Aggregation Functions and Personal Utility Functions in General Insurance

Transcription

1 Acta Polytechca Huarca Vol. 7, No. 4, 00 Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace Jaa Šprková Departmet of Quattatve Methods ad Iformato Systems, Faculty of Ecoomcs, Matej Bel Uversty Tajovského 0, Baská Bystrca, Slovaka, Pavol Kráľ Departmet of Quattatve Methods ad Iformato Systems, Faculty of Ecoomcs, Matej Bel Uversty Tajovského 0, Baská Bystrca, Slovaka Isttute of Mathematcs ad Computer Scece, UMB ad MÚ SAV Ďumberska, 974 Baská Bystrca, Slovaka, Abstract: The model of a utlty fucto s forms s a very terest part of moder decso mak theory. We apply a basc cocept of the persoal utlty theory o determato of mmal et ad mal ross aual premum eeral surace. We troduce specfc values of ross aual premum o the bass of a persoal utlty fucto, whch s determed emprcally by a short persoal tervew. Moreover, we troduce a ew approach to the creato of a persoal utlty fucto by a fctve ame ad a areato of specfc values by mxture operators. Keywords: Utlty fucto; Expected utlty; Mxture operator; Geeral Isurace Itroducto Ths paper was maly spred by the books Moder Actuaral Rsk Theory [3] ad Actuaral models The Mathematcs of Isurace [3]. The authors of the above-metoed books assume utlty fuctos as lear utlty u ( w) = w, quadratc utlty u( w) = ( a w) c, power utlty u ( w) = w, etc. Lap [5] descrbes ad explas a applcato of the utlty fucto decso mak a really terest way. I ths book also the eerato of the utlty fucto 7

2 J. Šprková et al. Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace us formato extracted from a persoal tervew s explaed. A moder theoretcal approach to the utlty fucto s also descrbed by Norstad [8]. We ca fd a very terest dscusso about utlty fuctos [4]. A alteratve approach to the determato of a utlty fucto o the bass of the areato of specfc utlty values ca be foud [8]. However, real lfe people do ot behave accord to the theoretcal utlty fuctos. It s a psycholocal problem rather tha a mathematcal oe. The serousess ad also the ucertaty of a respodet's aswers deped o the stuato, o the form of the asked questos, o the tme whch the respodets have, ad o a lot of other psycholocal ad socal factors. I our paper we troduce the possblty of determ a persoal utlty fucto o the bass of a persoal tervew wth vrtual moey. Moreover, we recall ad apply oe type of areato operators [], the so-called mxture operators M, the eeralzed mxture operators M, ad the specally ordered eeralzed mxture operators M o the areato of socalled rsk eutral pots, see [6-7], [9-], [4-6]. Ths paper s orazed as follows: Secto we recall the basc propertes of utlty fuctos ad ther applcatos eeral surace. I Secto 3 we also recall mxture operators ad ther propertes, amely the suffcet codtos of ther o-decreas-ess. I Secto 4 we descrbe the persoal utlty fucto of our respodet who took part our short tervew. Us ths fucto we calculate the mal ross premum for a eeral surace polcy. I ths secto we also descrbe a alteratve approach, where a persoal utlty fucto s determed throuh the result of a fctve ame ad theoretcal utlty fuctos. The result utlty fucto s the used for the computato of the mal ross premum. Moreover, we evaluate the mmal et aual premum by meas of the theoretcal utlty fucto for the surer. Fally, some coclusos ad dcatos as to our ext vestato about the metoed topc are cluded. Utlty Fuctos Idvduals ca have very dfferet approaches to rsk. A persoal utlty fucto ca be used as a bass for descrb them. I eeral, we ca detfy three basc persoaltes wth respect to rsk. The rsk-averse dvdual, who accepts favorable ambles oly, a rsk seeker, or other words rsk-lov dvdual, who pays a premum for the prvlee of partcpato a amble, ad the rskeutral dvdual, who cosders the face value of moey to be ts true worth. Throuhout most of ther lfe people are typcally rsk averse. Oly ambles wth hh expected payoff wll be attractve to them. The rsk-averse dvdual s 8

