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1 Supplemental Materal Decson Analyss wth Geographcally Varyng Outcomes: Preference Models Illustratve Applcatons Jay Smon, * Crag W. Krkwood, L. Robn Keller Operatons Research, Appendx A. Preferental Independence In decson analyss, specfyng a multattrbute value or utlty functon requres assessments from a decson maker, ths can be dffcult because t requres the determnaton of an n-dmensonal functon. To smplfy ths, researchers have establshed condtons on preferences under whch the form of the value or utlty functon s smplfed. One of these condtons s preferental ndependence. A subset of 1, 2, n, when n 3, s defned to be preferentally ndependent of ts complement f the rank orderng of alternatves wth no uncertanty that have common levels for the complementary attrbutes does not depend on those common levels. If ths property holds for all subsets of 1, 2,, n, then mutual preferental ndependence s sad to hold, n ths case: n V ( z, z,, z ) a v ( z ) 1 2, (A-1) n 1 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). Gorman (1968) derves several condtons related to mutual preferental ndependence. In partcular, t follows from hs results that f every par of attrbutes s preferentally ndependent of the remanng attrbutes then mutual preferental ndependence holds, or, even more specfcally, f {, +1 } s preferentally ndependent of ts complement for = 1, 2,, n-1, then mutual preferental ndependence holds. These results can substantally reduce the number of assessments that must be made n applcatons to establsh that (A-1) s vald. See Keeney Raffa (1976 Sectons ) for more detals. Appendx B: Addtonal Dscusson 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 * Defense Resources Management Insttute, Naval Postgraduate School, Monterey, CA 93943, rsmon@nps.edu W. P. Carey School of Busness, Arzona State Unversty, Tempe, A , crag.krkwood@asu.edu The Paul Merage School of Busness, Unversty of Calforna, Irvne, CA , LRKeller@uc.edu S-1

2 strct relaton ( more 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) Completeness: It must be true that q r or r q. (That s, any consequences can be compared.) (b) Transtvty: If q r r s, then q s. (c) Contnuty: If q r, then there exsts 0 such that for any z, max z q mples z r, max z r mples q z. (d) Dependence on each subregon: For each subregon, there exst q, r, z z. q, z, r z such that (e) Parwse spatal preferental ndependence: For any two subregons, q, q, s r, r, s for some s mples q, q, z r, r, z for all z, z. Ths condton mples mutual preferental ndependence, as descrbed above. Snce the ndependence descrbed here s for a sngle attrbute across multple subregons, t s labeled spatal to dstngush t from preferental ndependence across multple attrbutes, whch s consdered n Theorem 2. DEFINITION A-1. An attrbute level there exst vectors q (f) Homogenety: If vectors q, md z s called a tradeoffs mdvalue of z ' r such that z', q md z, r z md, z'', md z s a tradeoffs mdvalue of z ' r such that z', md z, q r. z '' n subregon f z '' n subregon, then for any subregon q r t must be true that z md, z'', q r. (An equvalent defnton s that when two amounts z, z have a tradeoffs mdvalue n a subregon also n a subregon, then they have the same tradeoffs mdvalue n subregon as n subregon.) A value functon exsts over the set of consequences f only f s complete, transtve, contnuous, as specfed n condtons (a)-(c). (Ths s a specal case of Debreu (1954, 1964), who uses a more general topologcal space.) It follows drectly from Debreu s (1960) theorem that for a regon wth three or more subregons, a value functon can be wrtten as: m V ( z1, z2,, z ) a v ( z ) m 1 f only f condtons (a)-(e) above hold, where v s a sngle attrbute value functon over (A-2) z, a 0 s the weght assocated wth subregon. (A-2) has the same form as (A-1), but the z n (A-2) S-2

