O a axiomatizatio of the quasi-arithmetic mea values without the symmetry axiom Jea-Luc Marichal Departmet of Maagemet, FEGSS, Uiversity of Liège Boulevard du Rectorat 7 - B31, B-4000 Liège, Belgium Email: jl.marichal@ulg.ac.be Revised versio Abstract Kolmogoroff ad Nagumo proved that the quasi-arithmetic meas correspod exactly to the decomposable sequeces of cotiuous, symmetric, strictly icreasig i each variable ad reflexive fuctios. We replace decomposability ad symmetry i this characterizatio by a geeralizatio of the decomposability. AMS (1991) subject classificatio: Primary 39B12, 39B22, Secodary 26B35. 1 Itroductio A cosiderable amout of literature about the cocept of mea (or average) ad the properties of several meas (such as the media, the arithmetic mea, the geometric mea, the power mea, the harmoic mea, etc.) has bee already produced i the 19th cetury ad has ofte treated the sigificace ad the iterpretatio of these specific aggregatio fuctios. Cauchy [5] cosidered i 1821 the mea of idepedet variables x 1,..., x as a fuctio M(x 1,..., x ) which should be iteral to the set of x i values: mi{x 1,..., x } M(x 1,..., x ) max{x 1,..., x }. (1) The cocept of mea as a average is usually ascribed to Chisii [6], who gives i 1929 the followig defiitio (p. 108): Let y = g(x 1,..., x ) be a fuctio of idepedet variables x 1,..., x represetig homogeeous quatities. A mea of x 1,..., x with respect to the fuctio g is a umber M such that, if each of x 1,..., x is replaced by M, the fuctio value is uchaged, that is, g(m,..., M) = g(x 1,..., x ). 1
Whe g is cosidered as the sum, the product, the sum of squares, the sum of iverses, the sum of expoetials, or is proportioal to [( i x 2 i )/( i x i )] 1/2 as for the duratio of oscillatios of a composed pedulum of elemets of same weights, the solutio of Chisii s equatio correspods respectively to the arithmetic mea, the geometric mea, the quadratic mea, the harmoic mea, the expoetial mea ad the atiharmoic mea, which is defied as ( ) ( M(x 1,..., x ) = x 2 i / x i ). i i Ufortuately, as oted by de Fietti [7, p. 378] i 1931, Chisii s defiitio is so geeral that it does ot eve imply that the mea (provided there exists a real ad uique solutio to the above equatio) fulfils the Cauchy s iterality property. The followig quote from Ricci [13, p. 39] could be cosidered as aother possible criticism to Chisii s view:... whe all values become equal, the mea equals ay of them too. The iverse propositio is ot true. If a fuctio of several variables takes their commo value whe all variables coicide, this is ot sufficiet evidece for callig it a mea. For example, the fuctio g(x 1, x 2,..., x ) = x + (x x 1 ) + (x x 2 ) + + (x x 1 ) equals x whe x 1 = = x, but it is eve greater tha x as log as x is greater tha every other variable. I 1930, Kolmogoroff [9] ad Nagumo [12] cosidered that the mea should be more tha just a Cauchy mea or a average i the sese of Chisii. They defied a mea value to be a ifiite sequece of cotiuous, symmetric ad strictly icreasig (i each variable) real fuctios M 1 (x 1 ) = x 1, M 2 (x 1, x 2 ),..., M (x 1,..., x ),... satisfyig the reflexive law: M (x,..., x) = x for all ad all x, ad a certai kid of associative law: M k (x 1,..., x k ) = x M (x 1,..., x k, x k+1,..., x ) = M (x,..., x, x k+1,..., x ) (2) for every atural iteger k. They proved, idepedetly of each other, that these coditios are ecessary ad sufficiet for the quasi-arithmeticity of the mea, that is, for the existece of a cotiuous strictly mootoic fuctio f such that M may be writte i the form M (x 1,..., x ) = f 1[ 1 f(x i ) ] (3) for all IN 0 (IN 0 deotes the set of strictly positive itegers). The quasi-arithmetic meas (3) comprise most of the algebraic meas of commo use, ad allow oe to specify f i relatio to operatioal coditioig, see Table 1. Some meas however do ot belog to this family: de Fietti [7, p. 380] observed that the atiharmoic mea is ot icreasig i each variable ad that the media is ot associative i the sese of (2). i=1 2
f(x) M (x 1,..., x ) ame x 1 xi arithmetic x 2 1 x 2 i quadratic log x xi geometric x 1 1 1 1 x i harmoic x α (α IR 0 ) e α x (α IR 0 ) ( 1 x αi ) 1 α [ 1 1 α l ] e α x i power expoetial Table 1: Examples of quasi-arithmetic meas The above properties defiig a mea value seem to be atural eough 1. For istace, oe ca readily see that, for icreasig meas, the reflexivity property is equivalet to Cauchy s iterality (1), ad both are accepted by all statisticias as requisites for meas. Associativity of meas (2) has bee itroduced first i 1926 by Bemporad [4, p. 87] i a characterizatio of the arithmetic mea. Uder reflexivity, this coditio seems more atural, for it becomes equivalet to M k (x 1,..., x k ) = M k (x 1,..., x k) M (x 1,..., x k, x k+1,..., x ) = M (x 1,..., x k, x k+1,..., x ) which says that the mea does ot chage whe alterig some values without modifyig their partial mea. More recetly, Marichal ad Roubes [11] proposed to call this property decomposability i order ot to cofuse it with the classical associativity property. Observe however that this cocept has bee defied for symmetric meas. Whe symmetry is ot assumed, it is ecessary to rewrite the decomposability property i such a way that the first variables are ot privileged. Marichal et al. [10] proposed the followig geeral 1 Note that Fodor ad Marichal [8] geeralized the Kolmogoroff-Nagumo s theorem above by relaxig the coditio that the meas be strictly icreasig, requirig oly that they be icreasig. The family obtaied, which has a rather itricate structure, aturally icludes the mi ad max operatios. 3
form, called strog decomposability : M k (x i1,..., x ik ) = x M ( i K x i e i + i/ K x i e i ) = M ( i K x e i + i/ K x i e i ) for every subset K = {i 1,..., i k } {1,..., } with i 1 < < i k (e i represets the vector of {0, 1} i which oly the i-th compoet is 1). Of course, uder symmetry, decomposability ad strog decomposability are equivalet. The aim of this paper is to show that symmetry is ot ecessary i the Kolmogoroff- Nagumo s characterizatio, provided that decomposability is replaced by strog decomposability 2. Thus we show that ay strogly decomposable sequece (M ) IN0 of cotiuous, strictly icreasig ad reflexive fuctios is a quasi-arithmetic mea value (3). 