Supermodular and ultramodular aggregation evaluators

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1 Supermodular ad ultramodular aggregatio evaluators Marta Cardi 1 ad Maddalea Mazi 2 1 Dept. of Applied Mathematics, Uiversity of Veice, Dorsoduro 3825/E, Veice, Italy mcardi@uive.it 2 Dept. of Pure ad Applied Mathematics, Uiversity of Padua, via Trieste, 63, Padua, Italy mmazi@math.uipd.it Received: July 2008 Accepted: April 2009 Abstract. This paper presets the costructio of a ew kid of evaluators, which are studied o a vector lattice. I particular, by usig a importat idetity o a vector lattice, we prove a characterizatio of supermodular property ad we costruct supermodular evaluators, briefly amed SM-evaluators. The i a particular lattice SM-evaluators become aggregatio fuctios. Similarly we costruct ultramodular evaluators, briefly amed UM-evaluators. Keywords. Evaluators, aggregatio fuctios, supermodularity, lattices, vector lattices. M.S.C. classificatio. 90B50, 91B82, 60A10, 60E15. J.E.L. classificatio. C02. 1 Itroductio Normalized scalar evaluators were characterized i [9] as real fuctios which obey oly two properties: boudary requiremets ad mootoicity. Evaluators o a complete lattice have bee studied i [3] with a ivestigatio about T L ad S L evaluators. Our purpose i this work is to study supermodular property for ormalized scalar evaluators o a complete vector lattice, which is ot ecessarily of fiite dimesio. I fact o vector lattices we have some simple but importat results ad i particular we study the complete vector lattice [0, 1] for studyig aggregatio of supermodular evaluators. First of all we recall some cocepts ad results largely discussed i [22]. The we study ormalized scalar evaluators from complete vector lattices ad we recall some defiitios give i [4]. A particular complete vector lattice is [0, 1] ad it is iterestig for our aalysis to study aggregatio operators, their compositios ad relatioships with evaluators ad the particular case of supermodular aggregatio evaluators.

2 2 Marta Cardi ad Maddalea Mazi 2 Lattices ad Vector Lattices A partially ordered set (poset) is a set X o which there is a biary relatio that is reflexive, atisymmetric ad trasitive. The set R = {x = (x 1,..., x ) : x i R 1 for i = 1,..., } with the orderig relatio where x x i R if x i x i i R 1 for i = 1,..., is a partially ordered set. If two elemets, x ad x, of a partially ordered set X have a supremum (ifimum) i X, it is their joi (meet) ad is deoted x x (x x ). A partially ordered set that cotais the joi ad the meet of each pair of its elemets is a lattice. For ay positive iteger, R is a lattice with x x = (x 1 x 1,..., x x ) ad x x = (x 1 x 1,..., x x ) for x ad x i R. If X is a subset of a lattice X ad X cotais the joi ad meet (with respect to X) of each pair of elemets of X, the X is a sublattice of X. For example the closed itervals of a lattice are sublattices. Moreover, a lattice i which each oempty subset has a supremum ad a ifimum is complete. If X is a sublattice of a lattice X ad if, for each oempty subset X of X, sup X (X ) ad if X (X ) exist ad are i X, the X is a subcomplete sublattice of X. A sublattice of R is subcomplete if ad oly if it is compact (Theorem i [22]). Moreover a vector lattice is a ordered vector space whose uderlyig poset is a lattice. We use the term Riesz space to mea a real vector lattice, i.e., a vector lattice over R. It is well kow that spaces R of all vectors [x 1,..., x ] with real compoets, R d of all ifiite sequeces x 1, x 2, x 3,... of real umbers, ad R c of all real fuctios f(x) defied o the iterval 0 x 1 are complete vector lattices. It is also clear that ay subspace of a vector lattice, which is a sublattice as well, is a vector lattice. That is, if a subset cotais with ay f ad g also f +g, f g, f g ad every λf, the it is a vector lattice relative to the same operatios. We recall also that i a geeral vector lattice L V we have the followig idetity (see page 207 i [19]), x, y L V x + y = x y + x y. (1) 3 Supermodularity ad ultramodularity o vector lattices Now we cosider a vector lattice L V ad a vector sublattice X L V. We recall also the followig defiitio of supermodularity. Defiitio 1. A fuctio f : X R is said supermodular if for every x, y X f(x y) + f(x y) f(x) + f(y). The cocept of supermodularity is strictly coected to the cocept of icreasig differeces, so that we give the followig defiitio ad result.

