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1 e copanon ONLY AVAILABLE IN ELECTRONIC FORM Electronc Copanon Decson Analyss wth Geographcally Varyng Outcoes: Preference Models Illustratve Applcatons by Jay Son, Crag W. Krkwood, L. Robn Keller, Operatons Research, (ctaton nforaton) E copanon to: Decson Analyss wth Geographcally Varyng Outcoes: Preference Models Illustratve Applcatons Jay Son * Crag W. Krkwood L. Robn Keller APPENDIX A: PREFERENTIAL INDEPENDENCE. In decson analyss, specfyng a ultattrbute value or utlty functon requres assessents fro a decson aker, ths can be dffcult because t requres the deternaton of an n-densonal functon. To splfy ths, researchers have establshed condtons on preferences under whch the for of the value or utlty functon s splfed. One of these condtons s preferental ndependence. A subset of 1, 2, n, when n 3, s defned to be preferentally ndependent of ts copleent f the rank orderng of alternatves wth no uncertanty that have coon levels for the copleentary attrbutes does not depend on those coon levels. If ths property holds for all subsets of 1, 2,, n, then utual preferental ndependence s sad to hold, n ths case n V ( z, z,, z ) a v ( z ) 1 2, (A-1) n 1 * Defense Resources Manageent Insttute, Naval Postgraduate School, Monterey, CA 93943, rson@nps.edu W. P. Carey School of Busness, Arzona State Unversty, Tepe, A , crag.krkwood@asu.edu The Paul Merage School of Busness, Unversty of Calforna, Irvne, CA , LRKeller@uc.edu S-1

2 where v (z ) s called the sngle attrbute value functon over z, a 0 s called the weghtng constant for the th attrbute (Debreu 1960). Goran (1968) derves several condtons related to utual preferental ndependence. In partcular, t follows fro hs results that f every par of attrbutes s preferentally ndependent of the reanng attrbutes then utual preferental ndependence holds, or, even ore specfcally, f {, +1 } s preferentally ndependent of ts copleent for = 1, 2,, n-1, then utual preferental ndependence holds. These results can substantally reduce the nuber of assessents that ust be ade n applcatons to establsh that (A-1) s vald. See Keeney Raffa (1976 Sectons ) for ore detals. APPENDIX B: ADDITIONAL DISCUSSION OF THEOREM 1. Let ( at least as preferred as ) be a preference relaton over, whch s the set of possble consequences (where a consequence s a vector of levels across all subregons), wth the correspondng strct relaton ( ore preferred than ) ndfference relaton ( equally preferred to ). Let q, r, s denote arbtrary vectors n. For z, we defne z as the vector of attrbute levels n all subregons except subregon, z as the vector of attrbute levels n all subregons except subregons. Consder the followng condtons on : (a) Copleteness: It ust be true that q r or r q. (That s, any consequences can be copared.) (b) Transtvty: If q r r s, then q s. (c) Contnuty: If q r, then there exsts 0 such that for any z, ax z q ples z r, ax z r ples q z. (d) Dependence on each subregon: For each subregon, there exst q, z, r z such that S-2

3 q, r, z z. (e) Parwse spatal preferental ndependence: For any two subregons, q, q, s r, r, s for soe z z for all z, s ples q, q, r, r, z. Ths condton ples utual preferental ndependence, as descrbed above. Snce the ndependence descrbed here s for a sngle attrbute across ultple subregons, t s labeled spatal to dstngush t fro preferental ndependence across ultple attrbutes, whch s consdered n Theore 2. DEFINITION A-1. An attrbute level there exst vectors q d z s called a tradeoffs dvalue of z ' r such that z', q d z, r z d, z'', q r. z '' n subregon f (f) Hoogenety: If vectors q, d z s a tradeoffs dvalue of z ' r such that z', d z, z '' n subregon, then for any subregon q r t ust be true that z d, z'', q r. (An equvalent defnton s that when two aounts z, z have a tradeoffs dvalue n a subregon also n a subregon, then they have the sae tradeoffs dvalue n subregon as n subregon.) A value functon exsts over the set of consequences f only f s coplete, transtve, contnuous, as specfed n condtons (a)-(c). (Ths s a specal case of Debreu (1954, 1964), who uses a ore general topologcal space.) It follows drectly fro Debreu s (1960) theore that for a regon wth three or ore subregons, a value functon can be wrtten as: V ( z, z,, z ) a v ( z ) 1 2 (A-2) 1 f only f condtons (a)-(e) above hold, where v s a sngle attrbute value functon over z, a 0 s the weght assocated wth subregon. Whle (A-2) has the sae for as (A-1), the z n (A- 2) represent the levels of the sae attrbute n dfferent subregons, rather than the levels of dfferent attrbutes 1, 2,, n as n (A-1). Harvey (1986, p. 1126, Condton E) defnes the Equal Tradeoffs S-3

