The Probability of a Cyclical Majority
Author(s): Frank DeMeyer and Charles R. Plott
Source:
Econometrica,
Vol. 38, No. 2 (Mar., 1970), pp. 345-354
Published by: The Econometric Society
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Econometrica, Vol. 38,
No. 2
(March, 1970)
THE PROBABILITY
OF
A
CYCLICAL MAJORITY
BY FRANK DEMEYER AND CHARLES R. PLOTT1
CONSIDER
A
COMMITTEE or society attempting
to order the
alternatives
(X1, X2, X3)
by use of majority
rule. Each individual is assumed to have a
strong ordering
(called
a
profile) on the alternatives.
"Indifference"
is
not a
property of the profiles.
The committee is said to "prefer"
Xi
to
Xj,
denoted
XiCXj
if
Xi
is preferred
to
Xj
on a majority of the individual profiles. It is well known that if certain indivi-
dual
profiles are chosen,
the
resulting
"social
ordering" may
be
cyclical, i.e.,
XiCXj,
XjCXk,
XkCXi.
Such a result is called a "cycle."
Two aspects of this problem have been of interest. The first is that of placing
conditions
on individual
profiles necessary
and
sufficient
for the
resulting
social
ordering to contain a cycle (see [1, 2,
6,
8, 9, 11, 12, 13]).
The
second
is
that
of
obtain-
ing the probability that certain types
of
cycles occur-given
that individuals are
allowed to
choose at random among all possible profiles. This probability depends
upon the number of people (always assumed to be odd) and the number of alter-
natives.
There are three different probabilities of interest. We let n
=
2in +
1
be the odd
number of
individuals (m is a positive integer) and
we let
r
>
3 be the number of
alternatives.
The
probabilities
of interest are:
(i) Q(m, r): the probability that one issue is preferred by a majority to all other
issues;
(ii) P(m, r):
the
probability that the social ordering is completely transitive
(contains
no
cycle);
(iii) Z(m, r):
the
probability that one issue is preferred by a majority to all other
issues and
the
complete
social
ordering contains a cycle.
Very
little
is known about these functions. Duncan Black [2]
found
that
P(1, 3)
=
.9444, ..., by complete enumeration. David Klahr [7] found Q(1, 4)
=
.8888, ...,
by
enumeration.
Monte Carlo
techniques
were used
[3, 7]
to
estimate
Q(m, r)
for
small values
of the
variables. All
of these
probabilities are
for the case
where
choices over
the profiles are equally likely.
Our
analysis
will
proceed
as
follows. In
Section
1,
we will derive the
special
case for
P(m, 3)
=
Q(m, 3)
and the
choices
are
completely
random.
This is done
in
order to
acquaint
the reader with the notation used
in the
following
sections.
In Section
2,
we derive
Q(m, r).
In Section
3,
we
derive
P(m, r).
In the
final
section,
we
present
some numerical values for
Q(m, r)
and
P(m, r).
Before
continuing,
we can deal
with
Z(m, r) directly.
We
simply
observe
that
if
the social
ordering
is
completely
transitive
(contains
no
cycle),
then one
issue
1
This paper
was delivered at the
meeting
of the
Econometric
Society, Chicago,
1966.
The
material
in
Section
2
has been treated independently in two papers published since the writing of this paper
[5, 10]. The authors wish to thank Otto Davis, Morton Kamien, and David
Klahr
for
their
comments
and
suggestions.
345
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346
F.
DEMEYER AND
C. R.
PLOTT
is
preferred
by a
majority
to
all
others. It
follows
directly
that
(1)
Z(m, r)
=
Q(m, r)
-
P(m, r).
1.
P(m, 3)
=
Q(m, 3)
FOR THE
EQUALLY LIKELY
CASE
Since the notation of
the
following sections
becomes rather
cumbersome, we
shall
first derive
the
function
for
the
special
case of
three
alternatives and
equally
likely choices
over
the
possible
profiles.
Observe
P(m, 3)
=
Q(m,
3) since
in the
case of
three
alternatives
a
cycle
occurs if
and
only
if
no
alternative
is
preferred by
a
majority
to all
others.
Let
Q(Xi)
be the
probability that
Xi
is
preferred
by
a
majority to the
other two.
Then
3
(2)
Q(m,
3)
=
E
Q(Xi).
Since, by
assumption,
the
choices over
profiles are
equally
likely,
(3) Q(X1)
=
Q(X2)
=
Q(X3)
= Q(m,
3)
3
Consequently,
in order to
compute
Q(m, 3)
we need
only
find
Q(X1)
and
multiply
by
3.
The n
=
2m
+
1
individuals
choose at random from the
elements of Sr
=
S3=
{U1,
. *,
,76}.
The
set
Sr
contains all
possible
orderings
(profiles)
a,
of the
r alter-
natives. It
thus
contains r!
elements
as
enumerated.
af 1 2
U3 U4
U5
U6
Xl
Xl
X2
X3
X2
X3
X2
X3
X3
X2 X1
Xl
X3
X2
X1 X1 X3
X2
Let
Ui,
where 0
<
Ui
<
2m
+ 1, be
the
number
of
voters who select
profile
vi
E
S3,
1
<
i
<
3!.
Ui
will,
of
course, always be an
integer.
We know
(4)
U1
+
U2
+
...+
U6
=
2m
+
1,
0
<
Ui
<
2m
+
1,
or
3!