3 Supermodular ad ultramodular aggregatio evaluators 3 Defiitio 2. A fuctio f : X R is said to have icreasig differeces o X iff f(x + h + h ) f(x + h) f(x + h ) f(x), for all x X ad all h, h X + with h h, where X + = {x X : x 0} is the positive coe of X. Theorem 1. Let X L V ad f : X R. The f is supermodular o X if ad oly if f has icreasig differeces o X. Proof. ( ) Pick x ad y X ad set h = x x y, h = y x y. The h h = (x x y) (y x y) = (x y) (x y) = 0, i.e. h h. Sice x + y = x y + x y, for all x, y L V, replacig x by h + x y ad y by h + x y we get x y = x y + h + h. Sice f(a + h + h ) f(a + h) f(a+h ) f(a) for h ad h X + as chose ad every a L V, takig a = x y we get supermodularity. ( ) Pick a L V ad h, h X + such that h h. Let x = a + h ad y = a + h. Clearly, x y = a + h h = a. Hece, x y = a + h h = a + h + h h h = a + h + h. Usig the defiitio of supermodularity we get the property of icreasig differeces. Now we cosider the ultramodular (or directioally covex) fuctios, a class of fuctios that geeralizes scalar covexity ad that aturally arises i some ecoomic ad statistical applicatios (see [13] ad [20]). Defiitio 3. A fuctio f : X R is said to be ultramodular iff for all x X with h, k X +. f(x + h + k) f(x + k) f(x + h) f(x) Propositio 1. A fuctio f : X R is ultramodular if ad oly if f(x 3 ) f(x 1 ) f(x 4 ) f(x 2 ) for all collectios {x 1, x 2, x 3, x 4 } of vectors i X such that x 1 x 2 x 4 ad x 1 + x 4 = x 2 + x 3. Proof. By defiitio ad replacig x 4 by x + h + k, x 3 or x 2 by x + h or x + k, ad x 1 by x, where h, k X +, we immediately get the result. 4 Supermodular ad ultramodular evaluators As i [4] we cosider a complete sublattice (X,,, ), which is i our case a vector sublattice too with the least ad the greatest elemets ad, respectively ad we will focus o the evaluatio of elemets from L V by real umbers i the uit iterval. Defiitio 4. A fuctio φ : X [0, 1] is said to be a evaluator o X iff it satisfies the followig properties:

4 4 Marta Cardi ad Maddalea Mazi φ( ) = 0, φ( ) = 1, φ(x) φ(y) for all x, y X such that x y. A evaluator φ is called existetial if for arbitrary x X, φ(x) = 0 x =. A evaluator φ is called uiversal if for arbitrary x X, φ(x) = 1 x =. The stadard compariso of real umbers allows us to compare evaluators o the same system of objects. I this case we utilize the usual poitwise orderig of fuctios. This meas that if φ 1 ad φ 2 are two evaluators o X, we say that φ 1 is greater tha φ 2, with otatio φ 2 φ 1 if for all x X, φ 2 (x) φ 1 (x). The greatest evaluator is the existetial evaluator φ E defied for all x X by { 0 if x =, φ E (x) = 1 otherwise. The smallest evaluator is the uiversal evaluator φ U defied for all x X by { 1 if x =, φ U (x) = 0 otherwise. Now we itroduce the modular, supermodular ad ultramodular evaluators o the geeral complete vector sublattice X L V. Defiitio 5. A operatio SM : X [0, 1] is said to be a supermodular evaluator iff it is a evaluator ad it satisfies the followig property: SM(x y) + SM(x y) SM(x) + SM(y). I the case of equality i the above equatio, we have the modular evaluator. Defiitio 6. A operatio U : X [0, 1] is said to be a ultramodular evaluator (UM evaluator for short) iff U is a evaluator satisfyig the property: U(x 1 ) + U(x 4 ) U(x 2 ) + U(x 3 ) for all collectios {x 1, x 2, x 3, x 4 } of vectors i X such that x 1 x 2 x 4 ad x 1 + x 4 = x 2 + x 3. For a detailed study of the properties of supermodular ad ultramodular fuctios we refer to [13], [14], [20], [21] ad [22] as well as the refereces therei. Oe of the most importat cosequece is that the t-orms are SM evaluators, while the t-coorms are ot. It is kow that aggregatio of evaluators yields a evaluator (see Propositio 1 i [4]). Now we cotiue the discussio about aggregatio of several kids of evaluators ad we will focus o aggregatio of supermodular evaluators. We

