A class of multiattribute utility functions

Transcription

1 Economic Teory DOI / x EXPOSITA NOTE Andrá Prékopa Gergely Mádi-Nagy A cla of multiattribute utility function Received: 25 July 2005 / Revied: 8 January 2007 Springer-Verlag 2007 Abtract A function uz i a utility function if u z >0. It i called rik avere if we alo ave u z <0. Some autor, owever, require tat u i z >0ifi i odd and u i z <0ifi i even. Te notion of a multiattribute utility function can be defined by requiring tat it i increaing in eac variable and concave a an -variate function. A tronger condition, imilar to te one in cae of a univariate utility function, require tat, in addition, all partial derivative of total order m ould be poitive if m i odd and negative if m i even. In ti paper, we preent a cla of function in analytic form uc tat eac of tem atifie ti tronger condition. We alo give arp lower and upper bound for E[uX 1,...,X ] under moment information wit repect to te joint probability ditribution of te random variable X 1,...,X aumed to be dicrete and repreenting wealt. Keyword Multiattribute utility function Mixed utility function Expected utility JEL Claification Number D8 C61 C63 Partially upported by OTKA grant F and T in Hungary. A. Prékopa B RUTCOR, Rutger Center for Operation Reearc, Rutger Univerity, 640 Bartolomew Road, Picataway, NJ , USA prekopa@rutcor.rutger.edu G. Mádi-Nagy Matematical Intitute, Budapet Univerity of Tecnology and Economic, Műegyetem rakpart 1-3., 1111 Budapet, Hungary gnagy@mat.bme.u

2 A. Prékopa and G. Mádi-Nagy 1 Introduction Te mot general definition of a utility function uz, z 0 only require tat it ould be an increaing function, u z >0. It i called rik avere, if we alo ave u z <0 wic mean tat te function i alo concave. Pratt 1964 and Arrow 1970 tre te importance of utility function wit decreaing rik averion. If we take te Arrow Pratt meaure of abolute rik averion: u z u z, 1.1 ten te above requirement implie tat u z >0. More generally, we may require tat te nonzero derivative u m+1 z doe not cange ign 1.2 or 1 n 1 u n z >0, n = 1,...,m or 1 n 1 u n z >0, n = 1, 2, Utility function atifying 1.4 are called mixed by Caballe and Pomanky A imilar but tronger condition on utility function wa introduced by Cander For economic jutification ee Ingeroll Relation 1.4 mean tat u z i a completely monotone function. In view of ti, a utility function atifying 1.3 will be called monotone of order m + 1. If uz i a mixed utility function wit u0 = 1, ten, by a well known teorem [ee Feller 1971, p. 439], it admit te repreentation uz = 0 e z df, 1.5 were F i a c.d.f. on [0,. Example of mixed utility function are uz = a log 1 + z, uz = ae bz, b were a > 0, b > 0. Te next example i, on te oter and, a utility function tat atifie 1.3 uz = z x m x 2 [1 Fx 1 ] dx 1...dx m, 1.6 were F i a c.d.f. Te integral in 1.6 i finite if and only if 0 x m+1 dfx <.

3 A cla of multiattribute utility function Te value uz a a imple economic interpretation. Firt we remark tat, a it i eay to ow, te following equation old true z x m x 2 [1 Fx 1 ] dx 1 dx m = z x z m+1 dfx. 1.7 If F i te c.d.f. of a random demand and te upply i equal to z, ten te value in 1.7 i te expected penalty of te unerved demand wit penalty function { 0 if x z qx = x z m+1 if x > z. If m = 0 ten it i te expected unerved demand. Te value 1.7 alo appear in tocatic ordering teory ee Saked and Santikumar Te random variable X dominate te random variable Y in order m, in ymbol: X m Y,iff z x z m dfx z x z m dgx, 1.8 were F, G are te c.d.f of X and Y, repectively. By Eq. 1.7, te inequality 1.8 can be expreed in term of te m-fold integral. Brockett et al apply a utility function uz wit u 4 < 0 in connection wit an inurance problem. Tey aume te exitence of te firt tree moment of te random lo L, and apply te Markov Krein teorem concerning Cebyev ytem. Te reult i tat if we take maxeul or mineul on toe et of c.d.f. tat ave te precribed tree moment, ten te extremal ditribution doe not depend on te pecial utility function. It i uniquely determined by te requirement tat u 4 < 0. More generally, if 1.2 old true, and te firt m moment of a random variable X are known, ten, by te ue of te metodology of te Markov moment problem ee Krein and Nudelman 1977 we can obtain extremal ditribution to repreent maxeux and mineux. Toe are called upper and lower principal repreentation, repectively, of te moment µ 1,...,µ m. Again, under te mentioned condition, tee extremal ditribution do not depend on te utility function uz. Tu, in order to create upper and lower bound for te expectation EuX taken wit te true ditribution, we only ave to ae te value of te utility function uz at te upport of te extremal ditribution. More elegant i te bounding procedure if X a an unknown dicrete ditribution wit known upport. In ti cae te metodology of te dicrete moment problem ee Prékopa 1990 can be applied to obtained bound for EuX.Tat poibility obviouly carrie over to utility function atifying 1.3 or 1.4. Multiattribute utility function ave alo been extenively reearced, ee, e.g., Keeney and Raiffa 1976, Dyer and Sarin 1979, Dyer et al In mot