3 represent the levels of the same 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 Comparsons condton, he notes that a condton that s mathematcally equvalent to homogenety (our condton (f)) s equvalent to hs Equal Tradeoffs Comparsons condton. He then shows (Harvey, 1986, p , proof of Theorem 1) that an ntertemporal 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 Theorem 1. Value functons are only determned to wthn a postve monotonc transformaton, so any postve monotonc transformaton of the value functon forms n all of the Theorems 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 z z z, vectors q * r, z, z, q r, mples z *, z, q r, where 0.5. Ths condton ensures that sngle-attrbute value functons wll be lnear by the followng argument: 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 (Small, 2007, Secton 2.3), the soluton s v() z az b for constants a b when vz ( ) s contnuous. (An anonymous revewer provded valuable suggestons that helped us to develop the materal ust presented n ths secton. Ths same revewer also provded many valuable suggestons for the other theorem conecture presentatons n ths supplemental materal.) Appendx C: Addtonal Dscusson of Conecture 1 Let have correspondng strct ndfference relatons as prevously defned, let q, r, s denote arbtrary consequences. Consder the followng condtons on : (a ) Completeness: It must be true that q r or r q. (That s, any consequences can be compared.) (b ) Transtvty: If q r r s, then q s. S-3

4 (c ) Contnuty: If q r, then there exsts 0 such that for any consequence z, xy, qx mples z r, max z x, r x, max z x, y, xy, mples 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, = rx, 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, qx, r x, = rx, for all xy,, qx, = rx, qx, rx, xy,, then t must 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) attrbute level md z q( x, y) r( x, y) that have constant levels z z, respectvely, wthn P. An s a tradeoffs mdvalue for z z wth respect to P f there exst functons 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 md 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. (f ) Homogenety: If md z s a tradeoffs mdvalue 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 mdvalue for z z wth respect to any other proper subregon for whch a tradeoffs mdvalue for z z exsts. Note that tradeoffs mdvalues are unlkely to exst wth respect to very large subregons, snce the requred q( x, y) r( x, y) md z may 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 small relatve to the entre regon. We present a plausblty argument for why condtons (a ) through (f ) mght mply (3), dscuss where dffcultes arse n the proof. The condtons stated n Conecture 1 are analogous to those used n Theorem 1, (3) s analogous to (1). Condtons (a )-(e ) are straghtforward adaptatons of condtons (a)-(e). (Condton (e ) s analogous to mutual preferental ndependence, rather than parwse preferental ndependence, because n the nondscrete case there are not exstng dscrete subregons wth whch to create pars.) Condton (f ) s analogous to condton (f). where To establsh the plausblty of Conecture 1, start wth (1) defne A s the area (for example, n square mles) of subregon. Then (1) can be rewrtten as a / A n (1), S-4

5 m 1, z2,, zm ) 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 unform grd, where the two dmensons of the grd are desgned by x y, where the x y dmensons of each cell n the grd are desgnated by x y, respectvely, so that the area A of any cell s xy. It s easest to vsualze ths partton f A s rectangular, though the analyss can be extended to any regon that s bounded by a pecewse smooth curve, as establshed n the references gven below. If v z ) dd not vary wthn a grd cell, then f the assumptons for Theorem 1 are assumed to hold, (A-3) can be wrtten as m V ( z1, z,, z ) ( x, y ) v[ z( x, y )] xy, (A-4) where 2 x m 1 y desgnate some specfed but arbtrary pont wthn grd cell, ( x, y ), v z( x, y )] v( z ). Equaton (A-4) has the form of a specal case of a Remann sum 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 almost everywhere (that s, except on a subset of A wth measure zero), then ( x, y) v[ z( x, y)] s Remann ntegrable over A f the boundary of A s a pecewse smooth curve. (For proofs of ths, see Apostol 1962, Secton 2.12, or Trench 2003, Theorem ) If ( x, y) v[ z( x, y)] s ntegrable, the Remann sum n (A-4) wll converge to a unque value (whch by defnton s the ntegral) as the partton of A s made fner so that m Thus, n the lmt, (A-4) would become ( approaches nfnty both x y approach zero. V( z) ( x, y) v[ z( x, y)] dxdy, (A-5) A where V(z) s the value from a decson-makng perspectve assocated wth the dstrbuton of the attrbute over the regon of nterest. The converse of the model result would follow by drect substtuton from (3). Note that n (3), ( x, y) n (A-5) has been replaced wth a(x, y) to make the notaton more parallel to (1). However, the unts for a n (1) (3) are dfferent. The plausblty argument gven above s not a proof, there are two prmary dffcultes nvolved n provng Conecture 1. The frst s that the partton of the regon used n (1) s fxed, as s the number of subregons. Therefore, we cannot be sure that assessed preferences for dfferent parttons, such as the changng parttons n (A-4) as m ncreases, wll yeld the same sngle-subregon value functon v, hence yeld a sngle common lmt for any possble Remann sum. Ths means t s not defensble to examne the lmt of (1) as m goes to nfnty, because a common lmt of vz ( ) ndependent S-5

