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Axioms 2018, 7(2), 35; doi:10.3390/axioms7020035
Quantiles in Abstract Convex Structures
Department of Economics, Ca’Foscari University of Venice, Sestiere Cannaregio 873, 30123 Venezia, Italy
Received: 14 May 2018 / Accepted: 25 May 2018 / Published: 28 May 2018
In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex spaces to formalize the notion of quantiles. We also put in evidence that many important models of decision problems can be viewed as convex spaces-based models. Several properties of aggregation operators are translated into this general setting, and independence and invariance are used to provide axiomatic characterizations of quantiles.
Keywords:convex space; aggregation operator; invariance; independence; quantile
The aim of this paper is to propose a general unified framework for defining aggregation operators. Our framework is abstract and algebraic in nature and in this framework we generalize some results of the literature [1,2,3,4].
We consider convex structures where the notion considered here (see ) is not restricted to the context of vector spaces. The basic idea of our approach is to describe the space of alternatives in terms of a “topological” relation. We can prove that lattices, median spaces and interval spaces are convex spaces and also that to every property spaces (see [1,2,4]) is associated with a convex structure.
We then focus on aggregation operators where X is convexity space and A is a nonempty set. We study aggregation operators that satisfy properties of monotonicity and independence and we consider aggregation operators that are based on decisive subsets of A. Moreover, we consider operators that are componentwise compatible with the structure of convexity space of X.
We propose also a particular version of Arrow’s theorem thus considering a link between aggregation theory and social choice theory as in . It appears that there are many connections between the work presented here with the results of [3,4,7,8,9,10,11,12]. Applications of these types of results can be found in in [4,13,14].
The structure of the paper is as follows. In Section 2 we introduce convex spaces and we provide the necessary definitions. Section 3 is devoted to describe some important examples. Finally in Section 3 and Section 4 we study some classes of agggregation operators acting on abstract convex structures.
2. Abstract Convex Structures
The notion of convexity is a basic mathematical structure that is used to analyze many different problems and there are in the literature various kinds of generalized, topological, or axiomatically defined convexities. There are generalizations that are motivated by concrete problems and those that are stated from an axiomatic point of view, where the notion of abstract convexity is based on properties of a family of sets.
In this paper the general notion of abstract convexity structure that is studied in  is considered.
A family of subsets of a set X is a convexity on a set X if ∅ and X belong to and is closed under arbitrary intersections and closed under unions of chains.
The elements of are called convex sets of X and the pair is called a convex space.
Moreover, the convexity notion allows us to define the notion of the convex hull operator, which is similar to that of the closure operator in topology.
If X is a set with a convexity and A is a subset of X, then the convex hull of is the set
This operator enjoys certain properties that are identical to those of usual convexity: for instance is the smallest convex set that contains set A. It is also clear that C is convex if and only if .
The convex hull of a set is called an n-polytope and is denoted by A 2-polytope is called the segment joining .
A convexity C is called N-ary if whenever for all where F has at most N elements. A 2-ary convexity is called an interval convexity.
We also consider biconvex spaces, i.e., triples of the form where are two convexities on a set X, called the lower and the upper convexity. Obviously every convex space can be viewed as a biconvex space .
If are convex spaces with convexities , respectively, we consider the following definition of a compatible map between two convex spaces.
A map is convex if for every and such that when for
For a general theory of convexity we refer to .
3. Some Examples
We present some examples and classes of convex spaces. First of all we note that every real vector space together with the collection of all convex sets in the usual meaning, is a 2-arity convex space.
Ordered spaces The usual convexity on can be defined in terms of ordering as follows: a set C is convex if and only if when and implies . We can define in the same way a convexity on a partially ordered set (see , p. 6). Such a convexity is called the order convexity.
Lattices If is a lattice we denote by and the collections of all ideals and all filters respectively (the empty set and the whole lattice are treated as (non-proper) ideals and filters). Since the union of a chain of filters (ideals) is a filter (ideal), these are two convexities on L that will be called the lower and the upper lattice convexity respectively. Moreover there exists a convexity generated by the least convexity containing all ideals and filters. This convexity will be called the lattice convexity on L.
Please note that if L is linearly ordered then G equals the order convexity. The convexity of the dual lattice is the same as the original one.
It is possible to consider lattices as convex spaces (with the lattice convexity) as well as bi-convex spaces (with the lower and upper lattice convexities). It is easy to check that a proper halfspace is either a prime filter or a prime ideal. It can be proved also that the lattice convexity is an interval convexity and that
Median spaces A median space is a convexity space X with an interval convexity such that for each there exists a unique point in . We call it the median of and denote by . This defines a map , called the median operator on X. In any convexity space, every point in is called a median of . There is a natural way to define the structure of a median space by means of the median operator (see ).