4 A. Prékopa and G. Mádi-Nagy cae relatively imple um of product of ingle attribute utility function provide u wit multiattribute one. However, large collection of analytic formula tat can erve for application doe not exit. In ti paper our purpoe i to improve on te ituation and introduce a cla of multiattribute utility function in uc a way tat we aume te knowledge of ingle attribute utility function, eac atifying relation 1.4 and ten couple tem into one -attribute utility function. Te rik avere multiattribute utility function may be defined in uc a way tat uz 1,...,z i increaing in eac variable and concave a an -variate function. In addition, we may require te nonzero derivative or i 1+ +i uz 1,...,z 1 z i doe nor cange ign if i 1 + +i = m i 1+ +i 1 i 1+ +i uz 1,...,z 1 z i > 0, 1 i 1 + +i m or 1 i 1+ +i 1 i 1+ +i uz 1,...,z 1 z i > 0, 1 i 1 + +i Tee are multivariate counterpart of relation 1.2, 1.3 and 1.4, repectively. Our cla of multiattribute utility function i given by Definition 1.1 Let k 1andD ={z 1,...,z e g j z j > 2, j = 1,...,}. We define te utility function u a [ ] uz 1,...,z := log ke g 1z 1 1 e g z 1 1, 1.12 were for every z 1,...,z n D te following condition old g j z j>0 g i j z j 0, if i > 1 and i odd g i j z j 0, if i i even j = 1,..., Te g j z j function can be coen, e.g., from te following type of function a log 1 + b z, were, a > 0, b > 0, ae bz, were, a > 0, b > 0, a n z n + +a 1 z + a 0, wit uitably coen coefficient. It i obviou tat function 1.12 are trictly increaing in eac variable in teir domain. It i alo true, but no longer obviou, tat te function are concave. We prove it in Sect. 2, togeter wit oter propertie of te function In Sect. 3 we preent numerical example for bounding expected utilitie under moment information.

5 A cla of multiattribute utility function 2 Propertie of te multiattribute utility function 1.12 Teorem 2.1 Te function 1.12 are concave on D. Proof In view of 1.13, it follow tat eac function e g j z j, j = 1,..., i logconcave on D. By Lemma in Prékopa 1995 ti old for te function e g j z j 1 a well. Ti implie tat ke g 1z 1 1 e g z 1 i logconcave on D.Foreveryz D it value i greater tan 1, ence te repeated application of te lemma prove te aertion. Te next teorem preent our central reult. Teorem 2.2 For every z = z 1,...,z D we ave i 1+ +i uz 1,...,z 1 z i > 0, if i 1 + +i i odd. and 2.1 i 1+ +i uz 1,...,z 1 z i < 0, if i 1 + +i i even. Firt we prove te aertion for te cae of g j z j = z j, j = 1,...,. Lemma 2.3 Property 2.1 old for uz := u 0 z := log [ ke z 1 1 e z 1 1 ]. 2.2 We need te following Aertion 2.4 Conider te function f z = log ke z 1 1, were k 1 contant and z > log2. We aert tat df dz = 1 + k + 1 ke z k + 1 d i f i dz i = 1 i 1 a i k + 1 ke z, k + 1 i = 2, 3,..., 2.3 were te number a i are defined by a i 1 = 1, i = 2, 3,... a 2 1 = a 2 2 = 1 a i a i i = a i ai 1, = 2,...,i 1 = a i 1 i 1 i