6 of the partton s not guaranteed to exst. In the condtons for Conecture 1, we address ths wth the contnuty condton (c ). Ths s meant to guarantee convergence as the partton becomes fner, but we cannot establsh that t does ths n the manner requred for a unque lmt to exst. A second dffculty s establshng that the weghtng functon ax, y s Remann ntegrable. It must be true that v[ z( x, y )] s Remann ntegrable, because z( x, y ) s Remann ntegrable by assumpton, condtons (c ) (d ) ensure that v s contnuous bounded. The weghtng functon must be bounded, otherwse a volaton of condton (d ) would occur. However, t must also be establshed that the weghtng functon s contnuous almost 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 mpose ths property on ax, y. In practcal applcatons, t s dffcult to thnk of a realstc weghtng functon that s not contnuous almost 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 tme decsons. Appendx D: Addtonal Dscusson of Theorem 2 We thank a revewer for pontng out that the result n Theorem 2 can be developed n ether of two ways: By usng results from Gorman (1968) then applyng homogenety condtons, or by applyng preferental ndependence condtons across attrbutes to Harvey s (1995) value model. We use the former approach here. Modfy the notaton presented earler so that n now desgnates the vector of n attrbutes,, 1 2, n subregon (referred to as the attrbute vector for subregon ), the vector, 2 m desgnates 1,, of the th attrbute across the m subregons (referred to as the subregon 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 denote arbtrary consequences. Consder the followng condtons on, whch are analogous to those used n Theorem 1. (a ) Completeness: It must be true that q r or r q. (That s, any consequences can be compared.) (b ) Transtvty: If q r r s, then q s. (c ) Contnuty: If q r, then there exsts 0 such that for any z, z r, max z r mples q z., max z q, mples S-6

7 (d ) Dependence on each attrbute-subregon combnaton: For each subregon attrbute, there exst consequences,, q, z r, z. (e ) Preferental ndependence: q z r z such that 1) Subregon preferental ndependence: For any subregon,,,,, z, q z r z for all 2) Attrbute preferental ndependence: For any attrbute,,,,, q z r z for all z. q s r s for some s mples q s r s for some s mples DEFINITION A-3. Preferences over the regon of nterest wth respect to a set of attrbute vectors,, 1 m for subregons 1,, m are multattrbute homogeneous f, when two alternatves that dffer only n the attrbute levels for a specfed subregon are ndfferent, then the same ndfference relaton holds for those same attrbute levels n any subregon. (In ths defnton, the scalar attrbute for each subregon consdered n Theorem 1 s replaced wth an attrbute vector for each subregon). (f ) Homogenety: s multattrbute homogeneous wth respect to,, 1 m. Condton (f ) plays a role for multple attrbutes smlar to the sngle-attrbute homogenety condton (f) prevously defned for Theorem 1. Condton (f) assumes there s a sngle attrbute that has a tradeoffs mdvalue, but when there are multple attrbutes there s not an unambguous meanng for the tradeoffs mdvalue. Therefore, we express homogenety here consderng levels of multple attrbutes. Conversely, the defnton gven here requres consderatons of tradeoffs between attrbutes, therefore s not applcable to the sngle-attrbute consequences used n Theorem 1. We frst show that condton (e ) mples an addtve form for V( z ). The frst step s to show that the set consstng of any par of attrbute-subregon combnatons s preferentally ndependent of ts complement. Let ac bd represent two such combnatons, where a b are arbtrary dstnct subregons, c d are arbtrary dstnct attrbutes. From condton (e ), each of a, b, c, d s preferentally ndependent of ts complement. From Theorem 1 n Gorman (1968), the unon of a d s preferentally ndependent of ts complement, as s the unon of b c. The ntersecton of these two unons s ac, bd, snce both unons are preferentally ndependent of ther complements, Theorem 1 n Gorman also mples that, must be preferentally ndependent of ts complement. Snce the choces of a, b, c, d were arbtrary, any par of attrbute-subregon combnatons s preferentally ndependent of ts complement for dstnct attrbutes subregons. It s then straghtforward to also use Theorem 1 n Gorman to show that a par of attrbute-subregon combnatons s preferentally ndependent of ts complement n the case where ether the attrbute or the ac bd S-7