2 The result We first show that, for ay strogly decomposable sequece (M ) IN0 the two-place fuctio M 2 fulfils the bisymmetry fuctioal equatio 3 of reflexive fuctios, M 2 (M 2 (x 1, x 2 ), M 2 (x 3, x 4 )) = M 2 (M 2 (x 1, x 3 ), M 2 (x 2, x 4 )). (4) To do this, we eed some itermediate results. Let us first itroduce the followig practical otatio: for all k IN 0, we defie k x := x,..., x (k times). For istace, we have M 5 (2 x, 3 y) = M 5 (x, x, y, y, y). Now, we preset a techical lemma which is adapted from Nagumo [12, 1]. It will be very useful as we cotiue. Lemma 2.1 Cosider a strogly decomposable sequece (M ) IN0 The we have, for all k, IN 0 with 2, of reflexive fuctios. M k. (k x 1,..., k x ) = M (x 1,..., x ), (5) M k. (x 11,..., x 1k ;... ; x 1,..., x k ) = M (M k (x 11,..., x 1k );... ; M k (x 1,..., x k )), (6) M (x 1,..., x ) = M (x,..., x 1), where x j = M 1 (x 1,..., x j 1, x j+1,..., x ). (7) Proof. Let us fix k, IN 0 with 2. We the have M k. (k x 1,..., k x ) = M k. ((k.) M (x 1,..., x )) (strog decomposability) = M (x 1,..., x ) (reflexivity) 2 Oe could thik that strog decomposability implies symmetry. Nevertheless, the sequece of osymmetric meas defied for all IN 0 by M (x 1,..., x ) = x 1 is strogly decomposable. 3 The bisymmetry property (also called mediality) is very easy to hadle ad has bee ivestigated from the algebraic poit of view by usig it mostly i structures without the property of associativity i a certai respect, it has bee used as a substitute for associativity ad also for symmetry. For a list of refereces see [2, 6.4] (see also [3, Chap. 17]). 4
which proves (5). Next, we have M k. (x 11,..., x 1k ;... ; x 1,..., x k ) = M k. (k M k (x 11,..., x 1k );... ; k M k (x 1,..., x k )) (strog decomposability) = M (M k (x 11,..., x 1k ),..., M k (x 1,..., x k )) (by (5)) which proves (6). Fially, by (5), we have M (x 1,..., x ) = M ( 1) (( 1) x 1,..., ( 1) x ) ad by usig strog decomposability with subset K j = {j, + j, 2 + j,..., ( 2) + j} for j = 1,...,, we obtai M ( 1) (( 1) x 1,..., ( 1) x ) = M ( 1) (x,..., x 1;... ; x,..., x 1). Therefore, we have which proves (7). M (x 1,..., x ) = M ( 1) (x,..., x 1;... ; x,..., x 1) = M 1 (( 1) M (x,..., x 1)) (by (6)) = M (x,..., x 1) (reflexivity) We ow have at had all the ecessary tools to establish that M 2 is a solutio of the bisymmetry equatio. Propositio 2.1 For ay strogly decomposable sequece (M ) IN0 of reflexive fuctios, the two-place fuctio M 2 fulfils the bisymmetry fuctioal equatio (4). Proof. We have successively, M 2 (M 2 (x 1, x 2 ), M 2 (x 3, x 4 )) = M 4 (x 1, x 2, x 3, x 4 ) (by (6)) = M 4 (M 2 (x 1, x 3 ), M 2 (x 2, x 4 ), M 2 (x 1, x 3 ), M 2 (x 2, x 4 )) (strog decomposability) = M 2 (M 2 (M 2 (x 1, x 3 ), M 2 (x 2, x 4 )), M 2 (M 2 (x 1, x 3 ), M 2 (x 2, x 4 ))) (by (6)) = M 2 (M 2 (x 1, x 3 ), M 2 (x 2, x 4 )) (reflexivity) which proves the result. Now, cosider a strogly decomposable sequece (M ) IN0 of cotiuous, strictly icreasig ad reflexive fuctios. Sice M 2 fulfils the bisymmetry equatio, it must have a particular form. Actually, it has bee proved by Aczél [1] (see also [2, 6.4] ad [3, Chap. 17]) that the geeral cotiuous, strictly