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EXISTENCE OF CONDORCET
EQUILIBRIUM
Sadik Gokturk
St. JohnÕs University
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EXISTENCE OF CONDORCET
EQUILIBRIUM
Abstract
We study the necessary and sufficient conditions for the existence of equilibrium for majority rule. A new and an alternative set of conditions for the
existence of equilibrium for majority rule are proposed, which increases the three
Plott conditions by three. They are an angle, a convex cone, and a cardinality
condition, which, incidentally, seem to be well-suited to put into some sort of
order the confusion concerning the celebrated Plott conditions (the assumptions
of strict pair-wise symmetry, an odd number of voters, and the existence of a
centrally located voter) and the situations when some or all of Plott’s conditions
are not satisfied. It turns out that all these six conditions have to be checked in
order to ascertain the necessary and sufficient conditions that have to be satisfied for all possible preference profiles and all possible numbers of voters. Since
there are a number of relationships between these conditions, those relationships
also need to be explored, which of course indeed are explored. We also study the
question whether or not the expansion and/or contraction of the membership
of the decision-making body has a stabilizing or a destabilizing influence on the
equilibrium of the resulting decision-making body. This latter stability aspect
has some highly interesting applications, notably to customs union theory of
international trade and other alliances.
Presented at 15th International Conference, Istanbul, Turkey, May 2005.
International Trade and Finance Association: International Trade and Finance Association 15th International Conference
We study the necessary and sufficient conditions for the existence of
equilibrium for majority rule. A new and an alternative set of conditions for the
existence of equilibrium for majority rule are proposed, which increases the three
Plott conditions by three. They are an angle, a convex cone, and a cardinality
condition, which, incidentally, seem to be well-suited to put into some sort of
order the confusion concerning the celebrated Plott conditions (the assumptions of
strict pair-wise symmetry, an odd number of voters, and the existence of a
centrally located voter) and the situations when some or all of Plott’s conditions
are not satisfied. It turns out that all these six conditions have to be checked in
order to ascertain the necessary and sufficient conditions that have to be satisfied
for all possible preference profiles and all possible numbers of voters. Since there
are a number of relationships between these conditions, those relationships also
need to be explored, which of course indeed are explored. We also study the
question whether or not the expansion and/or contraction of the membership of
the decision-making body has a stabilizing or a destabilizing influence on the
equilibrium of the resulting decision-making body. This latter stability aspect has
some highly interesting applications, notably to customs union theory of
international trade and other alliances.
The purpose of this work then is to show that an equilibrium, reached by a
majority, exists for a community of n voters, with respect to m issues, under a
certain set of assumptions. Since however geometric arguments are used
extensively, m is restricted to 2. But, it turns out that the case when m > 2 does
not present any insurmountable problems. As it is well known, the literature goes
back to Marquis de Condorcet (1785), Borda (1781), and Dodgson (1873). More
recently, Black (1948, 1958), Black and Newing (1951) have laid the foundations
for the study of this problem. But, it was Plott (1967), who set the tone for the
modern analysis. By now the literature is vast, cross-fertilizing such areas as
social choice, game theory, positive political theory, matching theory, and
economics. Since it is not the intention of this work to review the voluminous
literature that followed, the reader can consult the carefully selected references at
the end for the state of the art of this subject.
VOTERS, PREFERENCES, PREFERENCE PROFILES, AND ISSUES
We assume that there are n voters and m policy variables (or issues)
(where m = 2, when a geometric reasoning is used). Let N denote the set of
voters. Thus, N = {1, 2,…, n}. An issue is considered to be a public good à la
Samuelson, for, once its magnitude (value or quantity) is determined, it is equally
available to all the voters. The issue space is the positive orthant
of an mdimensional Euclidean space. When the issue space is 2-dimensional, that is if m
= 2, then the issue space is simply the positive quadrant of a 2-dimensional
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Cartesian graph. Clearly, the axes measure continuously the quantities of these 2
policy variables. While most of the conclusions of this work are valid for the mdimensional Euclidean space, most of our analysis makes use of 2-dimensional
geometry.
The voter i is presumed to have a preference relation i (or a utility
function) on the set of issues. This preference relation can be lexicographic,
Euclidean, or any other. We assume a Euclidean metric and, therefore, circular
preferences, that are spatial. Thus, in a 2-dimensional issue space, the preference
relation of any voter i can be represented by concentric circles, where the center
of them is the ideal (or bliss) point of that voter. One clearly has in a 3dimensional issue space concentric globes. In an m-dimensional issue space,
there will be a corresponding representation, which, however, cannot be easily
visualized geometrically, because of its complexity. They naturally will be mdimensional counterparts of concentric circles (or globes).