3 Acta Polytechca Huarca Vol. 7, No. 4, 00 maral utlty dmshes as the beefts crease, so that the rsk-averse dvdual s utlty fucto exhbts a decreas postve slope as the level of moetary payoff becomes hher. Such a fucto s cocave, see Fure. The behavor of a rsk-lov dvdual s opposte. The rsk-lov dvdual prefers some ambles wth eatve expected moetary payoffs. Ther maral utlty creases. Each addtoal euro provdes a dsproportoately reater sese of well-be. Thus, the slope of the rsk-lov dvdual s utlty fucto creases as the moetary chae mproves. Ths fucto s covex (see F. ). The utlty fucto for a rsk-eutral dvdual s a straht le. The utlty s equal to the utlty of expected value. Rsk-eutral dvduals buy o casually surace sce the premum chare s reater tha the expected loss. Rsk-eutral behavor s typcal for persos who are eormously wealthy. Of course, a lot of people may be rsk averse ad rsk lov at the same tme, deped o the rae of moetary values be cosdered, whch ca be llustrated us the behavor of the persoal utlty fucto of our respodet.. The Persoal Utlty Fucto The fudametal proposto of the moder approach to utlty s the possblty to obta a umercal expresso for dvdual prefereces. As people usually have dfferet approaches to rsk, two persos faced wth a detcal decso may actually prefer dfferet courses of acto. I ths secto we wll dscuss utlty as a alteratve expresso of payoff that reflects persoal approaches. Suppose that our respodet ows captal w, ad that he values wealth by the utlty fuctou. The ext Theorem, or other words Jese`s equalty, descrbes the propertes of the utlty fucto ad ts expected value [3], (see also Fure ). It ca be wrtte as follows. Theorem [3] (Jese's equalty) If u ( x) s a covex fucto ad X s a radom varable, the the expected utlty s reater or equals to a utlty value E[ u( X )] u( E[ X ]) () wth a equalty f ad oly f u ( x) s lear wth respect to X or var ( X ) = 0. From Jese`s equalty ad Fure t follows that for a cocave utlty fucto t holds E [ u( w X )] u( E[ w X ]) = u( w E[ X ]). () I ths case the decso maker s called rsk averse. He prefers to pay a fxed E X stead of a rsk amout X. amout [ ] 9

4 J. Šprková et al. Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace Fure Cocave utlty fucto - rsk averse approach Fure Covex utlty fucto - rsk lov approach I the ext part we llustrate whether to buy surace or ot by evaluat a dvdual's decso. Now suppose that our respodet has two alteratves, to buy surace or ot. Assume he s sured aast a loss X for a premum P. If he s sured, ths meas a certa alteratve. Ths decso ves us the utlty u w P. value ( ) If he s ot sured, ths meas a ucerta alteratve. I ths case the expected E u w X. utlty s [ ( )] From Jese's equalty () we et E [ u( w X )] u( E[ w X ]) = u( w E[ X ]) = u( w P). (3) Sce a utlty fucto u s a o-decreas cotuous fucto, ths s equvalet to P P, where P deotes the mum premum to be pad. 0

5 Acta Polytechca Huarca Vol. 7, No. 4, 00 Ths so-called zero utlty premum s a soluto of the follow utlty equlbrum equato E [ u( w X )] = u( w P ). (4) The dfferece ( P ) w s also called the certaty equvalet - CE. I [] 3 the certaty equvalet s defed as follows. Defto The certaty equvalet s that payoff amout that the decso maker would be wll to receve exchae for udero the actual ucertaty, tak to accout ts beefts ad rsks. Remark We recall that the expected utlty s calculated by meas of the wellkow formula E [ u( X )] = u( x ) p where X ( x x,..., ), (5) =, x s a vector of the possble alteratves ad =,,...,, are respectve probabltes. p, for Expected utltes ca be calculated as fucto values of a lear fucto, whch s assed uquely by pots A ad B, where pot A represets the worst outcome ad B the best outcome. Remark [ 5 ] Whe possble moetary outcomes fall to the decso maker's rae of rsk averse, the follow propertes hold (see Fure ): ) Expected payoffs EP = E[ w X ] are reater tha ther couterpart certaty equvalet CE = w P. ) Expected utltes [ u( w X )] expected moetary payoff ( ) E wll be less tha the utlty of the respectve u w P. 3) Rsk premums RP = EP CE are postve. If possble moetary outcomes fall to the decso maker's rae of rsk lov, the follow propertes hold (see Fure ): ) Expected payoffs EP E[ w X ] = are less tha ther couterpart certaty equvalet CE = w P. ) Expected utltes [ u( w X )] expected moetary payoff ( ) E wll be reater tha the utlty of the respectve u w P. 3) Rsk premums RP = EP CE are eatve.