8 subregon s common to both combnatons. Hence, from nductve applcaton of Gorman's results, also stated as a Corollary on page 114 of Keeney Raffa (1976), the ndependent, therefore: 1 1 are mutually preferentally m n V ( z) k v ( z ). (A-6) Snce the attrbutes are multattrbute homogeneous by condton (f ), t follows from an analogous argument to the one gven n the proof of Theorem 1 that sngle-attrbute value functons depend on the subregon, hence the followng equaton holds n ths case: 1 1 v cannot m n V( z) k v ( z ). (A-7) To show that k a b, hence (4) holds, frst assume 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 combnatons 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 most preferred level so that v ( z 1 ) 1, 2) another hypothetcal alternatve wth all the attrbute-subregon combnatons except 11 1 set to the same arbtrary levels as the frst alternatve, level so v ( z 1 ) 0, 11 set to the level 1 set to ts least preferred 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 common terms results n ( for any 1. However, by condton (f ), f ths equaton holds for k11 v1 z11) k1 subregon 1, then the same level z 11 must make the analogous equaton true for any subregon, hence k 1 v 1 ( z 11 ) k for any. Defne b v 1 ( z 11 ) a k 1. Substtutng these defntons nto k 1 v 1 ( z 11 ) k gves k ab. Substtute ths nto (A-7), (4) follows. The converse of the model result follows by drect substtuton from (4). Appendx E: Addtonal Dscusson of Conecture 2 Let have correspondng strct ndfference relatons as prevously defned, let q, r, s denote arbtrary consequences. Let for all xy, n the regon. Let z desgnate a Remann ntegrable functon such that, z z x y I z be defned as prevously. As n Conecture 1, assume that the boundary of the regon s a pecewse smooth curve. Consder the followng condtons on, whch are analogous to those used n Theorem 2: (a ) Completeness: It must be true that q r or r q. (That s, any consequences can be compared.) (b ) Transtvty: If q r r s, then q s. S-8

9 (c ) Contnuty: If q r, then there exsts 0 such that for any consequence z, x, y, mples z r, max z x, r x, max z x, y q x, y x, y, mples 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, x, xy,, q x, = r x, for all xy,, q r. (e ) Preferental ndependence: 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, x, r x, = r x, for all xy,, q x, = r x, x, x, xy,, then t must be true that q' r'. q = q q = r for all 2) Parwse attrbute preferental ndependence: For any two attrbutes, q, q, r, r, s s for some s mples 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 multattrbute homogeneous f q r, q( x, y) r ( x, y) for ( x, y) P ( x, y) q1,, q n r ( x, y) r1,, r n for ( x, y) P where q,, 1 q n r 1,, r n are constants mples 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 z ( x, y) r1,, r n for ( x, y) P then z z. z., q (f ) Homogenety: s contnuously multattrbute homogeneous over the regon. The development of Conecture 2 from Theorem 2 s analogous to the development of Conecture 1 from Theorem 1. A smlar plausblty argument can be gven for Conecture 2, n whch a Remann sum analogous to equaton (A-4) s developed. A further development analogous to the Conecture 1 reasonng leads to (5), drect substtuton from (5) yelds the converse of the result. However, analogous dffcultes to those nvolved n provng Conecture 1 arse here as well. Appendx F: Extensons of the Preference Models to Address Uncertanty In ths Appendx, we examne the case n whch the consequences of alternatves are uncertan. We assume 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, computed as the expected value of a sngle attrbute (or multattrbute) utlty functon. The preference condtons n Secton 3 can be extended to decsons under uncertanty to determne the requrements for an addtve utlty functon. The S-9

