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
Let S be an n-dimensional simplex with corners C , C , . . . , C . Let a stochastic
1 2
n 11
point P 5 X ,X , . . . ,X
[ S 1
1 2
n 11
be distributed with uniform density in S. In Eq. 4 below is shown a way to generate Eq. 1 by means of a standard generator for a rectangularly distributed variable. The X
i
are the barycentric coordinates, so that C 51, 0, . . . , 0[S, etc. By definition,
1
X 0 for all i and X 1 X 1 ? ? ? 1 X 5 1.
2
i 1
2 n 11
Since the barycentric coordinates are invariant under affine transformations, we may assume S is the standard simplex. Thus the X are stochastic variables identically
i
distributed on the unit interval [0,1]. Also all the pairs X , X with i ±j are identically
i j
distributed, and similarly for triples, etc. Thus, by Eq. 2 1
1 ]]
] EX 5
and CovX ,X 5 2 VarX .
3
i i
j i
n 1 1 n
Let ´ , ´ , . . . , ´ be independent stochastic variables uniformly distributed on [0,1].
1 2
n
Then in Eq. 1 we obtain a uniformly distributed P [S by defining
1 n 1 n
1 n 21 1 n
1 n 21
X ,X , . . . ,X 5 1 2
´ ,
´ ? 1 2
´ ,
´ ?
´
1 2
n 11 1
1 2
1 2
1 n 22 1 n
1 n 21
? 1 2 ´
, . . . , ´
? ´
? . . . ? ´ .
4
3 1
2 n
E . Stensholt Mathematical Social Sciences 37 1999 45 –57
47
To establish this claim, observe that the event X x may be written equivalently
1 n
´ 12x . Hence the event xX x1Dx has probability proportional to
1 1
n n
n 21
1 2 x 2 1 2 x 2 Dx
¯ n ? Dx ? 1 2 x The claim follows by induction on n, the P [S with X 5x forming an n21-
1
dimensional simplex. The density function for each barycentric coordinate becomes d
d
n n 21
] ]
ProbX x 5 1 2 1 2 x 5 n ? 1 2 x
. 5
i
dx dx
From Eqs. 3 and 5 we obtain n
2 1 ]]]]]
]]]]] VarX 5
, CovX ,X 5 if i ± j.
6
i 2
i j
2
n 1 1 ? n 1 2 n 1 1 ? n 1 2
Let Y be the sum of any t barycentric coordinates X , 1t n.
t i
THEOREM The density function of Y is
t
d n 2 1
n 2t t 21
] ProbY x 5 n ?
S D
? 1 2 x ? x
t
dx t 2 1
Fig. 1.
Comment: Eq. 4 defines a bijection between the interiors of the cube and the simplex which maps equal volumes to equal volumes. The theorem states that Y has a standard
t
form beta distribution with integer parameters n 112t, t. If the ´ , ´ , . . . , ´ are
1 2
n
ordered by size: ´
´ . . .
´ , then t-th order statistic has the same
s 1 s 2
s n
distribution as Y . See Johnson et al. 1995, Ch. 25, 26. Tovey 1997 shows directly
i
that the vector of differences of the order statistics, in the above notation ´
2 ´
, ´
2 ´
, . . . , ´
2 ´
[ S ´ 5 0,
´ 5 1,
s 1 s 0
s 2 s 1
s n11 s n
s 0 s n11
n 2t t 2n
Fig. 1. Illustration to the proof. The curve is ´
512x ?12r
.
t 11
48 E
. Stensholt Mathematical Social Sciences 37 1999 45 –57
is uniformly distributed. This gives an alternative way to generate P. Instead of the root extractions in Eq. 4 one then needs a sorting routine to find
s.
Proof of Theorem: For t 51, this is just Eq. 5. Without loss of generality, let
1 n 1 n 21
1 n 112t
Y 5 X 1 X 1 . . . 1 X 5 1 2 ´
? ´
? . . . ? ´
. 7
t 1
2 t
1 2
t
The proof now proceeds by induction on t, using the formula 12Y 512Y ?
t 11 t
1 n 2t
´ . Then, since
´ is rectangularly distributed on [0,1] and is also stochastically
t 11 t 11
independent of Y , we get
t
d d
n 2t t 2n
] ]
ProbY x 5
ProbY x and ´
1 2 x ? 1 2 Y
t 11 t
t 11 t
dx dx
x
d n 2 1
n 2t t 21
n 2t
] 5
E
S
n ?
S D
? 1 2 r ? r
? 1 2 1 2 x dx
t 2 1
t 2n
? 1 2 r
D
dr 5 see illustration
x
d n 2 1
n n 2 1
n 2t t 21
n 2t t
] n ?
S D
?
E
1 2 r ? r
dr 2
S D
? 1 2 x ? x
5 n ?
S D
dx t 2 1
t t
1 2
n 2t 21 t
? 1 2 x ? x j
3. A comparison of two ‘‘cultures’’