6 J. Šprková et al. Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace The surer wth a utlty fucto U ad captal W, wth surace of loss X for a premum P must satsfy the equalty E [ U ( W P X )] U ( W ) +, (6) ad hece for the mmal accepted premum U m ( W ) = E[ U ( W + P X )]. The Rsk Averso Coeffcet m P. (7) O the bass of equato (3) we ca evaluate a rsk averso coeffcet. Let μ ad σ be the mea ad varace of loss X. Us the frst terms the Taylor expaso of the utlty fucto u w μ, we obta u μ. u w X s ve by ( w X ) u( w ) + u ( w μ) ( μ X ) + u ( w μ) ( μ X ) The expected utlty from ( ) [ ( w X )] E u E u After some process we et ( w μ) + ( μ X ) u ( w μ) + ( μ X ) u ( w μ) E[ u( w X )] u( w μ) + σ u ( w μ). (8) The Taylor expaso of the fucto o the rht sde of equato (3) s ve by u ( w P ) u( w μ) + ( μ P ) u ( w μ) From the equalty of equatos (8) ad (9) we have u After some process we et ( μ) σ w + u ( w μ) u( w μ) + ( μ P ) u ( w μ) u ( w μ) P μ σ u ( w μ) where a rsk averso coeffcet ( w) s ve by. (9). (0), () r of the utlty fucto u at a wealth w μ

7 Acta Polytechca Huarca Vol. 7, No. 4, 00 ( w μ) ( w μ) u r ( w) =. () u P μ + r( w μ) σ. (3) From (3) you ca see that, f the sured has reater rsk averso coeffcet, the he s wll to pay reater premum. 3 Mxture Operators I ths part we revew some mxture operators troduced [6], [7], [9-]. Suppose that each alteratve x s characterzed by a score vector x= ( x,...x ) [ 0,], where N { } s the umber of appled crtera. A mxture operator ca be defed as follows: Defto A mxture operator : [ 0,] [ 0,] wehted by a cotuous weht fucto : [ 0,] ] 0, [ M ( x x ) M s the arthmetc mea ve by x,..., =, (4) where ( x,..., ) x s a put vector. Observe that due to the cotuty of weht fucto, each mxture operator M s cotuous. Evdetly, M s a dempotet operator, [], [6], [9-0]. Note that sometmes dfferet cotuous weht fuctos are appled for dfferet crtera score, whch leads to a eeralzed mxture operator, see [6], [9-0]. Defto 3 A eeralzed mxture operator M : [ 0,] [ 0,] M ( x x ) s ve by x,..., =, (5) 3

8 J. Šprková et al. Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace where ( x,..., x ) s a put vector ad (,..., ) = s a vector of cotuous weht fuctos. Obvously, eeralzed mxture operators are cotuous ad dempotet. A eeralzed mxture operator based o the ordal approach ca be defed as follows. Defto 4 A ordered eeralzed mxture operator ve by ( x() ) x() () M : [ 0,] [ 0,] M ( x,..., x ) =, (6) where (,..., ) ( ) = s a vector of cotuous weht fuctos ad x (),..., x( ) s a o-decreas permutato of a put vector. A ordered eeralzed mxture operator s a eeralzato of a OWA operator [9], correspod to costat weht fuctos = w, w [ 0,], w =. However, a mxture operator eed ot be o-decreas. Marques-Perera ad Pas [6] stated the the frst suffcet codto for a weht fucto order to a mxture operator (8) s to be o-decreas. It ca be wrtte as follows: Proposto Let : [ 0,] ] 0, [ be a o-decreas smooth weht fucto whch satsfes the ext codto: ( x) ( x) for all x [ 0,]. The : [ 0,] [ 0,] 0 (7) N, >. M s a areato operator for each We have eeralzed suffcet codto (7) our prevous work. I the ext part we recall more eeral suffcet codtos metoed [7], [4-6]. From (4) we see that s 4

9 Acta Polytechca Huarca Vol. 7, No. 4, 00 M x = ( + x ) x 0 (8) f ad oly f + α( + ( x β )) 0, (9) where α ( ) ad α β = ( ), ad thus ecessarly [ 0,] = x [( ) ( 0),( ) ( ) ] α. x x β ad Now t s easy to see that (7) mples (9). However, (9) s satsfed also wheever + ( x ) ( x β ) 0 for each x [ 0,] ad each [ ] (0) β 0,. Because ( x ) 0, (0) s fulfled wheever 0 ( x) ( x) ( x) for all [ 0,] x. We have just show a suffcet codto more eeral tha (7). Proposto Let : [ 0,] ] 0, [ be a o-decreas smooth weht fucto whch satsfes the codto: ( x) ( x) ( x) for all x [ 0,]. The : [ 0,] [ 0,] 0 () N, >. M s a areato operator for each Moreover, we have mproved suffcet codto (), but costraed by. Proposto 3 For a fxed N : 0, 0, be a o-decreas smooth weht fucto satsfy the codto: ( x) ( ) ( ) + ( x) ( x) ( x), >, let [ ] ] [ () 5