The vector of preferences for n voters
= ( 1, 2,…, n ) forms a
preference profile, which is distributed over the issue space. If the bliss points of
the preferences of all n voters were coincident, then it is said that the preferences
of all voters are perfectly homogeneous. In this case, democracy yields a
unanimous solution. If the preferences of n voters do not coincide, then it said
that the preferences of the n voters are heterogeneous. Clearly, the latter is the far
more interesting case, which also, alas, is beset with many problems, some
insurmountable, for a majority vote equilibrium may not exist for many such
cases. Since homogeneous preferences are trivial, we are concerned here with
those conditions that yield a solution, when preferences are heterogeneous.
Let the i denote the set of preferences of the ith voter. Then, the set of
preference profiles is given by = 1 × 2 ×…× n = ni=1 i.
We assume that the preferences of all n voters are (i.e., the preference
profile is) given for the purposes of this work. The different, and much more
difficult, case, when the preferences of any voter i, i, can vary (change), thus
forming the above-defined sets of preferences i, is deferred to a future work. It
should be noted that, since voters frequently undergo transformations in their
evaluations of the various policy issues, it becomes essential that the set of
preference profiles be studied for that latter problem.
Incidentally, another interesting problem to study is, what happens, when
the composition and size of the set of n voters N change, i.e., new voters join the
decision-making body or existing voters leave the body. This work has some
things to say about that. The analysis of this aspect of the problem amounts to
finding conditions under which a stable situation will remain stable or will be
destabilized (or an unstable situation will remain unstable or will be stabilized).
This aspect of course can be studied when the number of voters n is constant or
not (i.e, the set of voters N is given, growing, or shrinking). Coalition formation,
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changes in the number of the members of a coalition, and changes in the
composition of a coalition belong to this kind of endeavor.
Another difficult problem that is not analyzed in this work is when the
number of m issues, i.e., the dimension of the non-negative orthant , changes.
Even more complex is the problem when, even if n does not change, the kinds of
issues change, i.e., the composition of the issue space changes. The problem of
course is much more complex, when the number of issues (the dimension of )
and the composition of the issue space changes simultaneously (i.e., when one
has a different number and different kinds of issues).
MAJORITY RULE EQUILIBRIUM
Equilibrium for majority rule is said to exist, when no motion can be
found that will defeat an existing situation by a different majority, i.e., when an
existing situation is not dominated. If to the contrary, a majority prefers a motion
to the existing situation, the existing situation cannot be an equilibrium. In the
latter case, the system can and will cycle and, hence, there will not exist a solution
to the problem.
All this of course is reminiscent of the Paradox of Voting, Marquis de
Condorcet (1785), Arrow’s Impossibility Theorem (1963), etc. In fact, Black’s
solution of the Paradox of Voting (1948, 1958) by means of Single-peaked
Preferences can be considered as the foundation of the present work. Charlie
Plott’s (1967a, b) assumption of the presence of a Centrally Located Voter in his
seminal paper of course comes from Black’s assumption of Single-peaked
Preferences which the latter had extended to Spatial Theory of Voting. The
assumption of the existence of a centrally located voter turns out to be redundant
(in fact, contradictory), if one wants to admit the possibility that the number of
voters can be even. It will be recalled that Plott, and much of the voluminous
literature it gave rise to, is solely concerned with an odd number of voters. Could
that be due to fears that ties might result?
When n = 1, the solution is trivial. Naturally, that voter, being the only
one, prevails.
When n = 2, the two different bliss points are connected by a Pair-wise
Pareto Optimality curve, so that, starting out from any other point not on the line
or on either of the two bliss points, the two decision-makers move to a mutually
acceptable point, such as E in Figure 1, on the line Pair-wise Pareto Optimality
curve. The unanimously agreed upon point E clearly is Pareto Optimal, so that
they will not move away from it, once they are there.
When n = 3, there are two possibilities.
When n = 3 and the three bliss points form a triangle, where each of its
vertices is the bliss point of one of the voters, there is no equilibrium, because
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there always is a motion a different majority of two will prefer to the existing
situation. The system cycles all over the triangle in Figure 2, as shown by the
dashed lines.
When n = 3 and the third voter’s bliss point is on the Pair-wise Pareto
Optimality curve of the other two, the system converges to the bliss point of the
centrally located voter by a majority of 2 (as in Figure 3) and stays there, thereby
making that point an equilibrium. One can also say that equilibrium in this case
corresponds to the point which is an element of the set, which is the intersection
of the Pair-wise Pareto Optimality curve of voters 2 and 3 (a set!) and the set
{bliss point of voter 1}. That intersection set and the final equilibrium point,
however, coincide in this case. This latter reformulation will be quite useful
below.