Property-based domains A property-based domain (as defined in ) is a pair where X is a non-empty set and is a collection of non-empty subsets of X and if and there exists such that and . The elements of are referred to as properties and if we say that x has property represented by the subset H. This definition is slightly more general than that of  and of , in fact it is not assumed that the set X is finite and we do not consider that the set is a property if H is a property.
The “property space” model provides a very general framework for representing preferences and then aggregation of preferences. In every property-based domain we can define a convexity defined as follows. A subset is said to be convex if it is intersection of properties.
Arrowian framework The problem of preference aggregation can be viewed as a property-based domain and then as a convex space. We consider a set of alternatives A and a set of binary relations in A. We can consider different requirements on the set and so can be the set of preorders or the set of linear orders in A.
If we define for each pair the setthe family defines a property-based domain structure on the set . See  for more details on Arrowian framework.
4. Aggregation Functional over Convex Spaces
Aggregation operators are mathematical functions that are used to combine several inputs into a single representative outcome; see  for a comprehensive overview on aggregation theory. Aggregation operators play an important role in several fields such as decision sciences, computer and information sciences, economics and social sciences and there are a large number of different aggregation operators that differ on the assumptions on the inputs and about the information that we want to consider in the model.
If N is an arbitrary nonempty set and X is a convex space, then an aggregation functional is a map .
Our framework is very general, we do not assume that the sets X and A are finite or that the map is surjective. Moreover, we consider the case in which there are more than one equivalent solutions and also the case in which there are no solutions. For each , we denote by the constant c map in .
The following properties of an aggregation functional are key to our analysis.
- Monotonicity If and then where and if .
- Idempotence for every .
- Independence If and for all , if and only if we have that .
- Invariance For every convex map ,
5. Quantiles in Convex Spaces
We briefly consider aggregation functionals based on a complete lattices. As it is well known the quantile is a generalization of the concept of median and it plays an important role in statistical and economic literature. We study quantile in an ordinal framework and we consider an axiomatic representation of quantiles as in [1,7,8]. Here we provide a definition and characterization of quantiles for lattice-valued operators.
If A is a nonempty set and L a bounded lattice a non-additive measure on A with values in L is a function such that , and whenever .
If α is an element of L, then the lattice-valued quantile of level α is the functional defined by
It can be proved that this definition extends the well known definition of quantile for real-valued functions (see ).
We recall the definition of completely distributive lattice. A complete lattice L is said to be a completely distributive is the following distributive law holdsfor every doubly indexed subset of L. Please note that every complete chain (in particular, the extended real line and each product of complete chains) is completely distributive. Moreover, complete distributivity reduces to distributivity in the case of finite lattices.
A collection of sets is said to be an upper set in A if and implies that . Then we can prove the following results.
Let L be a completely distributive lattice. An aggregation functional is a lattice- valued quantile with respect to a non-additive measure if and only if there exists a upper set such thator if and only if there exists a upper set such that
Proof of Proposition 1
By Proposition 1 in  if L is a completely distributive lattice an aggregation functional is a lattice-valued quantile with respect to a non-additive measure if and only if there exists a upper set such thator if and only if there exists a upper set such that
Then we can prove that F is a lattice-valued quantile when if and only if there exists such that for every . Then we get
The second statement follows similarly. ☐
Since we know that the lattice convexity is an interval convexity we can prove the following characterization of lattice-valued quantiles.
If L is a completely distributive lattice, then an aggregation functional is a lattice- valued quantile with respect to a non-additive measure if and only if there exists an upper set such that
Then the elements in belong to a convex set C if and only if belongs to C for a “decisive ” or a “large enough”set.
Let us define quantiles in an abstract convex structures.
If N is an arbitrary nonempty set and X is a convex space, then a quantile is an aggregation functional defined bywhere is an upper set in A.
Furthermore, we can characterize from an axiomatic point of view quantiles in an abstract convex structure.
If N is an arbitrary nonempty set and X is a convex space, then a quantile is a monotone, idempotent and independent aggregation functional. Conversely an aggregation functional that is monotone and independent is a quantile.
Proof of Proposition 3
If C is a convex set in and f is an element of we define the set . Let F be a quantile, , f an element of such that and . If we define an element g of by and if then and so . Moreover, if we have that if F is a quantile since .