10 J. Šprková et al. Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace for all x [ 0,].The : [ 0,] [ 0,] M s a areato operator. I the ext proposto we troduce a suffcet codto for the o-decreas-ess of eeralzed mxture operators. Proof. Mmal value of + ( x ) ( x β ) for [ 0,]. e., t s + α Suppose that () holds. The α + β s attaed for β =,. Therefore, (9) s surely satsfed wheever + ( x ) ( x ) ( ) ( ). ( x ). e., (9) s satsfed ad thus s a ftt weht fucto. I the ext proposto we troduce a suffcet codto for the o-decreasess of eeralzed mxture operators. Proposto 4 For a fxed N, >,,,..., o-decreas smooth weht fuctos, such that j j ( x) () + ( x) ( x) ( x) =, let : [ 0,] ] 0, [, be a for all x [ 0,]. The M : [ 0,] [ 0,], where (,..., ) areato operator. (3) =, s a 4 Maxmal Premum Determed by a Persoal Utlty Fucto I practce, the utlty fucto ca be determed emprcally by a persoal tervew made by a decso maker. I our opo, there are at least two sutable ways to do ths. The frst oe s based o a tervew whch provdes us wth probabltes estmated by a tervewed subject; the secod oe o a ame wth kow probabltes where the tervewed subject ves us oly formato about a persoal break pot. The persoal break pot s the amout of wealth at whch our dvdual s chaed from rsk averse to a rsk seeker, or 6

11 Acta Polytechca Huarca Vol. 7, No. 4, 00 vce versa. A approprate curve for a rsk averse ad rsk lov part s the selected from the theoretcal utlty fuctos. 4. A Persoal Utlty Fucto a Probablty-oreted Approach Follow ths approach a persoal utlty fucto ca easly be costructed from the formato leaed from a short tervew us the classcal reresso aalyss. The decso maker ca use ths fucto ay persoal decso aalyss whch the payoff falls betwee 0 ad Now we recall the tervew, whch s compled as follows [4]. Let us suppose you are ower of a vestmet whch brs you zero payoff ow or a loss of However, you have a possblty to step asde from ths vestmet uder the pealty the amout of a sequece: A: 000, B: 5000, C: 0000, D: 5000, E: Your portfolo maaer ca provde you wth formato express the probablty loos the Thk. What would be the best probablty of the loss, so that you reta the above metoed vestmet? Oly a few well-proportoed raphc pots are requred. From our tervew we took the respectve perso's data pots ( 0,), ( 000,0.8) ( 5000,0.75), ( 0000,0.60), ( 5000,0,60), ( 5000,0.40), ( 30000,0.00),, ad created the approprate utlty fucto of our respodet as show F. 3. Ths curve has a terest shape that reflects our respodet s approach to rsk. The dfferet persoal utlty fuctos for our respodet were created us the IBM SPSS 8.0 system for the purpose of comparso. The mal premum P was calculated by u verse fucto to the utlty equlbrum equato (4) P = w u ( E[ u( w X )]) wth system Mathematca 5. (4) 7

12 J. Šprková et al. Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace Fure 3 A utlty fucto ad the expected utlty of our respodet (the fucto from Table ) Utlty fuctos are used to compare vestmets mutually. For ths reaso, we ca scale a utlty fucto by multply t by ay postve costat ad (or) trasfer t by add ay other costat (postve or eatve). Ths kd of trasformato s called a postve affe trasformato. All our results are the same wth respect to such a trasformato. Quadratc ad cubc utlty fuctos are wrtte Table. O the bass of statstcal parameters (adjusted R square, p- values) we ca assume that the cubc fucto s the best ftt fucto. Moreover, Table also cossts of approprate expected utltes expressed by lear fuctos. Remark 3 Expected utltes (for the utlty fuctos from Table ) ca be calculated by meas of a lear fucto whch s assed uquely by pots ( 30000, u ( 30000) ) ad ( 0, u ( 0) ), or by the formula (8), alteratvely. I both cases we et the same values for the expected utltes. I Fure 3 you ca see the persoal utlty fucto of our respodet, as well as three terest pots that are hhlhted (also Table 3). Maxmal premum P a represets the area where our respodet s rsk averse, ad P s, where he s rsk seek (lov). Table A utlty fucto ad the expected utlty A utlty fucto ad the expected utlty 0 u( x) =.80 0 x E[ u( x) ] = x u( x) =.30 0 x E[ u( x) ] = x x x x From Table you ca see that the sured perso s wll to pay more tha the expected loss to acheve hs peace of md''. p Probablty Table The expected utlty ad mal premum wth respect to a quadratc fucto E [ u] wth respect to quadratc fucto P ( ) E [ X ] ( ) m P ( )