When n = 4, there are four possibilities.
When n = 4 and the bliss points of the four voters form a rectangle (as in
Figure 4), then the intersection of the 2 Pair-wise Pareto Optimality curves is an
equilibrium, which is reached by, what I have called elsewhere, the Mechanism of
Alternating Majorities, the terminus of which no majority can overturn (Gokturk,
2002).
When n = 4 and the bliss points of three voters form a triangle and the
fourth voter’s is somewhere in the interior of the triangle (as in Figure 5), the bliss
point of the centrally located voter becomes the equilibrium, which no majority of
3 can overturn.
What if we still had a triangle, but the bliss point of voter 1 was on the
Pair-wise Pareto Optimality curve of, say, voters 3 and 4, as depicted in Figure 6?
Evidently, equilibrium becomes in this case the bliss point of 1, which cannot be
overturned by any majority of 3.
One has also to consider the simpler case, when all four bliss points lie on
the Pair-wise Pareto Optimality curve of, say, 1 and 4, as depicted in Figure 7. It
should also be noted that the entire Pair-wise Pareto Optimality curve of voters 2
and 3 lies on the Pair-wise Pareto Optimality curve of 1 and 4 in this case. Once
an equilibrium, such as E, is reached, it cannot be overturned by any majority of
3. Note that E cannot be between 3 and 4 or between 1 and 2. If the system
initially arrived at some point between, say, 3 and 4, then a majority of 3 exists,
that majority being {1, 2, 3}, that prefers the bliss point of 3 to the point between
3 and 4. However, once that point is reached, there won’t be any majority of 3
that can overturn that one. Thus, the equilibrium will be somewhere on the Pairwise Pareto Optimality curve of 2 and 3 (or, to put it somewhat more precisely,
the equilibrium will have to be in the set that is formed by the intersection of the 2
Pair-wise Pareto Optimality curves (read, sets)). Clearly, equilibrium is not
unique in this case, for there are infinitely many points in that set.
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Although the preference profiles for the four cases, when n = 4, are quite
different from each other, there is a common thread running through all four
cases. Using the set language introduced above for n = 4, one can say that
equilibrium exists in all manifestations of four preference profiles and is in each
instance in the set, each of which corresponds to the intersection of relevant Pairwise Pareto Optimality sets. It should also be noted that, oddly, when n = 4 (an
even number!), all the possible preference profiles have an equilibrium. It is then
said that the core is not empty and corresponds to the intersection sets of the
relevant Pair-wise Pareto Optimality curves (read, sets). That intersection set is
{E} in Figure 4, {bliss point of voter 1} in Figures 5 and 6, and {bliss point of
voter 2, bliss point of voter 3, and all the infinitely many points on their Pair-wise
Pareto Optimality curve} in Figure 7.
When n = 5, there are four cases.
When n = 5 and the bliss points of the 5 voters form a pentagon, there is
no equilibrium. The system cycles in the smaller shaded pentagon in Figure 8.
Evidently, the cause for this is that there are no 3 Pair-wise Pareto Optimality sets
that intersect in the interior of the pentagon, 3 being the requisite majority. Or, to
put it differently, the intersection of any 3 Pair-wise Pareto Optimality sets is
empty, except at the 5 vertices, where there cannot be any equilibrium.
When n = 5 and the bliss points of 4 of the voters form a rectangle and the
fifth one’s bliss point is in the interior of the rectangle at the intersection of the
Pair-wise Pareto Optimality curve of 2 and 4 with that of 3 and 5 (as in Figure 9),
then an equilibrium exists in this case and is at the intersection of the 2 Pair-wise
Pareto Optimality curves, which no majority of 3 can overturn. Rephrasing this in
the language of sets, we say that the equilibrium point (which is the bliss point of
voter 1) is an element of the set, which is formed by the intersection of the 2 Pairwise Pareto Optimality sets and the set {bliss point of voter 1}. Note that this
case and the case in Figure 3 above are the only cases satisfying Plott’s conditions
so far. So, it is quite evident already that there are quite a number of additional
cases where equilibrium exists.
When n = 5 and the bliss points of 3 of the voters form a triangle and the
bliss points of either or both of the remaining 2 are in the interior or on the sides
of the triangle (as in Figure 10), there is no equilibrium.