icreasig, reflexive real solutio of the bisymmetry equatio (4) is give by the quasi-liear mea. The statemet of this result is formulated as follows. Theorem 2.1 Let I be ay real iterval, fiite or ifiite. A two-place fuctio M : I 2 IR is cotiuous, strictly icreasig i each variable, reflexive ad fulfils the bisymmetric equatio (4) if ad oly if there exists a cotiuous strictly mootoic fuctio f : I IR ad a real umber θ ]0, 1[ such that M(x 1, x 2 ) = f 1[ θ f(x 1 ) + (1 θ) f(x 2 ) ], x 1, x 2 I. (8) 5
Accordig to this result, M 2 is of the form (8). By usig strog decomposability, we will show later that the umber θ occurig i this form must be 1/2, so that M 2 is symmetric. Next, we will show that every fuctio M is also symmetric. Before goig o, cosider two lemmas. the Lemma 2.2 If A correspods to the matrix θ θ 0 A = 1 θ 0 θ, θ ]0, 1[, 0 1 θ 1 θ lim i + Ai = 1 D with D = θ 2 + θ(1 θ) + (1 θ) 2. θ 2 θ 2 θ 2 θ(1 θ) θ(1 θ) θ(1 θ) (1 θ) 2 (1 θ) 2 (1 θ) 2 Proof. The eigevalues of A correspod to the solutios of det(a zi) = 0 or (z 1)[θ(1 θ) z 2 ] = 0. Three distict eigevalues are obtaied: z 1 = 1, z 2 = θ(1 θ), z 3 = θ(1 θ) ad A ca be diagoalized: = S 1 AS = diag ( 1, θ(1 θ), θ(1 θ) ). We also have the followig eigevectors: s 11 θ 2 s 12 θ S 1 = s 21 = θ(1 θ), S 2 = s 22 = θ 1 θ, s 31 (1 θ) 2 s 32 1 θ s 13 θ S 3 = s 23 = θ 1 θ. s 33 1 θ A ca be expressed uder the form: A = S S 1 ad Fially, settig s ij := (S 1 ) ij, we have A i = S i S 1, i IN 0. lim i + Ai = S ( lim i) S 1 = ( s i + 11S 1 s 12S 1 s 13S 1 ). Sice S 1 S = id, we oly have to determie s 11, s 12, s 13 such that ( s 11 s 12 s 13 ) S = ( 1 0 0 ) ad we ca see that s 11 = s 12 = s 13 = 1 D. 6
Lemma 2.3 Cosider a strogly decomposable sequece (M ) IN0 of fuctios. If M 2 is symmetric the, for all IN, > 2, M is also symmetric. Proof. Let us proceed by iductio over 2. Assume that M is symmetric for a fixed 2. By strog decomposability, we have M +1 (x 1,..., x +1 ) = M +1 (x 1, M (x 2,..., x +1 )) ad M +1 is also symmetric. Hece the result. = M +1 ( M (x 1,..., x ), x +1 ), Now, we ca tur to the mai result. Before statig it, recall the Kolmogoroff-Nagumo s theorem. Theorem 2.2 Let I be ay (fiite or ifiite) real iterval, ad (M ) IN0 be a decomposable sequece of cotiuous, symmetric, strictly icreasig ad reflexive fuctios M : I IR. The ad oly the there exists a cotiuous strictly mootoic fuctio f : I IR such that, for all IN 0, M (x 1,..., x ) = f 1[ 1 f(x i ) ], (x 1,..., x ) I. i=1 As already aouced, we show that the symmetry property is ot ecessary i Theorem 2.2 if we replace decomposability by strog decomposability. The statemet is the followig. Theorem 2.3 Let I be ay (fiite or ifiite) real iterval, ad (M ) IN0 be a strogly decomposable sequece of cotiuous, strictly icreasig ad reflexive fuctios M : I IR. The ad oly the there exists a cotiuous strictly mootoic fuctio f : I IR such that, for all IN 0, M (x 1,..., x ) = f 1[ 1 f(x i ) ], (x 1,..., x ) I. i=1 Proof. (Sufficiecy) Trivial (see Theorem 2.2). (Necessity) Cosider a strogly decomposable sequece (M ) IN0 of cotiuous, strictly icreasig ad reflexive fuctios M : I IR. By Propositio 