4 Coparsons condton, he notes that a condton that s atheatcally equvalent to hoogenety (our condton (f)) s equvalent to hs Equal Tradeoffs Coparsons condton. He then shows (Harvey, 1986, p , proof of Theore 1) that an nterteporal analog of (1) s a vald representaton of preferences f only f (A-2) hs condton E hold. Snce hs Condton E s equvalent to our condton (f), that proof also establshes our Theore 1. Value functons are only deterned to wthn a postve onotonc transforaton, so any postve onotonc transforaton of the value functon fors n all of the Theores Conectures wll also satsfy the specfed condtons. See Keeney Raffa (1976, Secton 3.3.4) Krkwood (1997, Secton 9.2) for further dscusson of ths pont. Wth a stronger condton n place of condton (f), a specal case of equaton (1) holds where vz ( ) s lnear. Consder the stronger condton of tradeoffs neutralty: for any subregon, attrbute levels z z, vectors q * r, z, z, q r, ples z *, z, q r, where * z z z 0.5. Ths condton ensures that sngle-attrbute value functons wll be lnear by the followng arguent: Snce tradeoffs neutralty s a specal case of condton (f), equaton (1) stll holds n ths case. By drect substtuton, when tradeoffs neutralty holds, v[0.5 ( z z )] 0.5 [ v( z) v( z )] for any z z. Ths s Jensen s equaton (Sall, 2007, Secton 2.3), the soluton s v() z az b for constants a b when vz ( ) s contnuous. (An anonyous revewer provded valuable suggestons that helped us to develop the ateral ust presented n ths secton. Ths sae revewer also provded any valuable suggestons for the other theore conecture presentatons n ths Appendx.) APPENDIX C: ADDITIONAL DISCUSSION OF CONJECTURE 1. Let have correspondng strct ndfference relatons as prevously defned, let q, r, s denote arbtrary consequences. Consder the followng condtons on : S-4

5 (a ) Copleteness: It ust be true that q r or r q. (That s, any consequences can be copared.) (b ) Transtvty: If q r r s, then q s. (c ) Contnuty: If q r, then there exsts 0 such that for any consequence z, xy, qx y ples z r, ax z x, y r x, y ax z x, y, xy, ples q z. (d ) Dependence on any subregon: For any proper subregon of the regon wth area greater than zero, there exst consequences q r such that qx, y = rx, y for all xy,, q r. (e ) Spatal preferental ndependence: For any proper subregon of the regon wth area greater than = zero, f there exst consequences q, r, q', r' such that q r, q x, y qx, y, = rx, y for all xy,, qx, y = rx, y qx, y rx, y r x y xy,, then t ust be true that q' r'. = for all DEFINITION A-2. Consder a proper subregon of the regon wth area greater than zero, consder two consequences z( x, y) z ( x, y) that have constant levels z z, respectvely, wthn P. An attrbute level d z s a tradeoffs dvalue for z z wth respect to P f there exst functons q( x, y) r( x, y) defned outsde P such that the followng two ndfference relatons hold: ) a consequence that s equal to z wthn P equal to q( x, y ) outsde P s ndfferent to a consequence that s equal to d z wthn P equal to r( x, y ) outsde P, ) a consequence that s equal to wthn P equal to q( x, y ) outsde P s ndfferent to a consequence that s equal to z wthn P equal to r( x, y ) outsde P. d z S-5

