DISPERSION OF GROUP JUDGMENTS

Transcription

1 ISAHP 5, Hoolulu, Hawa, July 8-, 3 DISPERSION OF ROUP JUDMENTS Thomas L. Saaty Saaty@atz.ptt.edu Lus. Vargas lgvargas@ptt.edu The Joseph M. Katz raduate School of Busess Uversty of Pttsburgh Pttsburgh, PA 56 Keywords: recprocal uform dstrbuto, geometrc mea, geometrc dsperso, group cohesveess, group laso, prcpal rght egevector, beta dstrbuto. Summary: To acheve a decso wth whch the group s satsfed, the group members must accept the judgmets, ad ultmately the prortes. Ths requres that (a) the judgmets be homogeeous, ad (b) the prortes of the dvdual group members be compatble wth the group prortes. There are three levels whch the homogeety of group preferece eeds to be cosdered: () for a sgle pared comparso (moogeety), () for a etre matrx of pared comparsos (multgeety), ad (3) for a herarchy or etwor (omgeety). I ths paper we study moogeety ad the mpact t has o group prortes.. Itroducto I all facets of lfe groups of people get together to mae decsos. The group members may or may ot be agreemet about some ssues ad that s reflected how homogeeous the group s ts thg. I the AHP groups mae decsos by buldg a herarchy together ad provdg judgmets expressed o a to 9 dscrete scale havg the recprocal property. Codo et al. (3) metoed that there are four dfferet ways whch groups estmate weghts the AHP: cosesus, vote or compromse, geometrc mea of the dvdual judgmets, ad weghted arthmetc mea. The frst three deal wth judgmets of dvduals whle the last deals wth the prortes derved from the judgmets. To acheve a decso wth whch the group s satsfed, the judgmets, ad ultmately the prortes, must be accepted by the group members. Ths requres that (a) the judgmets be homogeeous, ad (b) the prortes of the dvdual group members be compatble wth the group prortes. There are three levels whch the homogeety of group preferece eeds to be cosdered: () for a sgle pared comparso (moogeety), () for a etre matrx of pared comparsos (multgeety), ad (3) for a herarchy or etwor (omgeety). Moogeety relates to the dsperso of the judgmets aroud ther geometrc mea. The geometrc mea of group judgmets s the mathematcal equvalet of cosesus f all the members are cosdered equal. Otherwse oe would use the weghted geometrc mea. Aczel ad Saaty (983) showed that the oly mathematcally vald way to sythesze recprocal judgmets preservg the recprocal codto s the geometrc mea. If the group judgmets for a sgle pared comparso are too dspersed,.e., they are ot close to ther geometrc mea, the resultg geometrc mea may ot be used as the represetatve judgmet for the group. Multgeety relates to the compatblty dex of the prorty vectors. The closeess of two prorty vectors v = ( v,..., v ) T ad w= ( w,..., w ) T ca be tested through ther compatblty dex (Saaty, 994) gve by ev T W T e V = v v ad, where s the Hadamard or elemetwse product, ( j)