5 Supermodular ad ultramodular aggregatio evaluators 5 would like to kow which aggregatio of supermodular evaluators yields a supermodular evaluator. Let K 1,..., K m : X [0, 1], i = 1, 2,..., m be SM evaluators. A vector fuctio K : X [0, 1] m give by K(x) = (K 1 (x),..., K m (x)) is said to be a SM evaluator. Propositio 2. If ψ : [0, 1] m [0, 1] is a icreasig UM evaluator ad K: X [0, 1] m is a icreasig SM evaluator, the the fuctio H : X [0, 1]give byh(x 1,..., x ) = ψ(k)(x) = ψ(k 1 (x),..., K m (x)) is a SM evaluator. Proof. We cosider 3 vectors x, h, k such that h, k 0 ad h k. For all i = 1,..., m, K i (x + h + k) K i (x + k) K i (x + h) K i (x) ad the there exist s i, t i with s i t i 0 such that K i (x + h + k) = K i (x + k) + s K i (x + h) = K i (x) + t i. So there exist s, t vectors i R m such that s t 0 ad K(x + h + k) = K(x + k) + s K(x + h) = K(x) + t. Sice ψ is a icreasig UM evaluator ad K is icreasig i each variable oe has: ψ(k)(x + h + k) ψ(k)(x + k) = ψ(k(x + k)) + s) ψ(k(x + k)) ψ(k(x + k) + t) ψ(k(x + k)) ψ(k(x) + t)) ψ(k(x)) = ψ(k(x + h)) ψ(k(x)) = ψ(k)(x + h) ψ(k)(x). Corollary 1. Let A be a SM evaluator. If ϕ : [0, 1] [0, 1] is a cotiuous icreasig ad covex fuctio with ϕ(0) = 0 ad ϕ(1) = 1 the the fuctio is a SM evaluator. A ϕ (x) := ϕ (A(x 1,..., x )) Proof. It is obvious that A ϕ ( ) = 0 ad A ϕ ( ) = 1. The, it suffices to apply the above theorem to the fuctio H(x 1,..., x ) = ψ(k 1 (x)), with ψ = ϕ ad K 1 = A. I fact scalar covex fuctios are ultramodular ad so ψ is icreasig ad ultramodular.

6 6 Marta Cardi ad Maddalea Mazi 5 The [0, 1] case A particular ad importat kid of evaluator is the aggregatio fuctio. Let N, 2. A ary aggregatio fuctio is a mappig A from N [0, 1] ito [0, 1] that satisfies the followig properties: (A1) A(0,..., 0) = 0 ad A(1,..., 1) = 1; (A2) A is icreasig i each compoet. Each aggregatio fuctio A ca be caoically represeted by a family (A () ) N of -ary operatios, e.g., fuctios A () : [0, 1] [0, 1], give by A () (x 1,..., x ) = A(x 1,..., x ). Each fuctio A () is a evaluator o the complete vector lattice ([0, 1],,, ). If A(x 1,..., x ) = 0 implies that x i = 0 for i = 1,...,, we say that aggregatio operator A does ot have zero divisors. I this case, fuctio A () is a existetioal evaluator ad A is a existetioal aggregator. If A(x 1,..., x ) = 1 implies that x i = 1 for i = 1,...,, fuctio A () is a uiversal evaluator ad A is a uiversal aggregator. Now we study particular families of SM evaluators o the complete vector lattice ([0, 1],,, ) used for the aggregatio of evaluated elemets from [0, 1]. Propositio 3. The quasi arithmetic mea M f (x) := f 1 ( f(x1 ) + + f(x ) where f : [0, 1] R is a cotiuous strictly mootoe fuctio, is a SM evaluator if ad oly if f 1 is covex. Proof. M f is a SM aggregatio evaluator if, ad oly if, for ay couple of itegers α, β, such that 1 α < β, M f (x α, x β ) = M f (a 1,..., a α 1, x α, a α+1,..., a β 1, x β, a β+1,..., a ) is a SM evaluator, if ad oly if f 1 is covex. Due to wellkow characterizatio of quasi-arithmetic meas M f bouded from above by the arithmetic mea M (see lemma 1 i [5]) we have the ext result. Corollary 2. M f is a SM aggregatio evaluator if ad oly if M f M. A similar result holds for weighted quasi-arithmetic meas, that is with - dimesioal weightig vector w = (w 1,..., w ) [0, 1] such that ( i=1 w i = 1 ad W f (x) := f 1 P ) i=1 w if(x i ) we have: Corollary 3. W f is a SM aggregatio evaluator if ad oly if W f W. ),

