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A generalization of Moulin’s Pareto extension theorem
* John A. Weymark
Department of Economics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z1
Received 14 October 1998; accepted 25 November 1998
Abstract
In this note, it is shown that for any strongly complete and quasitransitive binary relation B on a set X, there exists a set of linear orders on X such that B is simultaneously the union of these linear orders and the Pareto extension relation generated by them. No assumptions about the cardinality
´
of X are made. This result generalizes a theorem established by Herve Moulin for finite X.
2000 Elsevier Science B.V. All rights reserved.
Keywords: Binary relations; Pareto extension
JEL classification: C60
1. Introduction
The Pareto extension relation derived from a set of strongly complete binary relations extends the corresponding strong Pareto relation by incorporating the Pareto
noncompar-1
able alternatives into the symmetric part of the Pareto extension relation. Theorem 1 in Moulin (1985) shows that when the set of alternatives X is finite, any binary relation B on X that is strongly complete and quasitransitive is the Pareto extension relation for some set of linear orders on X. Moulin’s proof of this theorem makes essential use of the finiteness of X. Subsequently, Moulin (1988, p. 310) observed that B is also the union of the linear orders used to construct the Pareto extension relation. In other words, for any strongly complete and quasitransitive binary relation B on a finite set X, there exists a set of linear orders on X such that B is simultaneously the union of these linear orders and
*Tel.: 1001-604-822-4121; fax:1001-604-822-5915. E-mail address: weymark@econ.ubc.ca (J.A. Weymark)
1
All undefined terms used in this section are defined in Section 2.
the Pareto extension relation generated by them. In this note, I show that this theorem is valid even when the set of alternatives is not finite.
2. Preliminaries
2
Let X be the set of alternatives. A binary relation B on X is a subset of X 5X3X. It
is often convenient to write xBy instead of (x, y)[B. Given B, a number of derived binary relations may be defined. For all x, y[X, define
xP yB ↔xBy∧ ¬[ yBx],
xI yB ↔xBy∧yBx,
xN yB ↔ ¬[xBy]∧ ¬[ yBx],
xR yB ↔xBy∨xN y,B
¯
xBy↔ ¬[xBy],
21
xB y↔yBx,
xD yB ↔ ¬[ yBx].
These relations are, respectively, the asymmetric part of B, the symmetric part of B, the noncomparable part of B, the completion of B, the complement of B, the inverse of
2
B, and the dual of B. A number of useful properties follow from these definitions,
]21
including: (i) D 5B , (ii) D 5P <N , (iii) D 5B, (iv) D 5R , (v) D 5
B B B B DB PB B B>B9
D <D , and (vi) P 5P . See Duggan (1997).
B B9 RB B
A binary relation B on X is
• reflexive if ;x[X, xBx,
• irreflexive if ;x[X,¬[xBx],
• asymmetric if;x, y[X, xBy→ ¬[ yBx],
• antisymmetric if ;x, y[X, xBy∧yBx→x5y,
• complete if;x, y[X, x±y→xBy∨yBx,
• strongly complete if;x, y[X, xBy∨yBx,
• transitive if ;x, y, z[X, xBy∧yBz→xBz,
• quasitransitive if P is transitive.B
Clearly, (i) B is strongly complete if and only if B is reflexive and complete, (ii) B is
2
Unfortunately, there is no standard terminology for the names of some of these relations. I follow Duggan
21
asymmetric if and only if B is irreflexive and antisymmetric, and (iii) RB is strongly complete. Further, B5P if and only if B is asymmetric. See Duggan (1997). If B isB transitive and irreflexive, then B is asymmetric. See Lemma 7.1 in Fishburn (1973).
The following combinations of properties are considered in the next section. A binary relation B on X is a
• linear order if B is strongly complete, antisymmetric, and transitive,
• strict partial order if B is irreflexive and transitive,
• strict linear order if B is irreflexive, complete, and transitive,
• tournament if B is complete and asymmetric.