When n = 5 and all 5 of the bliss points are lined up on a line, an
equilibrium exists and coincides with the bliss point of the centrally located voter,
as depicted in Figure 11, which is obtained by rotating one of the Pair-wise Pareto
Optimality curves in Figure 9 until a single line is obtained. Evidently, these two
cases are topologically equivalent. Once again, the equilibrium point (the bliss
point of the centrally located voter) turns out to be an element of that set which
happens to be the intersection of the 2 Pair-wise Pareto Optimality sets and {bliss
point of the centrally located voter}.
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The discussion so far shows clearly that there is absolutely no universal
need that the number of voters to be odd. In fact, it is the even cases above (n = 2
and n = 4) which alwayshave an equilibrium (which continues to be true, when n
becomes 6, 8, 10,…), while the odd cases have no equilibrium (with the trivial
exception of n = 1) for most possible preference profiles. It is because of this that
many researchers have concluded that majority rule is a very fragile thing.
In fact, when the number of voters is odd, the only preference profile that
admits a solution is the one postulated by Plott, i.e., that there be strict pair-wise
symmetry between (n – 1)/2 pairs of voters and that the remaining voter’s bliss
point be at the intersection of the pair-wise symmetry curves.
Since equilibrium exists for all possible preference profiles, when the
number of voters is even, what restrictions, if any, are to be placed on them?
Clearly, both of the Plott conditions are not applicable at the same time. The case
in Figure 4 clearly requires strict pair-wise symmetry, but there is no centrally
located voter. The case in Figure 5 has a centrally located voter, but there is no
strict pair-wise symmetry à la Plott. In Figure 5, there is of course a strict pairwise symmetry between voter 1 and 2 (or 3, or 4), as well as between any pair on
the vertices, but this kind of symmetry is not a Plott-type symmetry. The case in
Figure 6 can be interpreted both ways. There is strict symmetry between 1 and 2
(respectively 3 and 4), and the two curves intersect at the bliss point of voter 1,
who looks like being a centrally located voter, equilibrium being at his bliss point.
Thus, it looks like the two Plott conditions are being satisfied. On the other hand,
because of the symmetry relationships between 1 and 2 (or 3, or 4), Figure 6 and
Figure 5 are topologically equivalent, so that equilibrium is at the intersection of
those three curves, although the symmetry curve for 1 and 3 forms one part of the
symmetry curve for 3 and 4, while the symmetry curve for 1 and 4 makes up the
remainder. The case in Figure 7 does not have a centrally located voter, though
the strict symmetry condition is satisfied.
Since evidently the Plott conditions are not universal, what are the
universal necessary and sufficient conditions for majority rule equilibrium to
exist?
UNIVERSAL CONDITIONS FOR THE EXISTENCE OF MAJORITY RULE
EQUILIBRIUM
The research reported in this paper yields 3 totally new conditions for the
existence of majority rule equilibrium. In addition, Plott’s 3 conditions (viz.,
strict pair-wise symmetry, the existence of a centrally located voter, and an odd
number of voters) are extended as to include also non-strict pair-wise symmetry,
non-existence of a centrally located voter, and the possibility of an even number
of voters. There might be others, though I strongly doubt it. Thus, we have 6
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universal conditions for the study of the problem of the existence of majority rule
equilibrium.
It should first be remarked that the celebrated Plott conditions, that there
be (1) strict pair-wise symmetry between opposing types of preferences, (2) a
centrally located voter, and (3) an odd number of voters, are, by definition,
necessary and sufficient for the existence of equilibrium for majority rule for an
odd number of voters. In fact, the first two in the above list cannot be otherwise,
if the object of attention indeed is only the case with only an odd number of
voters. The question then is, what if there is an even number of voters, and/or
there is no strict pair-wise symmetry, and/or there is no centrally located voter?
Will there not exist an equilibrium under those circumstances? While the
previous section gave us some idea about these matters, the purpose of this
section is to study this aspect, as well as to introduce the 3 new conditions,
systematically.
The 6 universal conditions for the existence of equilibrium for majority
rule must be the following:
1. Weak or Strong Pair-wise Pareto Optimality Condition
As suggested above, the present author discovered that for an even number of
voters, not only does a majority rule equilibrium always exists, but, surprisingly
perhaps, one cannot find any preference profile, where an equilibrium will not
exist. While the preference profiles depicted in Figures 4, 6, and 7 correspond to
what might be called Strong Pair-wise Pareto Optimality, which is similar to
Plott’s strict pair-wise symmetry, the preference profile in Figure 5 clearly
corresponds to a very different situation. What matters in that latter case
ultimately is the relationship between the centrally located voter 1 and the others,
and not the relationship between 2 and 3, or 3 and 4, or 4 and 2, for the
equilibrium in that case is at the intersection of three Pair-wise Pareto Optimality
curves (1 and 2, 1 and 3, and 1 and 4), which happens to be at the bliss point of
voter 1. Thus, that point is not obtained by the intersection of two pair-wise
symmetry curves, as in the case of Strong Pair-wise Pareto Optimality. For want
of a better name for that, we have called that situation as the Weak Pair-wise
Pareto Optimality Condition. Clearly, this condition will be relevant for certain
preference profiles, when the number of voters is even.