13 Acta Polytechca Huarca Vol. 7, No. 4, Table 3 The expected utlty ad mal premum wth respect to a cubc fucto p Probablty E [ u] wth respect to cubc fucto P ( ) E [ X ] ( ) m P ( ) , We determe the mmal premum by meas of (7) wth respect to the utlty fucto for surer U ( x) = l x wth hs basc captal W = ad loss X = The equato ca be rewrtte as follows: m m ( W ) p U ( W + P X ) + ( p) U ( W P ) U = + (5) ad hece W = m ( ) p m W P X ( W P ) ( + + p). (6) 9

14 J. Šprková et al. Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace We determed dvdual mmal premums wth correspod probablty wth the system Mathematca A Persoal Utlty Fucto a Game-based Approach Our expectato that our subject ca approprately estmate probabltes s the ma drawback of the prevous approach. I fact, we ca doubt whether somebody wthout approprate kowlede about probabltes ca provde us wth relable aswers. I order to avod ths problem we ca assume a ame wth probabltes whch are easy to uderstad, e.. ames based o co toss. Let us assume the follow ame. You have two possbltes: ether to toss a co wth two possble results, head meas you wll et 0, tal meas you wll et oth; or to choose 5 wthout play. What s amout of moey for whch you wll start (stop) play? It s easy to see that the expected value s the same both cases ad we make our decso about play wth respect to our persoal utlty fucto. The pot at whch we ve up (stop) play s the above metoed break pot. For smplcty we wll assume the quadratc utlty fucto. Ths approach allows us to combe easly persoal utlty fuctos to a roup utlty fucto us areato operators. The roup utlty fucto ca represet a specfc roup of customers of our surace compay. Let us assume three utlty fuctos based o dfferet break, a utlty fucto for x = 9900 u ( x) = a utlty fucto for x = 9800 u ( x) = A utlty fucto for x = 9650 u ( x) = ( x 9900) ( x 9900) ( x 9800) ( x 9800) ( x 9650) ( x 9650) for 0 x 9900, (7) for 9900 x for 0 x 9800, (8) for 9800 x for 0 x (9) for 9650 x To costruct the combed utlty fucto we ca use for example a ordered eeralzed mxture operator =,, M wth weht vector (, 3 ) where ( x) = 0.x 0. 8, ( x) = 0.5x 0. 5 ad ( x) = 0.75x Let us ote that the selected weht fuctos satsfy the codtos requred for of odecreas areato operators. 0

15 Acta Polytechca Huarca Vol. 7, No. 4, 00 Values 9900, 9800, 9650 we trasform to the ut terval ad areate them by meas of M. We obta a areated value M = , ad after trasformato we have pot of dvso x = , where the sured s eutral to rsk. O the bass of ths dvso pot we ca create a ew combed utlty fucto u ( x) = ad approprate expected utlty 0 ( x 987,4) ( x 987,4) for 0 x 987,4 (30) for 987,4 x x for 0 x 987,4 E ( x) = (3) for 987,4 x Table 4 The expected utlty ad mal premum wth respect to a fucto (8) w [ ( )] X X E u w X w P P m P ,00, ,00 0, ,00 0, ,90 7, ,00 0, ,30 60, ,59 987,4 0, ,40 7,60 7, ,00 0, ,70 076, ,00 0, ,90, ,00 0, , , ,00 0, ,0 9355, ,00 0, , , ,00 0, , , ,00 0, , , ,00 0, ,37 0, ,00 0, ,79 33, ,00 0, , , ,00 0, ,8 683, ,00 0, , , ,00 0, , , The mmal premum we evaluated o the bass of formula (7) wth the utlty fucto for surer U ( x) = l ( x +) wth the system Mathematca. From Table 4 ad also from the formula (7) you ca see that the mmal premum s ve by the sze of the expected loss. A ewly-aed utlty fucto would be requred for evaluat a decso wth more extreme payoffs or f our respodet's