6 A. Prékopa and G. Mádi-Nagy Proof of Aertion 2.4 It i trivial for i = 1 and can eaily be cecked for i = 2. For te proof of te general cae we ue induction. Aume tat te equation in te econd line of 2.3 old for ome i 2. It follow tat d i+1 f i dz i+1 = 1 i a i k k + 1ke z ke z k + 1 ke z k i = 1 i a i k k + 1 ke z k + 1 ke z k k ke z k + 1 [ i = 1 i a i k + 1 i ] ke z + a i k + 1 k ke z k + 1 i = 1 i a i + ai 1 1 k + 1 ke z k i a i k + 1 i+1 i i ke z k + 1 i+1 = 1 i a i+1 k + 1 ke z. k + 1 Hence te aertion. Aertion 2.4 implie tat te odd order derivative of f are poitive and te even order derivative of f are negative. It i wort to mention tat if we apply te equation in te econd line of 2.3 for te cae of i = 1, ten we obtain te formula in te firt line witout te 1. Proof of Lemma 2.3 Let z 2,...,z fixed and k 1 = ke z 2 1 e z 1. Ten u 0 z 1,...,z can be written in te form u 0 z 1,...,z = log k 1 e z By Aertion 2.4 we ave u 0 k = 1 + z 1 k 1 e z 1 k i 1u 0 i 1 z i = 1 i1 1 a i 1 k k 1 1 e z, 1 k i Similar formula can be written up for te derivative of u 0 z 1,...,z wit repect to any of it variable. Ti implie tat te aertion of te lemma old for tat pecial cae, were we take derivative only wit repect to a ingle variable.

7 A cla of multiattribute utility function To prove te lemma for mixed derivative we rewrite te equation in 2.6 and take te derivative of order i 2 wit repect to z 2 etc. Let u introduce te notation We ave te equation Uing ti, 2.6 can be written a: k 2 = ke z 1 1e z 3 1 e z 1. k 1 e z 1 k = k 1 e z = ke z 1 1e z 2 1 e z 1 1 = k 2 e z = k 2 e z 2 k u 0 = 1 + k k z 1 k k 2 e z 2 k i 1u 0 i 1 z i = 1 i 1 1 a i 1 k1 + 1 k k k 2 e k, 2 k i Now, we fix te value z 1, z 3,...,z and conider te function 2.7 a function of te ingle variable z 2. If we take te firt derivative wit repect to z 2, ten, for te term correponding to te ubcript we obtain k k k 2 + 1k 2 e z 2 z 2 k 2 e z = 2 k k 2 e z 2 k k 2 e z 2 k k = k 2 e z 2 k k k 2 e z 2 k k k 2 e z 2 k [ k = k 2 e z 2 k ] k k 2 e z. 2 k If we take te furter derivative until order i 2,wecaneetat i 1+i 2 u 0 1 z i = 1 i1+i2 1 poitive value. 2 2 Proceeding ti way, along te derivative wit repect to z 3,...,z, te lemma follow.

8 A. Prékopa and G. Mádi-Nagy In te general cae, we ue te following Aertion 2.5 If te univariate function gz a te property g z >0 g i z 0, g i z 0, if i > 1 and i odd if i i even 2.8 on te et D and te function fz a te property f i z >0, f i z <0, if i i odd if i i even 2.9 on te et {gz z D}, ten2.9 i alo true for teir compoition. i.e. [ f gz] i > 0, if i i odd [ f gz] i < 0, if i i even 2.10 on te et D. Proof of Aertion 2.5 It i eay to ee by induction tat i [ f g z] i = l=1 l+k 1 + +k l =i f l gzg k 1+1 z g k l+1 z Te condition in te econd um i equvivalent to l 1 + k 1 + +k = i If i i odd even i 1 i even odd te number of odd term in te um 2.12 i alway even odd by 2.8 and 2.9 te number of nonpoitive term i even odd in eac product of 2.11 eac product of 2.11 i nonnegative nonpoitive. Since te term f i gzg z g z i poitive negative and it i alway in te um 2.11, it follow tat [ f gz] i > 0< 0. Proof of Teorem 2.2 Introduce te following function u k z 1,...,z := u 0 g 1 z 1,...,g k z k, z k+1,...,z, k = 0, 1,..., Te teorem will be proved if we prove tat te function 2.13 ave te property 2.1. We prove te lat aertion by induction. For k = 0 relation 2.1 old, by Lemma 2.3. Aume tat 2.1 old for k, i.e., i 1+ +i u k z 1,...,z 1 z i = 1 i 1+ +i 1 poitive value. 2.14