2. Odd or Even Number of Voters Condition
As indicated already, we cannot ignore the situations involving an even number of
voters.
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3. The Presence or Absence of a Centrally Located Voter Condition
Clearly, we cannot also ignore preference profiles not containing a centrally
located voter.
4. Cardinality Condition
To illustrate this new condition, let n = 6. The six bliss points are depicted in
Figure 12. H is a hyperplane that goes through the bliss point of voter 1,
separating the remaining 5 bliss points into two mutually exclusive sets: S = {2,
6} and T = {3, 4, 5}. The cardinality of S is 2, since the number of elements in S
is 2, while the cardinality of T is 3, since the number of voters in T is 3.
Therefore, the net cardinality c = |T| - |S| = 3 – 2 = 1. Incidentally, as will be
shown below, one cannot find any other net cardinality other than 1 in this case.
If the net cardinality was much bigger than 1, there would be many more bliss
points on one side of H than the other side. Rotating the Pair-wise Pareto
Optimality curve of voters 1 and 6 clockwise until the bliss point of 6 is on the
right side of H, we obtain such a situation. Figure 13 shows such a case. Can
equilibrium continue to be at the bliss point of voter 1 under these circumstances?
Since there are now 4 voters on the right side of H and only 1 on its left, the net
cardinality c = 4 – 1 = 3. Thus, equilibrium cannot be at the bliss point of voter 1.
This can be shown as follows. If the starting point in Figure 13 was the bliss
point of voter 1, then a motion to move in an easterly fashion would win the
approval of {3, 4, 5, 6}, which constitute the needed majority of 4 in this case.
The equilibrium will be somewhere on the Pair-wise Pareto Optimality curve of
voters 3 and 6, between point A (which is the intersection between that Pair-wise
Pareto Optimality Curve and the pair-wise Pareto Optimality curve of voters 1
and 5) and point B (which is the intersection between the Pair-wise Pareto
Optimality curve of voters 3 and 6 and the Pair-wise Pareto Optimality curve of
voters 1 and 4). Since E now is an equilibrium for the time being, and not the
bliss point of voter 1 any more, the old H now has become redundant and is
replaced by H’ that goes through E. This temporary situation is depicted in
Figure 14. Since 4 voters are on one side of H’ and only 2 on the other, the net
cardinality c = 2. Thus a requisite majority {1, 2, 5, 6} exists and will take the
system in a northwesterly direction now. Thus, E cannot be the final equilibrium
position. As can be expected, the cycling process goes on ad infinitum in this
case. The reason for this of course is that the 3 Strong Pair-wise Pareto
Optimality curves do not intersect at one point, which would have become the
final equilibrium. Thus, the necessary and sufficient conditions in this case are
that (1) such an intersection occurs at a single point and (2) there is no bliss point
at that intersection. The equilibrium clearly is that intersection point. Figure 15
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summarizes all this, where a hyperplane H is drawn through that point, separating
the bliss points into 2 sets, each of which contains exactly 3 bliss points. Thus, he
net cardinality c = 3 – 3 = 0. Therefore, equilibrium for majority rule exists in
this case.
We thus can summarize that the necessary and sufficient condition for the
existence of majority rule equilibrium in general is that the net cardinality number
to be either 0 or 1. More formally, we say that c is an element of the set {0, 1},
for the majority rule equilibrium to exist. Figure 15 shows the case for c = 0,
while Figure 12 illustrates the case for c = 1. These are the only possibilities for
equilibrium to exist, when n = 6. When n increases to 8, 10,…, etc., this
condition still is valid, though the geometry becomes messier and messier.
It should also be noted that, since for an odd number of voters, there has to
be Strong Pair-wise Pareto Optimality between pairs of voters and a centrally
located voter at the intersection of them, the cardinality condition is automatically
satisfied, for c = 0 in this case. Thus, we say that for a majority vote equilibrium
with an odd number of voters to exist, the necessary and sufficient condition is
that c = 0.