For a set@ of binary relations on X, for all x, y[X, define
B@ and B@ are, respectively, the strong Pareto relation and the Pareto extension relation for the set @. It is straightforward to verify that if each B[@ is strongly
complete, then for all x, y[X,
Interpreting the members of @ as individual weak preference relations, (i) x is strictly
preferred to y according to the Pareto extension rule if and only if x strictly Pareto dominates y and (ii) x is indifferent to y according to the Pareto extension rule if and only if either x and y are Pareto indifferent or they are Pareto noncomparable. See Sen (1970) for a detailed discussion of the strong Pareto and Pareto extension relations.
1 2 2 1
strongly complete, then B@ is an extension of B@.
I conclude this section with some further notation. For a set@ of binary relations on
3. The Pareto extension theorem
If the asymmetric factor P of a binary relation B on X is the intersection of the binaryB relations in a set@ of asymmetric binary relations on X, then each of the members of@
is an extension of P . Lemma 1 shows that if B is strongly complete, then B is equal toB
3
the union of the completions of the members of @.
Lemma 1. If B is a strongly complete binary relation on X, then B5 < R for any B9[@ B9
As the following example demonstrates, the assumptions of Lemma 1, while sufficient to ensure that B is the union of the members of 5@, do not imply that the Pareto
extension relation for 5@ is equal to B.
2 1 2 1 2
If the members of @ in Lemma 1 are complete, the symmetric factors of their
completions all equal DX. With this extra structure, the binary relation B is the Pareto extension relation for 5 , not just the union of the members of 5 .
@ @
Lemma 2. If B is a strongly complete binary relation on X and @ is a set of
PE
Equivalently, for all x, y[X,
xBy↔'B9[@ for which [xP y∨xI y].
Duggan (1999) identifies a number of situations in which a binary relation B can be expressed as the union of the members of a class of binary relations derived from B.
4
↔;B9[@, xI x
Szpilrajn (1930) has shown that any strict partial order has a strict linear order extension. As noted by Dushnik and Miller (1941), it follows from Szpilrajn’s Theorem that any strict partial order is the intersection of all the strict linear orders that extend it. The Dushnik–Miller Theorem is used to help prove the following generalization of Moulin’s Pareto extension theorem.
Theorem. If B is a strongly complete and quasitransitive binary relation on X and@ is
the set of all strict linear orders that extend P , then each B9[5 is a linear order on
B @
PE
X and B5 < B9 5B . B9[5@ 5@
Proof. Because B is strongly complete and quasitransitive, P is a strict partial order. ByB the Dushnik–Miller Theorem, P 5> B9. For each B9[@, because B9is a strict
B B9[@
linear order, its completion R is a linear order. Hence, by Lemma 1, B5< R 5
B9 B9[@ B9
PE
< B9. Because a strict linear order is a tournament, by Lemma 2, B5B . h
B9[5@ 5@
For a strongly complete and quasitransitive binary relation B on a finite set, Moulin (1985) also establishes bounds on the minimal number of linear orders needed to express
B as a Pareto relation. As is evident from the proof of the Theorem, this number is the
minimal number of strict linear orders required to obtain P as their intersection. TheB minimal cardinality of a set of strict linear extensions of a strict partial order is called the
dimension of the strict partial order. The dimensionality of strict partial orders is
considered in some detail by Fishburn (1985) and Trotter (1992).
Acknowledgements
I am grateful to the Social Sciences and Humanities Research Council of Canada for ´
References
Duggan, J., 1997. Notes on binary relations, unpublished manuscript, Wallis Institute of Political Economy, University of Rochester.
Duggan, J., 1999. A general extension theorem for binary relations. Journal of Economic Theory, in press. Dushnik, B., Miller, E.W., 1941. Partially ordered sets. American Journal of Mathematics 63, 600–610. Fishburn, P.C., 1973. The Theory of Social Choice, Princeton University Press, Princeton.
Fishburn, P.C., 1985. Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, John Wiley and Sons, New York.
Moulin, H., 1985. Choice functions over a finite set: A summary. Social Choice and Welfare 2, 147–160. Moulin, H., 1988. Axioms of Cooperative Decision Making, Cambridge University Press, Cambridge. Sen, A.K., 1970. Collective Choice and Social Welfare, Holden–Day, San Francisco.
Szpilrajn, E., 1930. Sur l’extension de l’ordre partiel. Fundamenta Mathematicae 16, 386–389.