2 ( w wj) W =. Note that for a recprocal matrx A = ( a ) wth prcpal egevalue λ max ad correspodg rght egevector w= ( w,..., w ), e T A W T e = λ max. Thus, oe ca test the compatblty of each dvdual vector wth that derved from the group judgmets. A homogeeous group should have compatble dvduals. It s clear that homogeety at the pared comparsos level mples compatblty at the group level, but the coverse s ot always true. At the herarchy or etwor level, t appears that t s more meagful to spea of compatblty tha of homogeety. The ma thrust of ths paper s to study moogeety. Dsperso judgmets leads to volatos of Pareto Optmalty at both the parwse comparso level ad/or the etre matrx from whch prortes are derved. Ramaatha ad aesh (994) explored two methods of combg judgmets herarches but they volated the Pareto Optmalty Prcple for parwse comparsos (Saaty ad Vargas, 3), ad hece, they correctly cocluded that the geometrc mea volates Pareto Optmalty. Pareto Optmalty at the parwse level s ot suffcet to esure Pareto Optmalty at the prorty level. Fudametally, Pareto Optmalty meas that f all dvduals prefer A to B the so should the group. The group may be homogeeous some pared comparsos ad heterogeeous others thus volatg Pareto Optmalty. The degree of volato of Pareto Optmalty ca be measured by computg compatblty alog the rows, whch yelds a vector of compatblty values. What does oe do whe a group s ot homogeeous all ts comparsos? Lac of homogeety (heterogeety) o some ssues may lead to breag up the group to smaller homogeeous groups. How should oe separate the group to homogeeous subgroups? Sce homogeety relates to dsperso aroud the geometrc mea, ad dsperso tself volves ucertates, how much of the dsperso s ate ad how much s ose that whe fltered oe ca spea of true homogeety? I other words, how does oe separate radom cosderatos from commtted belefs? Dsperso at the sgle pared comparso level affects the prortes obtaed by each group member dvdually ad could lead to volatg Pareto Optmalty. Should oe combe or sythesze the prortes of the dvduals to obta the group prorty or should oe combe ther judgmets? Here we develop a way to test moogeety,.e., how homogeeous the judgmets of the members of a group are for each judgmet they gve respose to pared comparsos. Ths s doe by dervg a measure of the dsperso of the judgmets based o the geometrc mea. Computg the dsperso aroud the geometrc mea requres a multplcatve approach rather tha the usual addtve expected value used to calculate momets aroud the arthmetc mea. Ths leads to a ew multplcatve or geometrc expected value used to defe the cocept of geometrc dsperso. The geometrc dsperso of a fte set of values s gve by the geometrc mea of the ratos of the values to ther geometrc mea, f the rato s greater tha, or the recprocal, f the rato s less tha or equal to. Ths measure of varablty or dsperso of the judgmets aroud the geometrc mea allows us to (a) determe f the geometrc mea of the judgmets of a group ca be used as the sytheszed group judgmet, (b) f the geometrc mea caot be used, dvde the group to subgroups accordg to ther geometrc dsperso, ad (c) measure the varablty of the prortes correspodg to the matrx of judgmets sytheszed for the group. Basa (988) developed a test of moogeety usg a lelhood rato crtero appled to the etre matrx of judgmets, ot a sgle judgmet. I addto, Basa s test s ot teded to aswer the questo: should the geometrc mea be used as the group judgmet? Our test exames the homogeety of each set of pared comparsos, ad t wll allow us to detfy dvdual pared comparsos o whch the group dverges. I geeral, uless a group decdes through cosesus whch judgmets to assg respose to a pared comparso, the dvdual members may gve dfferet judgmets. We eed to fd f the dsperso of ths set of judgmets s a ormal occurrece the group behavor. To do ths, we compare the dsperso of the group wth the dsperso of a group provdg radom resposes to the pared comparso. Thus, we assume that a dvdual s parwse comparso judgmets about homogeeous elemets s cosdered radom, ad expressed o a dscrete /9,, /,,,, 9 scale of sevetee equally lely values. A sample cossts of a set of values selected at radom from the set of sevetee