The above arguments yield the following:
Theorem. Let n be either even or odd. The necessary and sufficient condition for
a majority rule equilibrium to exist is that the net cardinality c belongs to the set
{1, 0}.
Remark 1. Note that the Cardinality Condition is a lot less messy and quite more
elegant than the combined conditions of Strong Pair-wise Pareto Optimality and
the existence of a centrally located voter.
Remark 2. Plott’s conditions (pair-wise symmetry condition, the requirement of a
centrally located voter, and an odd number of voters) are automatically satisfied
by the Cardinality Condition, since c = 0 in this case.
Remark 3. Since there is no strict Pair-wise Pareto Optimality, when c = 1, the
Condition of Strong Pair-wise Pareto Optimality becomes redundant for that
case. Thus, strict pair-wise symmetry is not a universal condition.
Remark 4. The Cardinality Condition that is developed in this work is not
discussed in the literature.
Remark 5. In a sense, the Cardinality Condition measures which half of the
playing field is crowded relatively more and rules out certain situations of
crowding for the majority rule equilibrium to exist.
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5. Cone Condition
Convex cones were first used to study the existence of majority rule equilibrium
by Slutsky (1979). We use again the preference profile of Figure 12 to construct
the cones. By extending the 5 Pair-wise Pareto Optimality curves into the other
(negative) side of the bliss point of voter 1, we obtain 5 convex cones, one of
which (the one containing the bliss point of 3) is shaded in Figure 16. The reader
will notice that each convex cone contains exactly 2 bliss points, viz., the bliss
point of voter 1 at its vertex and the other one at the other end of the
corresponding Pair-wise Pareto Optimality curve. Thus, the bliss points of voters
1 and 3 are contained in the convex cone that is formed by the Pair-wise Pareto
Optimality curve of voters 1 and 5 and that of voters 1 and 6, i.e., they are
contained in the shaded convex cone. Similarly, the other 4 convex cones each
contain exactly 2 bliss points.
Now, consider a preference profile as in Figure 13. Figure 17 shows that
preference profile, along with its convex cones. The reader will notice that the
convex cones now contain an unequal number of voters. 2 of them contains the
bliss points of 3 voters each, 2 others contain the bliss points of 1 each, and 1
convex cone contains 2 bliss points. Since this preference profile was seen
above to be the case of non-existence of equilibrium, clearly the fact that each
convex cone contains different (some cones containing the same number as some
other cones) numbers of voters is the culprit. Thus, for a majority vote
equilibrium to exist, the number of bliss points of voters in each convex cone has
to be the same (and equal to 2 in this case).
The discussion above yields:
Theorem. For majority rule equilibrium to exist, each convex cone must contain
the same number of bliss points of voters, including the voter at the vertex of each
cone. That number is 2.
Remark. In a sense, this condition is focusing the attention on the relative
crowdedness of a particular “neighborhood,” as defined by the cones. The
convex cone containing the bliss points of voters 5 and 6, on the one hand, and
the one containing the bliss points of voters 3 and 4, on the other hand, are very
crowded (3 in each), while 2 others are almost empty, containing only voter 1 at
their vertices (as in Figure 17). The focus of attention thus is on the relative
emptiness or crowdedness of certain areas of the issue space with respect to the
preferences of the voters. Thus, the convex cones are in some sense much finer
than the hyperplane above in this regard. After all, there are 5 convex cones
here, while there are only 2 halfspaces in Figures 13 and 14. But, both have their
uses.
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6. Angle Condition
We use once again the preference profile in Figure 12 to describe the Angle
Condition. We define as the relevant angle for this condition the angle, which is
the sum of n/2 adjacent angles, each angle being between 2 Pair-wise Pareto
Optimality curves. For n = 6, there are 3 adjacent angles that form the sum.
There are 5 such sums. All are pictured in Figure 18. (For an odd n, this sum
becomes the sum of (n – 1)/2 adjacent angles.) If 26 denotes the angle between
the 2 Pair-wise Pareto Optimality curves for voters 2 and 6 (with voter 1), the
other 2 relevant angles are 56 and 23. Thus, the relevant sum is
= 35. The
reader might have noticed that all 5 such sums of angles (i.e., sums of 3 adjacent
angles) are greater than 180°. Thus, for a majority rule equilibrium to exist, none
of these 5 sums of angles should be less than 180°.
Now, consider the preference profile of Figure 13 again. In this case of
Figure 19, the sum of the 3 adjacent angles between voters 3 and 4, 4 and 5, and 5
and 6 clearly is less than 180°, which evidently is responsible for the nonexistence
of equilibrium for majority rule in this case. The other 4 sums of adjacent angles
are not drawn in Figure 18 for clarity’s sake; but all 4 are greater than 180° in this
case, as the reader can verify.
