NAIS Journal

Aggregation of Preferences
Milan Vlach
Professor, The Kyoto College of Graduate Studies for Informatics

In several past decades various abstract models have been

We should not conclude from the example that the simple

created for the purpose of assisting individuals or group of indi-

majority rule is not applicable at all. For instance, if

viduals in arriving at good decisions. Here we are concerned

player 1 prefers x to y, y to z, and x to z,

with some questions that arise when a group of individuals has

player 2 prefers y to x, x to z, and y to z,

to select one alternative from a given set of possible alterna-

player 3 prefers x to z, z to y, and x to y,tives. Depending on the context the alternatives might be called

then we have the situation in which the simple majority rule pro-strategies, commodity bundles, feasible solutions and so forth.

duces perfectly acceptable result: namely,
x is better than y, y is better than z, x is better than z.

If each member of the group has logically consistent preferences and if the preferences of all members are sufficiently

Before proceeding to more detailed analysis, we need tosimilar, then there is usually easy to find a satisfactory solution

introduce some definitions and notation. Let X denote the set

by using a suitable voting procedure. Unfortunately, in many cases

whose elements are to be evaluated in terms of preferences in

the latter condition is not satisfied. Then one of the possibilities

some particular situation. We shall call the elements of X alter-for resolving conflict among preferences of different members

natives. If a player prefers an alternative x to alternative y, weis to employ a suitable rule for aggregating the individual prefer-shall write briefly x y. We assume that each player is rational

ences into a group preference, and then to select the best alterna-in the sense that his or her preferences satisfy certain conditions

tive on the basis of the resulting group preference. To illustrate

which reflect some reasonable requirements of consistency. For

some obstacles and difficulties we may encounter, we first con-example, it would be rather strange to allow the existence ofsider the following simple example.

two alternatives, say x and y, such that a player prefers alterna-

Let X = {x, y, z} be a set consisting of three different alter-

tive x to alternative y and at the same time he or she prefers al-natives x, y and z, and let N = {1, 2, 3} be a group consisting ofternative y to alternative x. In other words, we shall assume thatthree members, which we will call players. Suppose that

preference relations of all players are asymmetric in the sense

player 1 prefers x to y, y to z, and hence x to z,

that, for all x, y ∈ X , we haveplayer 2 prefers z to x, x to y, and hence z to y,if xy, then not x(asymmetry)y.

player 3 prefers y to z, z to x, and hence y to x.

As a consequence, we obtain that preference relations of all

We wish to aggregate this profile of three individual preference

players are irreflexive in the sense that, for all x ∈ X, we have
not x x.

relations over set X into a group preference relation over the(irreflexivity)

same set X in such a way that the individual preferences are tak-It would also be rather unreasonable to allow that there are

en into account.

three alternatives, say x, y, and z, such that a player prefers x to

Let us apply one of the common aggregation rules, namely, the simple majority rule. In our particular situation we ob-y, y to z, and simultaneously z to x. Therefore we assume thatthe preference relations are transitive, that is,
if x

tain

y and yz, then xz.

(transitivity)

x is better than y because 1 and 2 vote for x over y,

We also need the property of negative transitivity, which meansy is better than z because 1 and 3 vote for y over z,

thatz is better than x because 2 and 3 vote for z over x.

if not xy and not yz, then not x

z. (neg. transitivity)

We now have rather unpleasant situation. Each individual pref-

Let us now return to the problem of aggregation of individ-

erence relation is without any contradiction however the result-ual preferences into a group preference. We have already dem-ing group preference relation is self-contradictory. It is cyclic

onstrated that the simple majority rule may aggregate transitive

and no alternative is best. In other words, all participants arepreference relations into an intransitive relation, even in a veryrational, however as a group they are rather irrational. This phe-

simple situation involving only three alternatives and three play-

nomenon is known as the Condorcet voting paradox.

ers. In this connection, two natural questions arise. First, wheth67

Aggregation of Preferences

er such an undesirable property is typical for the simple majori-simple majority voting is acyclic. Let us call such sets of orders

ty rule, and how to avoid it. Second, whether there exist someacyclic sets. It is surprisingly difficult to specify the largest (inreasonable rules without that drawback.

the sense of the number of orders) acyclic sets when the num-

The following table gives a partial answer to the first ques-

ber of alternatives is greater than three. It is known that the larg-tion. Its entries present the fractions of the set of all n-tuples ofest acyclic sets for four alternatives consists of nine orders, andorders on the set of m alternatives for which the simple majority

the largest acyclic sets for five alternatives consists of twenty or-rule gives an intransitive relation.

ders. But it is not much known about the exact size of the largest acyclic sets when the number of alternatives is greater than

n

3

5

7

911

...

limit

3.056

.069

.075

.078

.080

...

.088

4

.111

.139

.150

.156

.160

...

.176

5.160

.200

.215

.230

.251

...

.251

6.202

.255

.258

.284

.294

...

.315

7.239

.299

.305

.342

.343

...

.369

...