3 values, oe for each member of the group. It s the dsperso of ths sample of umbers aroud ts geometrc mea that cocers us. Ths dsperso ca be cosdered a radom varable wth a dstrbuto. Because treatg the judgmets as dscrete varables becomes a tractable computatoal problem as the group sze creases, we assume that judgmets belog to a cotuous radom dstrbuto. For example, f there are fve people each choosg oe of 7 umbers the scale /9,,,, 9, there are 7 5 =,49,857 possble combatos of whch,47 are dfferet. Thus, the dsperso of each sample from ts geometrc mea has a large umber of values for whch oe eeds to determe the frequecy ad thus the probablty dstrbuto. To deal wth ths complexty, we use the cotuous geeralzato stead. Ths allows us to ft probablty dstrbutos to the geometrc dsperso for groups of arbtrary sze. Oce we have the cotuous dstrbuto of the geometrc dsperso, the parameters that characterze ths dstrbuto are a fucto of the umber of dvduals the group. To use the geometrc mea to sythesze a set of judgmets gve by several dvduals respose to a sgle parwse comparso, as the represetatve judgmet for the etre group, the dsperso of the set of judgmets from the geometrc mea must be wth some prescrbed bouds. To determe these bouds, we use the probablty dstrbuto of the sample geometrc dsperso metoed above. We ca the fd how lely the observed value of the sample geometrc dsperso s. Ths s doe by computg the cumulatve probablty below the observed value of the sample dsperso the theoretcal dstrbuto of the dsperso. If t s small the the observed value s less lely to be radom, ad we ca the fer that the geometrc dsperso of the group s small ad the judgmets ca be cosdered homogeeous or α-cohesve at that specfed α level. O the other had, f the dsperso s uacceptable, the we could dvde the group of dvduals to subgroups represetg smlarty judgmet. The remader of the paper s structured as follows. I secto we gve a summary of the geometrc expected value cocept ad ts geeralzato to the cotuous case that leads to the cocept of product tegral. I secto 3 we defe the geometrc dsperso of a postve radom varable ad apply t to the judgmets of groups. I secto 4 we approxmate the dstrbuto of the group geometrc dsperso. I secto 5 we setch how groups could be dvded to subgroups f the geometrc dsperso s large, ad secto 6 we show the mpact of the dsperso of a group s judgmets o the prortes assocated wth ther judgmets.. eeralzato of the eometrc Mea to the Cotuous Case Let X be a radom varable. ve a sample from ths radom varable x = ( x,..., x ), the sample geometrc mea s gve by x x = / ther absolute frequeces are equal to m,..., m wth gve by: by pˆ m / m m / = = = x x x. Let us assume that ot all the values are equally lely, ad = m =. The, the sample geometrc mea s. A estmate of the probabltes p P[ X x] =. Thus the geometrc expected value of a dscrete radom varable X s gve by: P[ X= x]lx P[ X= x ] x E[l X ] = = s gve E[ X] = x = e = e () x I the cotuous case, because PX [ = x] = for all x, we eed to use tervals rather tha pots, ad hece, we obta: E [ X] = lm x = x () P[ x < X x + x] f ( x) dx x x x

4 Equato () s ow as the product tegral (ll ad Johase, 99). If X s defed the terval (s,t], we have l E [ X] = lm P[ x< X x+ x]l x = f( x) l xdx I geeral, we have. x s x t ( st, ] f ( x)lxdx f ( x) dx D( X ) { E[l X ]} [ ] = = = e D( X ) E X x e where D( X) s the doma of the varable X ad f( x) dx =. D( X ) 3. The eometrc Dsperso of a Postve Radom Varable Usg the geometrc expected value, we defe a measure of dsperso smlar to the stadard devato. X Let σ be the geometrc dsperso of a postve radom varable X gve by σ ( X) = E, µ x f x > l x, x > l x x, x > where x =. For l x =, the e = ad f x l, x x, x x x µ x F ( µ ) σ ( X ) = exp { E[ l } = µ exp (l ) ( ) x f x dx µ. It s possble ow to wrte σ x = µ ω, l xf ( x) dx where the varable ω has a geometrc mea equal to ad a geometrc dsperso equal to e. 3.. eometrc Dsperso of roup Judgmets Let X, =,,..., be the depedet detcally dstrbuted radom varables assocated wth the judgmets. Let { X, =,,..., } be cotuous radom varables dstrbuted accordg to a recprocal uform RU [,9],.e., the varable Y = l X s a uform radom varable defed the terval 9 [ l9,l9]. The probablty desty fucto (pdf) of Y s gve by g( y) = I[ l9,l9] ( y), ad l9 hece, the pdf of X s gve by f( x) = I[ ( x).,9] 9 l9 x The sample geometrc dsperso s gve by: x l x x s( x,..., x) = = e = x = (4). ( x,..., x ) be the order statstcs correspodg to the sample { x, =,,..., },.e., Let [: ] [ : ] x[ h : ] x[ : ] f h. Let be a value for whch x[ : ] x for =,,...,. We have x[ : ] l s( x[: ],..., x[ : ] ) = l = l x l x [ : ] = x ad hece, we obta ( ) s( x,..., x) = s( x[: ],..., x[ : ] ) = x x[ : ]. For a group cosstg of dvduals, the dstrbuto of S( X,..., X ) s gve by PS [ s] P s P X = υ = = X = [ : ] [ υ ] (3)