We thus have proved:
Theorem. If there is a centrally located voter and, thus, there is no pair-wise
symmetry, and the sum of n/2 adjacent angles is greater than 180°, for an even
number of voters, then a majority rule equilibrium will exist.
Remark. For an even number of voters, if there is pair-wise symmetry for one or
more pairs and a centrally located voter, then some of the sums will equal 180°,
while others will still be greater than 180°. Then, a majority rule equilibrium will
exist.
Theorem. If there is pair-wise symmetry and (1) n is odd and there is a centrally
located voter, or (2) n is even and there is no centrally located voter, then the sum
of the relevant number of adjacent angles is equal to 180°, the relevant number of
adjacent angles being (n-1)/2 for (1) and n/2 for (2). Then, a majority rule will
exist for both cases.
Remark 1. The Angle Condition for the existence of majority rule equilibrium is
that the sum of the relevant number of adjacent angles 180°, where the relevant
number for (1) an even number of voters is n/2 and (2) an odd number of voters is
(n – 1)/2.
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Remark 2. The Angle Condition has not been considered in the literature and,
thus, is new.
Remark 3. In some sense, this condition looks at the proximity (hence, the
similarity of preferences) of voters to each other.
Remark 4. One may summarize the last 3 newly introduced universal conditions
as follows: If a halfspace contains an overwhelming majority of bliss points, such
as the halfspace in some of the figures above, which contains {3, 4, 5, 6} while the
other one only {2}, then the 2 most crowded convex cones will also be in that
halfspace, as well as those voters will be much closer to each other than anybody
else to each other.
THE RELATIONSHIP BETWEEN NECESSARY AND SUFFICIENT
CONDITIONS FOR THE EXISTENCE OF MAJORITY RULE EQUILIBRIUM
Cardinality Theorem. Strict pair-wise symmetry holds, if and only if the net
cardinality c = 0.
Proof. (1) For the case when n is an odd number, if one has (n – 1)/2 pairs of strict
pair-wise symmetry curves, intersecting at the bliss point of a centrally located
voter, then there exists a hyperplane H, going through the bliss point of the
centrally located voter, separating the bliss points into 2 sets, whose cardinalities
are equal. Thus, c = 0. (2) For the case when n is an even number, if one has n/2
pairs of strict pair-wise symmetry curves, intersecting at a unique point, then there
exists a hyperplane H, going through the intersection, separating the bliss points
into 2 sets, whose cardinalities are equal. Thus, c = 0. One shows similarly, if c =
0, then there must be strict pair-wise symmetry in both cases.
Remark. Clearly, majority rule equilibrium exists for both even and odd number
of voters, if there is strict pair-wise symmetry, which is equivalent to c = 0.
Theorem. If n is even and there is a centrally located voter, then a majority rule
equilibrium exists.
Proof. If n is even and there is a centrally located voter, then the number of the
remaining voters n – 1 is odd. Thus, there cannot be strict pair-wise symmetry.
Thus, the centrally located voter must be in the interior or the boundary of the
triangle (as in Figure 5, and also in Figure 6). The Separating Hyperplane
Theorem then yields an equilibrium for the majority in both cases. (Note that the
strict pair-wise symmetry curve for voters 1 and 2 does not intersect the pair-wise
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symmetry curve for voters 3 and 4 in Figure 5. But, there exists a hyperplane H,
going through the bliss point of the centrally located voter 1, separating the
remainder of bliss points into 2 sets, whose cardinalities are 2 and 1, respectively.
Therefore, c = 1, ensuring the existence of an equilibrium.)
Figure 20 summarizes all this.
Cone Theorem. The number of voters in each convex cone = 2, if and only if n is
even.
Proof. If n is an even number, one has either Figure 4 or 5. Constructing the
convex cones, as outlined above, one verifies that that number indeed is = 2 in
each convex cone. The converse is easily proved.
Theorem. If n is odd, then the number of voters in each convex cone is 3.
Proof. One can easily verify in Figure 9 that indeed the number of voters in each
convex cone is 3, counting the centrally located voter’s bliss point in each convex
cone.
Remark 1. Equilibrium for majority rule exists in both cases that are covered by
the two above theorems.
These are summarized by Figure 21 above.