6 (f ) Hoogenety: If d z s a tradeoffs dvalue for z z wth respect to a specfed proper subregon P of the regon wth area greater than zero, then t s also a tradeoffs dvalue for z z wth respect to any other proper subregon for whch a tradeoffs dvalue for z z exsts. Note that tradeoffs dvalues are unlkely to exst wth respect to very large subregons, snce the requred q( x, y) r( x, y) ay not exst for such cases. However, for the purposes of both practcal applcatons possble proofs of Conecture 1, we are concerned only wth subregons whch are sall relatve to the entre regon. We present a plausblty arguent for why condtons (a ) through (f ) ght ply (3), dscuss where dffcultes arse n the proof. The condtons stated n Conecture 1 are analogous to those used n Theore 1, (3) s analogous to (1). Condtons (a )-(e ) are straghtforward adaptatons of condtons (a)-(e). (Condton (e ) s analogous to utual preferental ndependence, rather than parwse preferental ndependence, because n the non-dscrete case there are not exstng dscrete subregons wth whch to create pars.) Condton (f ) s analogous to condton (f). To establsh the plausblty of Conecture 1, start wth (1) defne a / A n (1), where A s the area (for exaple, n square les) of subregon. Then (1) can be rewrtten as 1, z2,, z ) A v( z ) 1 V ( z. (A-3) Snce condtons (a ) through (f ) are analogous to condtons (a) through (f) requred for (1), t s plausble to use (1) as a startng pont for developng (3). Extend (A-3) to an attrbute that vares over the regon as follows: Partton the regon nto a unfor grd, where the two densons of the grd are desgned by x y, where the x y densons of each cell n the grd are desgnated by x y, respectvely, so that the area A of any cell s xy. Whle t s easest to vsualze ths partton f A s rectangular, the analyss can be extended to any regon that s bounded by a pecewse sooth curve, as establshed n the references gven below. If v z ) dd not vary wthn a grd ( cell, then f the assuptons for Theore 1 are assued to hold, (A-3) can be wrtten as S-6

7 V ( z1, z,, z ) ( x, y ) v[ z( x, y )] xy, (A-4) 2 1 where x y desgnate soe specfed but arbtrary pont wthn grd cell, ( x, y ), v z( x, y )] v( z ). Equaton (A-4) has the for of a specal case of a Reann su of [ ( x, y) v[ z( x, y)] over A, f v are both bounded, then ther product s also bounded. If these functons are contnuous alost everywhere (that s, except on a subset of A wth easure zero), then ( x, y) v[ z( x, y)] s Reann ntegrable over A f the boundary of A s a pecewse sooth curve. (For proofs of ths, see Apostol 1962, Secton 2.12, or Trench 2003, Theore ) If ( x, y) v[ z( x, y)] s ntegrable, the Reann su n (A-4) wll converge to a unque value (whch by defnton s the ntegral) as the partton of A s ade fner so that approaches nfnty both x y approach zero. Thus, n the lt, (A-4) would becoe V( z) ( x, y) v[ z( x, y)] dxdy, (A-5) A where V(z) s the value fro a decson-akng perspectve assocated wth the dstrbuton of the attrbute over the regon of nterest. The converse of the odel result would follow by drect substtuton fro (3). Note that n (3), ( x, y) n (A-5) has been replaced wth a(x, y) to ake the notaton ore parallel to (1). However, the unts for a n (1) (3) are dfferent. The plausblty arguent gven above s not a proof, there are two prary dffcultes nvolved n provng Conecture 1. The frst s that the partton of the regon used n (1) s fxed, as s the nuber of subregons. Therefore, we cannot be sure that assessed preferences for dfferent parttons, such as the changng parttons n (A-4) as ncreases, wll yeld the sae sngle-subregon value functon v, hence yeld a sngle coon lt for any possble Reann su. Ths eans t s not defensble to exane the lt of (1) as goes to nfnty, because a coon lt of vz ( ) ndependent of the partton s not guaranteed to exst. In the condtons for Conecture 1, we address ths wth the S-7