By the definition of quantile if F is a quantile, , and for all , if and only if we can easily prove that and we get . So we have proved that quantiles are monotone , idempotent and independent functionals.
We note that functional F is monotone and independent if and only if and for all , if then we have that .
We say that a set is decisive with respect to an element if there exists such that and . Being F monotone and independent a set U is decisive with respect to C if and only if for every such that , .
Since the functional F is monotone and independent then the set of decisive subset of N does not depend on the convex set C. If is the family of decisive subsets of N for every , if and only if . So we have proved that
The following proposition presents another property of quantiles in convex spaces.
f N is an arbitrary nonempty set and X is a convex space, then a quantile is an invariant aggregation functional.
Proof of Proposition 4
Let be a quantile and a convex map.
If being continuousand we get that . ☐
6. Concluding Remarks
We introduced a unified qualitative framework for studying aggregation operators. The approach presented in this paper has taken its inspiration from social choice theory and we generalize some results in social choice in certain respects. This setting has several appealing aspects, for it provides sufficiently rich structures studied in the literature , which allow the definition of quantiles from an ordinal point of view, and which do not depend on the usual arithmetical structure of the reals.
There are however many opportunities for much more detailed research in this area in particular from the point of view of aggregation theory. An obvious topic for future research is to analyze other aggregation functionals defined in convex spaces. There are several extensions avaiable within this framework, for instance, one could consider Sugeno type integral defined by a class of decisive sets.
Conflicts of Interest
The author declares no conflict of interest.
- Cardin, M. Sugeno Integral on Property-Based Preference Domains. In Advances in Fuzzy Logic and Technology 2017; Advances in Intelligent Systems and Computing Series; Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M., Eds.; Springer: Basel, Switzerland, 2017; Volume 641. [Google Scholar]
- Cardin, M. Aggregation over Property-Based Preference Domains. In Aggregation Functions in Theory and in Practice; Advances in Intelligent Systems and Computing Series; Torra, V., Mesiar, R., Baets, B., Eds.; Springer: Basel, Switzerland, 2017; Volume 581. [Google Scholar]
- Gordon, S. Unanimity in attribute-based preference domains. Soc. Choice Welf. 2015, 44, 13–29. [Google Scholar] [CrossRef]
- Nehring, K.; Puppe, C. Abstract Arrowian aggregation. J. Econ. Theory 2010, 145, 467–494. [Google Scholar] [CrossRef]
- Van de Vel, M.L.J. Theory of Convex Structures; North-Holland Mathematical Library Series; Elsevier: Amsterdam, The Netherlands, 1993; Volume 50. [Google Scholar]
- Candeal, J.C. An Abstract Result on Projective Aggregation Functions. Axioms 2018, 7, 17. [Google Scholar] [CrossRef]
- Cardin, M. A quantile approach to integration with respect to non-additive measures. In Proceedings of the International Conference on Modeling Decisions for Artificial Intelligence, Catalonia, Spain, 21–23 November 2012; Torra, V., Narukawa, Y., Lopez, B., Villaret, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2012; pp. 139–148. [Google Scholar]
- Cardin, M.; Couceiro, M. An ordinal approach to risk measurement. In Mathematical and Statistical Methods for Actuarial Science and Finance; Perna, C., Sibillo, M., Eds.; Springer: Milano, Italy, 2012; pp. 79–86. [Google Scholar]
- Chambers, C. Ordinal Aggregation and Quantiles. J. Econ. Theory 2007, 137, 416–443. [Google Scholar] [CrossRef]
- Halaš, R.; Mesiar, R.; Pócs, J. A new characterization of the discrete Sugeno integral. Inf. Fusion 2016, 29, 84–86. [Google Scholar] [CrossRef]
- Halaš, R.; Mesiar, R.; Pócs, J. Congruences and the discrete Sugeno integrals on bounded distributive lattices. Inf. Sci. 2016, 367, 443–448. [Google Scholar] [CrossRef]
- Leclerc, B.; Monjardet, B. Aggregation and Residuation. Order 2013, 30, 261–268. [Google Scholar] [CrossRef]
- Daniëls, T.; Pacuit, E. A General Approach to Aggregation Problems. J. Logic Comput. 2009, 19, 517–536. [Google Scholar] [CrossRef]
- Monjardet, B. Arrowian characterization of latticial federation consensus functions. Math. Soc. Sci. 1990, 20, 51–71. [Google Scholar] [CrossRef]
- Grabisch, M.; Marichal, J.L.; Mesiar, R.; Pap, E. Aggregation Functions Encyclopedia of Mathematics and its Applications; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
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