(5)
Ui
=
2m
+
1-
E
Ui.
i=2
We write
ai(Xj)
=
k in
case,
on
profile
i,
there
are k
-
1
alternatives
preferred
to
Xj.
That
is,
Xj
is ranked in the kth
place
on
profile
i.
If
oi(Xj)
<
oi(XI),
then
Xi
is ranked
higher
on
profile
i than is
XI.
Thus the committee
"prefers"
Xi
to
Xj
in
case a
majority
of the
voters choose
profiles
ai such
that
uI(Xi)
<
cr(Xj).
This is
written
XiCXj.
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CYCLICAL
MAJORITY
347
Define:
Ali
=
ai
eS3ai(X1)
<
Ui(Xj)},
j
=
2,3,
A'1
=
{ai
E
S3Joi
OA
lj},
Bij=
{ilJicAlj};
B'j
=
{ilaieA'jl}.
Where
the
profiles
are
indexed as enumerated above
A12
=
{O71,O2,O6};
A'12
=
{O3,O4,O5};
B12
=
{1,2,6};
B'12
=
{3,4,5};
A13
=
{Ol1,O2,O5};
A'13
=
{O3,
U4,
6};
B13
=
{1,2,5};
B'13= {3,4,6}.
If a
majority
of the
voters
prefer
X1
to both
X2
and
X3,
we have
(6.1) U1
+
U2 + U6
>
U3
+
U4
+
U5,
(6.2) U1
+ U2
+
U5
>
U3 + U4 +
U6,
or
(7)
uj
U> Euj,
j=2,3.
ic-Btj
i
c-B'
j
Observe that
(5)
and
(7) provide
necessary
and sufficient conditions for
X1
to be
preferred
to
X2
and
X3 by
a
majority.
Now
substituting (5)
into
(6),
we
get
(8.1) 2m
+
1
>
2U3 +
2U4 + 2U5,
(8.2) 2m
+
1
>
2U3
+
2U4 + 2U6,
which
simplifies
to
(9.1)
m
U3 + U4 +
U5,
(9.2)
m
U3 + U4 +
U6,
or
(10)
m
Eu,
j
=
2,3.
i
c-B'l
j
System (10)
simply stipulates
that less than half of
the
individuals choose
profiles
on which
either
X2
is
preferred
to
X1
or
X3
is
preferred
to
X1 .
Again, (5)
and
(10)
are
necessary
and
sufficient
for
XiCXi,
i
=
2,
3.
Any
solution to the
system
(5)
and
(10)
will
be a distribution of
the
voters
among
the
possible
profiles
such that
X1CXi,
i
=
2,
3. Further
if
voters choose
profiles
such that
X1CXi,
i
=
2, 3,
then that distribution
of
voters
will be a solution to
the
system
(5)
and
(10).
If
voters
choose
among
the
possible profiles
in
S3,
such
that
Ui
of
the voters
choose
profile
ai,
the
probability
that a
particular
U',.
.
.
Ut!
occurs is
given
by
the
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348
F.
DEMEYER
AND
C. R.
PLOTT
multinomial
formula
as
(11)
P(U1,...
U6)
=(m
U+
.
1)6!
01
*6
where
Oi
is the
probability
that
an
individual
chooses
vi.
Since, by
assumption,2
01
=,...,
=
06
=
l/r!
= 1/6,
we can
simplify
(11)
to
(2m
+
1)?
6
1
(12)
P(Ul,.
U6)=
()2mt
+
-
1
HU
We can
now find
Q(X1)
by
attaching
to
each
solution
to
(5)
and
(10)
the
number
dictated
by
(12)
and
summing
all such
numbers
over all solutions
to
(5) and
(10).
By substituting
(5)
into (12),
using
(3), and summing,
we
obtain
(13) 3Q(X1)-Q(m,3)=
3(2m
+
1)!
f(2)
f(6)
6
1
-(6)
U2=0***
U6=0
j=2
Uj!(2m
+
1
-
jUy
02m
+
1-
i,<kUi
if
k
B'1,
1=2,3,
f(k)
=
-
+
1
<k
Ui
2m
.
I
ill<k
i
Imin{
i*
1
t
M-li<k
Ui,
a1l1withkc-B'1a,1{2,3J,
which is the
desired
expression.
In
terms of
the
profiles
as
indexed,
this
is
(14)
Q(m,
3)
=
(6)2m
+1
rn
+
I
m
+
2m +
I-U2
M2
+
I-U2
min
-U2
min
-U2-U3
min<
-U3-U4
min?
-U3-U4-U5
2m+1
m
Mm-U3
M(m-U3-U4
(m-U3-U4
X
E
E
E
E
E
[U2!
U2=o
U3=0
U4=0
U5=O
U6=O
X
U3!U4!
U5!
U6!(2m
+
1
-
Si=2
U)!] .
2. Q(m,
r):
PROBABILITY
OF
PREFERENCE
FOR ONE
ISSUE
The
derivation
of Q(m,
r)
is a straight
forward
generalization
of the
analysis
contained
in
Section
1. We start
by
calculating
the
probability
Q(X,)
that
X,
is
preferred
by
a
majority
to all other
alternatives.
Observe
that
r
(15)
Q(m,r)
=
E
Q(Xs).
S=
Again
individuals
choose
from the
elements
of Sr
=
{'1,
. ..
,
r!}
with
Oi
being
the
probability
that any particular
individual
chooses profile
vi.
2
This
assumption
will
be
dropped
in Sections
2
and 3.
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