9 A cla of multiattribute utility function For k k + 1 conider te univariate function f z k+1 := 1 i 1+ +i k +i k+2 + +i i 1+ +i k +i k+2 + +i u k z 1,...,z 1 z i k k z i k+2 k+2 zi 2.15 for fixed z 1,...,z k, z k+2,...,z value. Ten, in view of 2.14, f z k+1 atifie 2.9. Alo, g k+1 z k+1 atifie 2.8, and by Aertion 2.5 f g k+1 z k+1 a property From ti we derive i 1+ +i u k+1 z 1,...,z 1 z i Ti prove te teorem. = 1 i 1+ +i k +i k+2 + +i f i k+1 g k+1 z k+1 = 1 i 1+ +i 1 poitive value Numerical example for bounding E [ux 1, X 2, X 3 ] Aume tat te upport of te random variable X j i te et Z j. Ten te upport of te random vector X = X 1,...,X T i a ubet of te et Z = Z 1 Z. Let p i i = PX 1 = z 1i1,...,X = z i, 0 i j n j, j = 1,...,, n 1 µ α1...α = E[X α 1 1 X α ]= i 1 =0 n i =0 z α 1 1i 1 z α i p i1 i, were α 1,...,α are nonnegative integer. Te number µ α1...α i called te α 1,...,α -order moment of te random vector X and te um α 1 + +α i called te total order of te moment. Suppoe tat te probability ditribution of X i unknown but known are all moment of total order at mot m and furter univariate moment of te marginal ditribution. More preciely, we aume tat te following moment are known: and E[X α 1 1 X α ],α 1 + α m E[X α k k ], m α k m k, k = 1,...,. 3.1 Let f z, z Z be a dicrete function and introduce te notation f i1...i = f z i1,...,z i. Our multivariate dicrete moment problem MDMP i te following LP: minmax n 1 n f i1...i p i1...i i 1 =0 i =0

10 A. Prékopa and G. Mádi-Nagy ubject to n 1 i 1 =0 n i =0 z α 1 1i 1 z α i p i1...i = µ α1...α for α j 0, j = 1,...,; α 1 + α m and for α j = 0, j = 1,...,k 1, k + 1,...,, m α k m k, k = 1,...,; p i1...i 0, all i 1,...,i. 3.2 Here te p i1...i are te deciion variable, everyting ele in te LP i given. Te optimum value of te minimization maximization problem 3.2 i a lower upper bound for E[ f X 1,...,X ]. Te bound are alo arp in te ene tat no better bound can be given baed on te moment 3.1. For a ummary of linear programming te reader i referred to Prékopa In ti ection, we conider utility function 1.12 for te cae were = 3 and g j z j i linear j = 1, 2, 3, i.e., uz 1, z 2, z 3 = log [ e α 1z 1 +a 1 1e α 2z 2 +a 2 1e α 3z 3 +a ] z 1, z 2, z 3 Z, 3.3 were Z i pecialized a follow: Aume tat Z = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. e α j z j +a j > 2, j = 1, 2, 3, for z 1, z 2, z 3 Z. 3.4 We know from Teorem 2.2 tat te odd order partial derivative of 3.3 are poitive, wile te even order derivative of it are negative at any point tat atifie 3.4. Ti mean tat te odd even order divided difference of 3.3 on Z are poitive negative. In te following numerical example we conider te MDMP 3.2 wit te objective function 3.3, were m, m j, j = 1, 2, 3 are even number. We give arp lower and upper bound for te expected value of te utility function 3.3 by te ue of te dual algoritm of linear programming. Conidering te reult of Teorem 4.1 in Mádi-Nagy and Prékopa 2004, te collection of vector correponding to te ubcript in {i 1, i 2, i 3 i 1 + i 2 + i 3 m or i k = 0, k = j, m i j m j, j = 1, 2, 3}, 3.5 i a dual feaible bai in te minimization problem 3.2. Similarly, te collection of vector correponding to te ubcript in {i 1, i 2, i 3 9 i i i 3 m or i k = 9, k = j, m 9 i j m j, j = 1, 2, 3}, 3.6