8 contnuty condton (c ). Whle ths s eant to guarantee convergence as the partton becoes fner, we cannot establsh that t does ths n the anner requred for a unque lt to exst. A second dffculty s establshng that the weghtng functon ax, y s Reann ntegrable. It ust be true that v[ z( x, y )] s Reann ntegrable, because z( x, y ) s Reann ntegrable by assupton, condtons (c ) (d ) ensure that v s contnuous bounded. The weghtng functon ust be bounded, otherwse a volaton of condton (d ) would occur. However, t ust also be establshed that the weghtng functon s contnuous alost everywhere, t s not clear precsely what condtons on the preference relaton wll guarantee ths. Condton (c ) establshes the contnuty of v, but does not pose ths property on ax, y. In practcal applcatons, t s dffcult to thnk of a realstc weghtng functon that s not contnuous alost everywhere, but we are unable to prove that condtons (a ) through (f ) establsh ths property. Harvey Østerdal (2011) provde addtonal dscusson of the steps necessary to establsh a result analogous to (3) n the context of contnuous te decsons. APPENDIX D: ADDITIONAL DISCUSSION OF THEOREM 2. We thank a revewer for pontng out that the result n Theore 2 can be developed n ether of two ways: By usng results fro Goran (1968) then applyng hoogenety condtons, or by applyng preferental ndependence condtons across attrbutes to Harvey s (1995) value odel. We use the forer approach here. Modfy the notaton presented earler so that now desgnates the vector of n attrbutes,, 2, 1 n n subregon (referred to as the attrbute vector for subregon ), desgnates the vector 1,, of the th attrbute across the subregons (referred to as the subregon, 2 vector for attrbute ), where z z are vectors of specfc levels of, respectvely. Let have correspondng strct preference ndfference relatons as prevously defned, let q, r, s S-8

9 denote arbtrary consequences. Consder the followng condtons on, whch are analogous to those used n Theore 1. (a ) Copleteness: It ust be true that q r or r q. (That s, any consequences can be copared.) (b ) Transtvty: If q r r s, then q s. (c ) Contnuty: If q r, then there exsts 0 such that for any z, ax z q, ples z r, ax z r ples q z., (d ) Dependence on each attrbute-subregon cobnaton: For each subregon attrbute, there exst consequences (e ) Preferental ndependence: q, z r, z such that q, r, z z. 1) Subregon preferental ndependence: For any subregon,,,,, q z r z for all z, 2) Attrbute preferental ndependence: For any attrbute,,,,, q z r z for all z. q s r s for soe s ples q s r s for soe s ples DEFINITION A-3. Preferences over the regon of nterest wth respect to a set of attrbute vectors,, 1 for subregons 1,, are ultattrbute hoogeneous f, when two alternatves that dffer only n the attrbute levels for a specfed subregon are ndfferent, then the sae ndfference relaton holds for those sae attrbute levels n any subregon. (In ths defnton, the scalar attrbute for each subregon consdered n Theore 1 s replaced wth an attrbute vector for each subregon). (f ) Hoogenety: s ultattrbute hoogeneous wth respect to,, 1. Condton (f ) plays a role for ultple attrbutes slar to the sngle-attrbute hoogenety condton (f) prevously defned for Theore 1. Condton (f) assues there s a sngle attrbute that has a S-9

10 tradeoffs dvalue, but when there are ultple attrbutes there s not an unabguous eanng for the tradeoffs dvalue. Therefore, we express hoogenety here consderng levels of ultple attrbutes. Conversely, the defnton gven here requres consderatons of tradeoffs between attrbutes, therefore s not applcable to the sngle-attrbute consequences used n Theore 1. We frst show that condton (e ) ples an addtve for for V( z ). The frst step s to show that the set consstng of any par of attrbute-subregon cobnatons s preferentally ndependent of ts copleent. Let ac bd represent two such cobnatons, where a b are arbtrary dstnct subregons, c d are arbtrary dstnct attrbutes. Fro condton (e ), each of a, b, c, d s preferentally ndependent of ts copleent. Fro Theore 1 n Goran (1968), the unon of a d s preferentally ndependent of ts copleent, as s the unon of b c. The ntersecton of these two unons s,, snce both unons are preferentally ndependent of ac bd ther copleents, Theore 1 n Goran also ples that, ust be preferentally ndependent of ts copleent. Snce the choces of a, b, c, d were arbtrary, any par of attrbute-subregon cobnatons s preferentally ndependent of ts copleent for dstnct attrbutes subregons. It s then straghtforward to also use Theore 1 n Goran to show that a par of attrbute-subregon cobnatons s preferentally ndependent of ts copleent n the case where ether the attrbute or the subregon s coon to both cobnatons. Hence, fro nductve applcaton of Goran's results, also ac bd stated as a Corollary on page 114 of Keeney Raffa (1976), the are utually preferentally ndependent, therefore: n V ( z) k v ( z ). (A-6) 1 1 Snce the attrbutes are ultattrbute hoogeneous by condton (f ), t follows fro an analogous arguent to the one gven n the proof of Theore 1 that sngle-attrbute value functons v cannot depend on the subregon, hence the followng equaton holds n ths case: S-10