10 prmary dfference s that the preferental ndependence condton dscussed n Secton 3 must 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 margnal probablty dstrbutons for each alternatve over the levels z 1, z 2,, z m of n each of the subregons 1,..., m. We frst consder the case analogous to Theorem 1, n whch the level for the sngle attrbute does not vary wthn each subregon, then consder the nondscrete case analogous to Conecture 1. In decson problems wth uncertanty, we maxmze expected utlty usng a utlty functon nstead of maxmzng a value functon. It s straghtforward to show that utlty s gven by m U( z, z,, z ) a u( z ) 1 2 (A-8) m 1 f only f addtve ndependence s satsfed, condtons analogous to (a)-(d) (f) n Appendx B are met (Fshburn 1965). See Keeney Raffa (1976, Sectons , pp ) Krkwood (1997, pp ) for further background, ncludng assessment procedures for utlty functons. 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 met. Wth these condtons, a plausblty argument for equaton (A-9) can be obtaned usng reasonng analogous to the dscusson n Appendx C. However, as n Conecture 1, ths plausblty argument would not consttute a proof. Results analogous to Theorem 2 Conecture 2, wth a multple attrbute utlty functon analogous assumptons can also be specfed. The requred preference assumptons for an addtve utlty functon are strong, however, may not be approprate n some decson stuatons. One possble approach to developng more general utlty functon forms wth less restrctve requrements would be to construct the utlty functon over the value functons that were developed n Secton 3 usng methods such as those presented by Dyer Sarn (1982) Matheson Abbas (2005). If the condtons needed for a value functon of the form 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 form: S-10

11 m U V ( z1, z2,, zm) U av( z ) (A-10) 1 Smlarly, n the nondscrete case, the utlty functon could be constructed over the value functon n (3) wth the form: 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 assessment procedures can be modfed to determne U. For example, a possble approach for assessng ths utlty functon s to dentfy the potental decson consequences wth the hghest lowest possble values, vsualze a hypothetcal bnary gamble between them 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 determned by fndng the value of p for whch the decson maker s ndfferent between the specfed consequence the gamble. By equatng expected utltes, the assessed p would be the utlty assocated wth the specfed consequence. Addtonal References See references lst n the man paper. Apostol TM (1962) Calculus, Volume II: Calculus of Several Varables wth Applcatons to Probablty Vector Analyss (Blasdell Publshng Company, New York). Debreu G (1960) Topologcal methods n cardnal utlty theory. In Arrow LJ, Karln S, Suppes P (eds.), Mathematcal Methods n the Socal Scences, 1959, (Stanford Unversty Press, Stanford, CA, 1960), pp Downloaded from on Nov. 29, Dyer JS, Sarn RK (1982) Relatve rsk averson. Management Scence 28(8): Fshburn, PC (1965) Independence n utlty theory wth whole product sets. Oper. Res. 13(1): Harvey CM, Østerdal LP (2011) Integral utlty models for outcomes over contnuous tme. Workng paper. Cted wth permsson of the authors. Matheson JE, Abbas AE (2005) Utlty transversalty: A value-based approach. J. Mult-Crtera Decson Anal. 13(5-6): S-11

12 Small CG (2007) Functonal Equatons How to Solve Them (Sprnger Scence+Busness Meda, New York). Trench WF (2003) Introducton to Real Analyss (Pearson Educaton. Upper Saddle Rver, NJ). Free Edton Downloaded on from S-12