1.000

m

five. The reader desiring to enjoy research in this area can find1

details in the paper by Peter C. Fishburn .
To answer the second question, we need to define ourproblem more precisely. Namely, we have to specify which aggregation rules are considered as reasonable.

...

...

...

...

...

1.000

1.000

1.000

1.000

1.000

...

...
limit

Let X be a given set of alternatives and let N be the set ofindividuals in society. We assume that both X and N are finite
sets containing at least two elements. For definiteness, let N be
the set {1, 2, . . . , n}. The set of all negatively transitive asym-We see that that the probability of encountering the para-

metric relations on X will be denoted by P, and ordered n-tuples ( 1, 2,..., n) of relations from P will be called profiles. We

dox increases rather rapidly with growing number of alterna-

assume that each person is completely free to have an arbitrary

tives. One way of avoiding the paradox is to limit the freedom

relation from P as his or her preference relationof players to choose arbitrary preferences.

ested in functions F that map the set of all possible profiles, thati. We are inter-

Then it is desirable to restrict the players freedom as lessis, the Cartesian product Pn=(P×P×...×P), into the set P. Suchas possible. This leads to an interesting problem, whose many

functions are called social welfare functions, and one is interest-aspects are still waiting for a satisfactory solution. To facilitate

ed in those functions that have some reasonable properties. As

its formulation, let us assume that the players are never indiffer-

examples of quite reasonable properties, consider the following

ent between distinct alternatives, so that their preferences arethree conditions on the behavior of social welfare functions.

represented by permutations of alternatives. In the simplest nontrivial case of three alternatives x, y and z, each player's preference relation is represented by one of the orders
x y z, x z y, y x z, y z x, z x y, z y x.
Let us forbid one of the orders, say xyz. Does it help? Not really, as we can see from the following preferences. Suppose thatCONDITION 1. If every member of a group prefers alternative x to alternative y, then the group also prefers x to y.
Formally speaking, we require that for all alternatives
x ,y ∈ X and all profiles p= ( 1, 2,..., n) in Pn, we have:if xiy for all i∈N , then xF(p)y.

player 1 prefers y to x, x to z, and y to z,
player 2 prefers x to z, z to y, and x to y,player 3 prefers z to y, y to x, and z to x.
Then in the simple majority voting y wins over x, x wins over zand z wins over y. Since similar results hold also for other or-CONDITION 2. There is no dictator.
A person i∈N is called a dictator when, for all alternatives x ,y ∈ X and all profiles p=( 1, 2,..., n) from Pn, we have:
if x

iy, then xF(p)y.

ders, we have to forbid more than one order to avoid the paradox. It is clear from the previous examples that we are not com-

It is easy to see that the second condition can be expressed

pletely free in excluding orders and that it is necessary to ex-

as follows: For each person i∈N , there exist alternatives x and y
in X, and a profile p=( 1, 2,..., n) in Pn such thatclude one of the orders x y z, z xy, y z x and one of the orders z y x,x zy, y x z. It turns out that this necessary condition is also suffi-xiy and not xF(p)y.

cient.
Thus, for example, the set consisting of the orders x y z,

CONDITION 3. The group preference between any twoz x y, x z y, y x z has the property that (independently of how

alternatives depends only on the individual preferences be-

many players we have) if the preference relation of each player

tween those two alternatives.

belongs to this set, then the group preference resulting from the68

In order to formalize this condition, we first intro-NAIS Journal

duce some notation. If Y is a subset of X, then it is possi=( ′n)ble that two profiles, say p=( 1, 2,..., n) and p′
1, ′2,..., ′are different when considered on X, but at the same time they
may be identical when considered on Y only. Let p/Y andF(p)/Y denote the reductions of p and F(p) from X to Y , respectively. Using similar notation for other profiles and relations,
we can now express the third condition as follows:nFor all x, y∈X and all p, p′∈P ,/ {x,y}, then F(p)/{x,y}=F(p′)/{x,y}if p/{x,y}=p′
It turns out that in spite of the fact that each condition isvery appealing on its own, they cannot be satisfied simultaneously.

1

Decision theory and discrete mathematics , Discrete Ap-plied Mathematics 68 (1996) 209-221.
2

Handbook of Social Choice and Welfare , Elsevier, Amster-dam, Volume 1, 2002; Volume 2, forthcoming.

THEOREM (Arrow). If X has more than two elements,nthen there is no F:P →P satisfying all three conditions.

Milan Vlach
Professor, The Kyoto College of Graduate Studies for Informatics

There is an extensive literature on the problem of aggrega-Bachelor of Science from Charles University, Czech Republic
Doctor of Natural Sciences from Charles University, Czech Republiction of individual preferences into a group or social preference.

Doctor of Philosophy from Charles University, Czech Republic

For information about various aspects and recent status of re-Doctor of Sciences from Czechoslovak Academy of Sciences

search, see the following handbook edited by K.J. Arrow, A.K.
2

Sen and K. Suzumura .

Formerly Professor, Charles University, Czech Republic
Formerly Professor of Information Science, Japan Advanced Institute ofScience and Technology (JAIST)69