11 n V( z) k v ( z ). (A-7) 1 1 To show that k a b, hence (4) holds, frst assue wthout loss of generalty that the subregons attrbutes are labeled so that the largest scalng constant s k 11. Consder two hypothetcal alternatves: 1) all the attrbute-subregon cobnatons except 11 1 are set to arbtrary levels, 11 s set to ts least preferred level so that v 1 ( z 11 ) 0 n (A-7), 1 s set to ts ost preferred level so that v ( z 1 ) 1, 2) another hypothetcal alternatve wth all the attrbute-subregon cobnatons except 11 1 set to the sae arbtrary levels as the frst alternatve, 1 set to ts least preferred level so v ( z 1 ) 0, 11 set to the level z 11 such that the two alternatves are equally preferred. Then equatng the values for each of these two alternatves calculated usng (A-7) cancellng coon ters results n ( for any 1. However, by condton (f ), f ths equaton holds for k11 v1 z11) k1 subregon 1, then the sae level z 11 ust ake the analogous equaton true for any subregon, hence k 1 v 1 ( z 11 ) k for any. Defne v 1 ( z 11 ) b a k 1. Substtutng these defntons nto k 1 v z k gves k ab. Substtute ths nto (A-7), (4) follows. The converse of the odel 1 ( 11 ) result follows by drect substtuton fro (4). APPENDIX E: ADDITIONAL DISCUSSION OF CONJECTURE 2. Let have correspondng strct ndfference relatons as prevously defned, let q, r, s denote arbtrary consequences. Let z desgnate a Reann ntegrable functon such that z x, y I for all, xy n the regon. Let z z be defned as prevously. As n Conecture 1, assue that the boundary of the regon s a pecewse sooth curve. Consder the followng condtons on, whch are analogous to those used n Theore 2: S-11

12 (a ) Copleteness: It ust be true that q r or r q. (That s, any consequences can be copared.) (b ) Transtvty: If q r r s, then q s. (c ) Contnuty: If q r, then there exsts 0 such that for any consequence z, x, y, ples z r, ax z x, y r x, y ax z x, y q x, y x, y, ples q z. (d ) Dependence on any attrbute n any subregon: For any attrbute proper subregon of the regon wth area greater than zero, there exst consequences q r such that x, y x, y xy,, q x, y = r x, y for all xy, (e ) Preferental ndependence:, q r. q = r for all 1) Spatal preferental ndependence: For any proper subregon of the regon wth area greater than zero, f there exst consequences q, r, q', r' such that q r, x, y x, y q = q r x, y = r x, y for all xy,, q x, y = r x, y x, y x, y xy,, then t ust be true that q' r'. q = r for all 2) Parwse attrbute preferental ndependence: For any two attrbutes, q, q, r, r, s s for soe s ples q, q, r, r, z z for all DEFINITION A-4. Consder a proper subregon of the regon wth area greater than zero, consder two consequences q r. Preferences over the regon are contnuously ultattrbute hoogeneous f q r, q( x, y) r ( x, y) for ( x, y) P, ( x, y) q1,, q n z. q x y r r r for (, ) 1,, n ( x, y) P where q,, 1 q n r 1,, r n are constants ples that for any consequences z z subregon such that z( x, y) z ( x, y) for ( x, y) P, z ( x, y) q1,, q n ( x, y) r1,, r n z for ( x, y) P then z z. (f ) : Hoogenety: s contnuously ultattrbute hoogeneous over the regon. S-12