Remark 2. Since Figure 9 depicts Plott’s Theorem, we now have found that for
Plott’s Theorem to hold, the above Theorem (the last one) must hold, i.e., these 2
theorems are equivalent to each other. Note that, since either theorem’s
hypothesis is that n = odd, the hypothesis that the number of voters in each
convex cone = 3 is equivalent to the assumption that there be strict pair-wise
symmetry and a centrally located voter. These are the necessary and sufficient
conditions for the existence of a majority rule equilibrium for Plott’s case.
Angle Theorem. If there is strict pair-wise symmetry, then the sum of the relevant
number of adjacent angles = 180°, where the relevant number is (1) n/2 when n is
even and (2) (n-1)/2, when n is odd.
Proof. It follows from Figures 4 and 9, that the relevant angle sums indeed are
180°.
It is clear that the converse does not hold.
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Theorem. If there is no strict pair-wise symmetry and n is even, then the sum of
n/2 adjacent angles 180°.
Proof. Follows from the two figures at the bottom left of Figure 22.
Remark. The rightmost figure at the bottom left of Figure 22 may be thought to
have been obtained by either (1) rotating one of the bliss points on the periphery
of the leftmost figure, or (2) by pulling one of the pair-wise symmetry curves at
the top figure, for n even, or (3) by deleting one of the voters on the periphery at
the bottom right figure, for n odd (While the first 2 are topologically equivalent to
each other, the last one is not, since a voter is being deleted.). The net result of
each is that a strict pair-wise symmetry curve intersects another one at one of its
extreme points, i.e., at a bliss point. Now, where does this fit? Since n = 4, it
does not fit to bottom right. Since there is strict pair-wise symmetry, it does not fit
to bottom left. Since the 2 sums of the 3 relevant angles are greater than 180°, it
does not fit to the top either. So, this hybrid has to be treated by itself, perhaps as
a corollary.
Corollary. If
180°, n is even, there is a centrally located voter, and there is
pair-wise symmetry (one of the pair-wise symmetries is between the centrally
located voter and another one, while the other one is between the other pair, so
that the ideal point of the centrally located voter is on the pair-wise symmetry
curve of the other pair), there exists an equilibrium for majority rule (See Figure
22).
Remark 1. There is no contradiction to have simultaneously strict pair-wise
symmetry and the existence of a centrally located voter in the case of an even
number of voters, if the preference profile is as in the above Corollary. Most of
the existing literature seems to see a contradiction here; but, the angle condition
clarifies this.
Remark 2. Clearly, equilibrium exists in both of the cases covered by the last 2
theorems and the above corollary, since none of the sums of the relevant number
of adjacent angles < 180°, as summarized by Figure 22.
Figure 23 puts together all the cases discussed separately in Figures 20,
21, 22. Since a majority rule equilibrium exists in all these cases, Figure 23 sorts
out which collections (subsets) of these conditions are to be satisfied in
conjunction with each other for equilibrium to exist for the various cases at hand.
It can be seen that Figure 23 is a valuable tool to ascertain the roles played by the
various collections of assumptions and the way they “hang together.” For
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instance, the celebrated Plott conditions (i.e., n = odd, strict pair-wise symmetry,
and the existence of a centrally located voter) are at the right bottom corner of
Figure 23. Figure 23 shows 3 additional characteristics about that case, viz., that
the net cardinality c = 0, the number of voters in each convex cone = 3, and the
sum of the (n-1)/2 adjacent angles = 180°. Thus, a total of 6 Conditions have to
be verified, in order to show the existence of majority rule equilibrium. However,
the reader should recall that, as shown above, there is an intimate relationship
between some these conditions. A similar statement can be made for the other 2
cases in Figure 23.
This is the most comprehensive study of the necessary and sufficient
conditions for the existence of majority rule equilibrium and their
interdependence.
STABILITY
Since the distribution of the locations of the bliss points do matter, stability (or the
lack of it) depends on which member (located where?) is removed from
(respectively, added to) the voting body. Clearly, if a pair-wise symmetry curve
is removed, by removing the corresponding member pair from the voting body, an
equilibrium will continue to exist, if there was an equilibrium to begin with, and
will not, if there existed no equilibrium to begin with. This statement of course is
still valid, if a new pair-wise symmetry curve was added, by admitting the
corresponding pair of voters to the voting body. Likewise, if the angle condition
was satisfied initially, but is violated when a member is removed (respectively,
admitted), a stable situation becomes unstable, and vice versa.
Thus, stable situations can stay stable, unstable situations can stay
unstable, stable situations can become unstable, or unstable situations can become
stable, by changing the number and the composition of the membership of N,
since equilibrium can continue to exist, can continue not to exist, existence can
lead into non-existence, or non-existence can lead into existence, depending on
these aforementioned catalogue of potential changes in N.
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