7 Supermodular ad ultramodular aggregatio evaluators 7 Proof. From covexity of f 1 ad Jese s iequality we have: ( W f (x) = f 1 i=1 w ) ( if(x i ) i=1 w if 1 ) (f(x i )) = i=1 = w ix i = W (x). Propositio 4. Let f i : [0, 1] [0, 1] be icreasig fuctios such that f i (0) = = 0 ad f i (1) = 1, i = 1, 2...,. If A is a SM evaluator, the the fuctio defied by A f1,...,f (x 1,..., x ) := A(f 1 (x 1 ),..., f (x )) is a SM aggregatio evaluator. Proof. It is obvious that A f1,...,f (0,..., 0) = 0, A f1,...,f (1,..., 1) = 1 ad A f1,...,f is icreasig i each place. Moreover, give x j 1 xj 2, j = 1,...,, oe obtais A f1,...,f (x 1 1, x 2 1,..., x 1 ) + A f1,...,f (x 1 2, x 2 2,..., x 2 ) A f1,...,f (x 1 2,..., x h 2, x h+1 1,..., x 1 ) + A f1,...,f (x 1 1,..., x h 1, x h+1 2,..., x 2 ), because A is a SM evaluator ad f i are icreasig. Corollary 4. Let A be a SM aggregatio evaluator ad ϕ : [0, 1] [0, 1] be a cotiuous strictly mootoe fuctio with ϕ(0) = 0 ad ϕ(1) = 1. The the followig statemets are equivalet: (a) ϕ is cocave; (b) the fuctio A ϕ (x) := ϕ 1 (A(ϕ(x 1 ),..., ϕ(x ))) is a SM aggregatio evaluator. Proof. (a) (b) If ϕ is cocave ad positive, the ϕ 1 is covex. By Corollary 1 ad Propositio 4 we have the result. (b) (a) If A ϕ (x) is a SM aggregatio evaluator, we ca cosider ( ) M ϕ (x) := ϕ 1 ϕ(x1 ) + + ϕ(x ). By Propositio 3, M ϕ (x) a SM aggregatio evaluator if ad oly if ϕ 1 is covex. So, ϕ is cocave. A special subclass of SM evaluators is that formed by modular aggregatio fuctios, i.e. those A s for which A(x y) + A(x y) = A(x) + A(y), for all x, y [0, 1]. For these operators the followig characterizatio holds.

8 8 Marta Cardi ad Maddalea Mazi Propositio 5. For a aggregatio operator A the followig statemets are equivalet: (a) A is modular; (b) there exist icreasig fuctios f i : [0, 1] [0, 1] such that f i (0) = 0, i = 1,...,, i=1 f i(1) = 1, ad A(x 1,..., x ) = f i (x i ); (2) (c) A is strogly additive, i.e., if x y = 0 ad x + y [0, 1], the A(x + y) = A(x) + A(y). Proof. (a) (b) If A is modular, set f i (x i ) := A(0,..., x i,..., 0), i = 1,...,. From modularity of A we get ad also, i=1 A(x) + A(0) = A(x 1, 0..., 0) + A(0, x 2,..., x ) = = f 1 (x 1 ) + A(0, x 2,..., x ), A(0, x 2,..., x ) + A(0) = A(0, x 2, 0..., 0) + A(0, 0, x 3..., x ) = = f 2 (x 2 ) + A(0, 0, x 3,..., x ), which implies (b) recursively. (b) (c) A(x + y) = A(x 1 + y 1,..., x + y ) = i=1 f i(x i + y i ). Sice x y = 0, the i=1 f i(x i + y i ) = i=1 (f i(x i ) + f i (y i )), i.e. A(x + y) = A(x) + A(y). (c) (a) By (1) x + y = x y + x y. So, A(x y) + A(x y) = A(x + y) = A(x) + A(y). 6 Coclusio Aggregatio of evaluators by a aggregatio operator yields a evaluator [4]. I this work we have show that aggregatio of supermodular evaluators yields a supermodular evaluator ad i several cases a supermodular aggregatio evaluator. I particular we have aalyzed supermodularity property for quasi-arithmetic meas, by usig appropriate fuctios o [0, 1]. So we have studied this kid of trasformatio i geeral to preset some iterestig applicatios i our future work. Refereces 1. Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloq. Publ. XXV, Amer. Math. Soc., New York City (1948)

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