2.1, M 2 fulfils the bisymmetry fuctioal equatio (4). By Theorem 2.1, there exists a cotiuous strictly mootoic fuctio f : I IR ad a real umber θ ]0, 1[ such that M 2 (x 1, x 2 ) = f 1[ θ f(x 1 ) + (1 θ) f(x 2 ) ], x 1, x 2 I. Defie Ω := f(i) = {f(x) x I}. The sequece (F ) IN0 defied by F (z 1,..., z ) := f [ M (f 1 (z 1 ),..., f 1 (z )) ], (z 1,..., z ) Ω, IN 0, is also strogly decomposable ad such that each F is cotiuous, strictly icreasig ad reflexive. Moreover, we have F 2 (z 1, z 2 ) = θ z 1 + (1 θ) z 2, z 1, z 2 Ω. (9) 7
Now, let us show that F 3 (z 1, z 2, z 3 ) = 1 D [ θ 2 z 1 + θ(1 θ) z 2 + (1 θ) 2 z 3 ], z1, z 2, z 3 Ω, (10) with D = θ 2 + θ(1 θ) + (1 θ) 2. We have successively F 3 (z 1, z 2, z 3 ) = F 3 (F 2 (z 1, z 2 ), F 2 (z 1, z 3 ), F 2 (z 2, z 3 )) (by (7)) = F 3 (θ z 1 + (1 θ) z 2, θ z 1 + (1 θ) z 3, θ z 2 + (1 θ) z 3 ) (by (9)) = F 3 ((z 1, z 2, z 3 ) A) where A is the matrix defied i Lemma 2.2. By iteratio, we obtai We the have F 3 (z 1, z 2, z 3 ) = F 3 ((z 1, z 2, z 3 ) A) = F 3 ((z 1, z 2, z 3 ) A 2 ) F 3 (z 1, z 2, z 3 ) = lim i + F 3((z 1, z 2, z 3 ) A i ) = F 3 ((z 1, z 2, z 3 ) lim = F 3 ((z 1, z 2, z 3 ) A i ) i IN 0. i + Ai ) (costat umerical sequece) (cotiuity) ( 1 [ ]) = F 3 3 θ 2 z 1 + θ(1 θ) z 2 + (1 θ) 2 z 3 (Lemma 2.2) D = 1 [ ] θ 2 z 1 + θ(1 θ) z 2 + (1 θ) 2 z 3 (reflexivity) D which proves (10). Now we show that θ must be 1/2. Strog decomposability implies By (9) ad (10), this idetity becomes F 3 (z 1, z 2, z 3 ) = F 3 (F 2 (z 1, z 3 ), z 2, F 2 (z 1, z 3 )). θ(1 θ)(1 2 θ)(z 3 z 1 ) = 0, that is θ = 1/2. Cosequetly, M 2 is symmetric. By Lemma 2.3, M is symmetric for all IN 0. We the coclude by Theorem 2.2. Refereces [1] J. Aczél, O mea values, Bull. Amer. Math. Soc. 54 (1948) 392 400. [2] J. Aczél, Lectures o fuctioal equatios ad applicatios, (Academic Press, New York, 1966). [3] J. Aczél ad J. Dhombres, Fuctioal equatios i several variables with applicatios to mathematics, iformatio theory ad to the atural ad social scieces, (Cambridge Uiv. Press, Cambridge, 1989). 8
[4] G. Bemporad, Sul pricipio della media aritmetica, Atti Accad. Naz. Licei (6) 3 (1926) 87 91. [5] A.L. Cauchy, Cours d aalyse de l Ecole Royale Polytechique, Vol. I. Aalyse algébrique, (Debure, Paris, 1821). [6] O. Chisii, Sul cocetto di media, Periodico di matematiche (4) 9 (1929) 106 116. [7] B. de Fietti, Sul cocetto di media, Gior. Ist. Ital. Attuari (3) 2 (1931) 369 396. [8] J.C. Fodor ad J.-L. Marichal, O ostrict meas, Aequatioes Math. 54 (1997) 308 327. [9] A.N. Kolmogoroff, Sur la otio de la moyee, Atti Accad. Naz. Licei Mem. Cl. Sci. Fis. Mat. Natur. Sez. (6) 12 (1930) 388 391. [10] J.-L. Marichal, P. Mathoet ad E. Tousset, Characterizatio of some aggregatio fuctios stable for positive liear trasformatios, Fuzzy Sets ad Systems, i press. [11] J.-L. Marichal ad M. Roubes, Characterizatio of some stable aggregatio fuctios, Proc. Iter. Cof. o Idustrial Egieerig ad Productio Maagemet, Mos, Belgium (1993) 187 196. [12] M. Nagumo, Über eie klasse der mittelwerte, Japa. J. Math., 7 (1930) 71 79. [13] U. Ricci, Cofroti tra medie, Gior. Ecoomisti e Rivista di Statistica 26 (1915) 38 66. 9