13 The developent of Conecture 2 fro Theore 2 s analogous to the developent of Conecture 1 fro Theore 1. A slar plausblty arguent can be gven for Conecture 2, n whch a Reann su analogous to equaton (A-4) s developed. A further developent analogous to the Conecture 1 reasonng leads to (5), drect substtuton fro (5) yelds the converse of the result. However, analogous dffcultes to those nvolved n provng Conecture 1 arse here as well. APPENDIX F: EXTENSIONS OF THE PREFERENCE MODELS TO ADDRESS UNCERTAINTY. In ths Appendx, we exane the case n whch the consequences of alternatves are uncertan. We assue that probabltes can be assgned to the possble consequences of each alternatve, we wsh to rank alternatves by ther overall desrablty usng ther expected utlty, coputed as the expected value of a sngle attrbute (or ultattrbute) utlty functon. The preference condtons n Secton 3 can be extended to decsons under uncertanty to deterne the requreents for an addtve utlty functon. The prary dfference s that the preferental ndependence condton dscussed n Secton 3 ust be replaced by a consderably stronger condton called addtve ndependence. DEFINITION A-5. Addtve ndependence wth respect to over the regon of nterest holds f the rank orderng for any set of alternatves depends only on the argnal probablty dstrbutons for each alternatve over the levels z 1, z 2,, z of n each of the subregons 1,...,. We frst consder the case analogous to Theore 1, n whch the level for the sngle attrbute does not vary wthn each subregon, then consder the non-dscrete case analogous to Conecture 1. In decson probles wth uncertanty, we axze expected utlty usng a utlty functon nstead of axzng a value functon. It s straghtforward to show that utlty s gven by U( z, z,, z ) a u( z ) 1 2 (A-8) 1 f only f addtve ndependence s satsfed, condtons analogous to (a)-(d) (f) n Appendx B are et (Fshburn 1965). See Keeney Raffa (1976, Sectons , pp ) Krkwood (1997, pp ) for further background, ncludng assessent procedures for utlty functons. S-13

14 Analogously to Conecture 1, we conecture that t s possble to extend (A-8) to stuatons where the sngle attrbute s defned at any locaton ( xy, ) wthn the regon of nterest. The correspondng utlty functon s gven by: U( z) a( x, y) u[ z( x, y)] dxdy (A-9) A f only f addtve ndependence, condtons analogous to (a )-(d ) (f ) n Appendx C are et. Wth these condtons, a plausblty arguent for equaton (A-9) can be obtaned usng reasonng analogous to the dscusson n Appendx C. However, as n Conecture 1, ths plausblty arguent would not consttute a proof. Results analogous to Theore 2 Conecture 2, wth a ultple attrbute utlty functon analogous assuptons can also be specfed. The requred preference assuptons for an addtve utlty functon are strong, however, ay not be approprate n soe decson stuatons. One possble approach to developng ore general utlty functon fors wth less restrctve requreents would be to construct the utlty functon over the value functons that were developed n Secton 3 usng ethods such as those presented by Dyer Sarn (1982) Matheson Abbas (2005). If the condtons needed for a value functon of the for gven n (1) hold, then n the case wth dscrete subregons, a utlty functon U could be constructed over the value functon n (1) wth the for: U V ( z1, z2,, z) U av( z ). (A-10) 1 Slarly, n the non-dscrete case, the utlty functon could be constructed over the value functon n (3) wth the for: U V ( z) U a( x, y) v[ z( x, y)] dxdy A. (A-11) These are less restrctve than (A-8) (A-9) n that they have unspecfed utlty functons U, hence requre less restrctve preference condtons than (A-8) or (A-9). Stard utlty functon assessent procedures can be odfed to deterne U. For exaple, a possble approach for assessng ths utlty S-14

15 functon s to dentfy the potental decson consequences wth the hghest lowest possble values, vsualze a hypothetcal bnary gable between the wth probablty p of the hghest-value consequence occurrng probablty 1-p of the lowest-value consequence occurrng. The utlty of the value placed on a specfed consequence could then be deterned by fndng the value of p for whch the decson aker s ndfferent between the specfed consequence the gable. By equatng expected utltes, the assessed p would be the utlty assocated wth the specfed consequence. Addtonal References Apostol, T. M Calculus, Volue II: Calculus of Several Varables wth Applcatons to Probablty Vector Analyss, Blasdell Publshng Copany, New York. Debreu, G Topologcal Methods n Cardnal Utlty Theory. In Matheatcal Methods n the Socal Scences, 1959, K. J. Arrow, S. Karln, P. Suppes (eds.), Stanford Unversty Press, Stanford, CA, 1960, pp Downloaded fro on Nov. 29, Dyer, J. S., R. K. Sarn Relatve Rsk Averson. Manageent Scence 28(8) Harvey, C. M., L. P. Østerdal, Integral Utlty Models for Outcoes over Contnuous Te. Workng paper. Cted wth persson of the authors. Matheson, J. E., A. E. Abbas Utlty transversalty: A value-based approach. Journal of Mult- Crtera Decson Analyss 13(5-6) Sall, C. G Functonal Equatons How to Solve The, Sprnger Scence+Busness Meda, New York. Trench, W. F Introducton to Real Analyss. Pearson Educaton. Upper Saddle Rver, NJ. Free Edton 1.05 downloaded on fro S-15