16 J. Šprková et al. Areato Fuctos ad Persoal Utlty Fuctos Geeral Isurace atttudes chae because of a ew job or lfestyle chae. Moreover, the utlty fucto must be revsed from the vewpot of tme. Coclusos We have show two approaches to creat a persoal utlty fucto ad we have calculated the mum premum aast the loss of wth respect to t. We thk that the persoal utlty fucto of a sured perso would be very mportat for a surer. O the bass of the persoal utlty fucto the surer would kow what approaches to rsk the customers have ad thus, how they wll behave towards ther ow wealth. Creat a utlty fucto for the surer s very dffcult. Moreover, our ext work we wat to vestate the surer's utlty fucto ad we wat to determe the mmal premum aast the loss of wth respect to a cocrete real surer's utlty fucto. Ackowledemet Ths work was supported by rat VEGA /0539/08. Refereces [] Calvo, T., Mesar, R., Yaer, R. R.: Quattatve Wehts ad Areato, Fellow, IEEE Trasacto o fuzzy systems, Vol., No., 004, pp [] Grabsch, M., Marchal, J.-L., Mesar, R., Pap, E.: Mooraph: Areato Fuctos, Acta Polytechca Huarca, 6(), 009, pp [3] Kaas, R., Goovaerts, M., Dhaee, J., Deut, M.: Moder Actuaral Rsk Theory, Kluwer Academc Publshers, Bosto, 00, ISBN [4] Khurshd, A.: The Ed of Ratoalty?, Abstracts, Teth Iteratoal Coferece o Fuzzy Set Theory ad Applcatos, FSTA 00, Slovak Republc, (00), pp. 0- [5] Lap, L. L.: Quattatve Methods for Busess Decsos, 5 th edto, Sa Jose State Uversty, 975, ISBN [6] Marques-Perera, R. A., Pas, G.: O No-Mootoc Areato: Mxture operators. I: Proceeds of the 4 th Meet of the EURO Work Group o Fuzzy Sets (EUROFUSE 99) ad d Iteratoal Coferece o Soft ad Itellet Comput (SIC 99), Budapest, Huary, (999), pp [7] Mesar R., Šprková, J.: Wehted Meas ad Weht Fuctos, Kyberetka 4,, 006, pp [8] Norstad, J.: A Itroducto to Utlty Theory, 005

17 Acta Polytechca Huarca Vol. 7, No. 4, 00 [9] Rbero, R. A., Marques-Perera, R. A.: Geeralzed Mxture Operators Us Weht Fuctos: A Comparatve Study wth WA ad OWA, Europea Joural of Operatoal Research 45, 003, pp [0] Rbero, R. A., Marques-Perera, R. A.: Wehts as Fuctos of Attrbute Satsfacto Values. I: Proceeds of the Workshop o Preferece Modell ad Applcatos (EUROFUSE), Graada, Spa, 00, pp [] Rbero, R. A., Marques-Perera, R. A.: Areato wth Geeralzed Mxture Operators Us Weht Fuctos. I: Fuzzy Sets ad Systems 37, 003, pp [] Resová, R.: Utlty Fucto a Theory ad Practce ( Slovak), Studet Scetfc Actvty, Baská Bystrca, 009 [3] Rotar V. I.: Actuaral Models The Mathematcs of Isurace, Chapma & Hall/CRC, Taylor & Fracs Group, 006 [4] Šprková, J.: Weht Fuctos for Areato Operator: 3 rd Iteratoal Summer School o Areato Operators, Uversta Della Svzzera Italaa, Luao, Swtzerlad, 005, pp [5] Šprková, J.: Mxture ad Quas-Mxture Operators, IPMU 006, Frace, 006, pp [6] Šprková, J.: Dssertato thess, Wehted Areato Operators ad Ther Applcatos, Bratslava, 008 [7] Šprková, J.: Utlty Fucto a Isurace, Proceeds from AMSE 009, Uherské Hradště, 009, pp [8] Šprková, J. Kráľ, P.: Areato Fuctos ad Utlty Fuctos Geeral Isurace, Abstract of Proceeds of 0 th Iteratoal Coferece o Fuzzy Set Theory ad Applcatos, Lptovský Já, Slovaka, 00, p. 4 [9] Yaer, R. R.: Geeralzed OWA Areato Operators, Fuzzy Optmzato ad Decso Mak 3, 004, pp