11 A cla of multiattribute utility function i a dual feaible bai in te maximization problem 3.2. Bot bae provide u wit bound, te firt one i a lower wile te econd one i an upper bound. Te bae, on te oter and, can erve a initial bae in te dual algoritm tat we carry out to obtain te bet bound. Example 3.1 Conider te function 3.3 wit parameter α 1 = α 2 = α 3 = a 1 = a 2 = a 3 = 1. Aume tat X 1, X 2, X 3 are independent and eac one a uniform ditribution on {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. In Table 1 and 2 we pecify te known moment and preent te lower and upper bound for EuX 1, X 2, X 3 wic are te optimum value of te LP wit objective function to be minimized and maximized, repectively. Example 3.2 Conider te function 3.3 wit parameter α 1 = 1.75, α 2 = 1.25, α 3 = 0.75, a 1 = 3, a 2 = 2, a 3 = 1 and te random variable X 1 = min X + Y 1, 9 X 2 = min X + Y 2, 9 X 3 = min X + Y 3, 9, were X, Y 1, Y 2, Y 3 ave Poion ditribution wit parameter 1, 2, 2.5, 3, repectively. Note tat X 1, X 2, X 3 are tocatically dependent. Te moment preented in Table 3, 4 are toe of te random variable X 1, X 2, X 3. Table 1 Bound of Example 3.1 in cae of m = 2 m m 1 m 2 m 3 Minimum Iteration Maximum Iteration Table 2 Bound of Example 3.1 in cae of m = 4 m m 1 m 2 m 3 Minimum Iteration Maximum Iteration Table 3 Bound of Example 3.2 in cae of m = 2 m m 1 m 2 m 3 Minimum Iteration Maximum Iteration Table 4 Bound of Example 3.2 in cae of m = 4 m m 1 m 2 m 3 Minimum Iteration Maximum Iteration

12 A. Prékopa and G. Mádi-Nagy Te reult are ummarized below. Looking at te table we may ay tat if te lower and upper bound are not cloe to te eac oter, i.e., we do not ave atifactory approximation to te value E[uX], ten it i adviable to increae te number of individual moment rater tan te number of mixed moment. Ti way we may expect to obtain better reult in a orter time. Reference Arrow, K.: Eay in te Teory of Rik Bearing, Cap. 3. Amterdam: Nort-Holland 1970 Brockett, P.L., Cox, Jr., S.H., Witt, R.C.: Inurance veru elf-inurance: a rik management perpective. J Rik Inur 53, Caballe, J., Pomanky, A.: Mixed rik averion. J Econ Teory 71, Cander, P.: Repetitive rik averion. Econ Teory 29, Dyer, J.S., Fiburn, P.C., Steuer, R.E., Walleniu, J., Ziont, S.: Multiple criteria deciion making, multiattribute utility teory: te next ten year. Manage Sci 385, Dyer, J.S., Sarin, R.K.: Meaurable multiattribute value function. Oper Re 27, Feller, W.: An Introduction to Probability Teory and it Application, vol. II, 2nd. New York: Wiley 1971 Ingeroll, Jr., J.E.: Teory of Financial Deciion Making. Totowa: Rowman and Littlefield 1987 Keeney, R.L., Raiffa, H.: Deciion wit Multiple Objective: Preference and Value Tradeoff. New York: Wiley 1976 Krein, M.G., Nudelman, A.A.: Te Markov Moment Problem and Extremal Problem. Tranlation of Matematical Monograp, vol. 50. American Matematical Society 1977 Mádi-Nagy, G., Prékopa, A.: On multivariate dicrete moment problem and teir application to bounding expectation and probabilitie. Mat Oper Re 292, Pratt, J.W.: Rik averion in te mall and in te large. Econometrica 32, Prékopa, A.: Te dicrete moment problem and linear programming. Dicr Appl Mat 27, Prékopa, A.: Stocatic Programming. Budapet: Akadémia Kiadó 1995 Prékopa, A.: A brief introduction to linear programming. Mate Sci 21, Saked, M., Santikumar, J.G.: Stocatic Order and Teir Application. San Diego: Academic Pre 1994