45 –57
simulation work with the purpose of comparing the properties of various election rules which define the result in terms of the profile. Consider 3 alternatives a, b, c parties,
candidates, propositions. A profile for these alternatives is a vector Eq. 9 in the standard 5-dimensional simplex S, where xyz is the fraction of the electorate with
preference order ‘‘x before y before z’’. This vector is multinomially distributed under the IC-assumption, and for a large electorate the distribution is concentrated around the
simplex center. It is shown that if a, b, c are three among a larger number of alternatives, there is a similar concentration under IAC. For an election rule symmetric under all
permutations of alternatives, all final rankings will then be equally probable, and any final ranking may be obtained for a profile arbitrarily close to the center.
Another common feature of IC and IAC is that the Condorcet paradox occurs much more often than it is seen in real life with large electorates. Section 4 suggests ways to
use the easy generation of beta-distributions to define variations of the IAC which give a much smaller frequency of the paradox. In these cultures the x . y . z voters agree more
with the x . z . y voters than with the z . y . x voters etc. Two such variations are studied with computer simulation. They have a much smaller paradox probability than
the standard IAC. The ‘‘pictograms’’ of the profiles, introduced in Stensholt 1996, visualize the reason for this reduction.
2. Some probabilities associated with a simplex