5 where υ = υ( A, x) represets the umber of occurreces of the evet A { X x}, ad t s also equal to the dex of the largest order statstc less tha or equal to the sample geometrc mea (alambos, 978). Let S ( x ) = P[ X x, X x,..., X x ]. Sce, < < < r S, ( x) = r P[ υ = r] r=, ad + r P[ υ = r] = ( ) S+ r, = r we have t t X + t PS [ s] = P s t ( ) υ = ( ) S t= X [: t] = Thus, the desty fucto s gve by:. t + t, t + t ( ) +, (5) f () s = f ( s ) t () S D D t t t= = that s a covex combato of desty fuctos of varables of the form ( X) ( Xh), = h=.e., the rato of products of recprocal uform varates. These desty fuctos are of the form ( l[ ] l[ ] a + a z + + a z ). z There are closed form expressos for the desty fucto of the geometrc dsperso for a group cosstg of three or less dvduals, but for groups larger tha three, t s cumbersome ad ot much precso s gaed from t. Istead, we approxmate them usg smulato. 4. Approxmatos of the eometrc Dsperso of roup Judgmets We computed the geometrc dsperso of radomly geerated samples of sze, uder the assumpto that the judgmets are dstrbuted accordg to a cotuous recprocal uform dstrbuto RU [,9]. We dd ths for groups cosstg of 4, 5,, 5,, 5, 3, 35, 4, 45, ad 5 dvduals. 9 We foud that as the group sze creases, the geometrc dsperso teds to become gamma dstrbuted (see Fgure ). cumulatve probablty amma Dstrbuto cgd_4 cumulatve probablty amma Dstrbuto cgd_ cumulatve probablty amma Dstrbuto cgd_5 Fgure. Emprcal evdece that the geometrc dsperso teds to a gamma dstrbuto The parameters of these gamma dstrbutos are gve Table. To exted these models to groups of ay sze, we ft regresso models to the parameters of the gamma dstrbutos. Regresso models of the shape (α) ad the scale (β) parameters versus appear to be surprsgly robust (see Fgure ).

6 Table : amma Dstrbuto Parameters α β α β x amma( αβγ,, ) = x e Γ ( α) shape scale shape scale α(shape) = * β(scale) =.58785*.664 (R-squared = ) (R-squared = ) Fgure. Shape ad scale models I addto, the average ad varace of the geometrc dsperso ca also be estmated from the parameters of these models: mea = exp( /) (R-squared = 99.9) varace = 5.947* (R-squared = ) Note that as teds to fty, the average geometrc dsperso teds to (99% C.I. (3.795,3.834)) ad the varace teds to zero (99% C.I. (4.9378E-9, 3.866E-9)). 4.. Statstcal Test for roup Dsperso Because we are ot comparg the geometrc dsperso to a specfc parameter value, but we wat to test f t s large or small comparso to the dsperso of totally radom judgmets, the hypothess test s compoud rather tha smple. Let s ad s to be the state of ature whch the judgmets are radom ad o-radom, respectvely. Let a be the acto assocated wth selectg s as the correct state of ature. Thus, f the decso s to tae acto a, the hypothess H s sad to be accepted (ad H rejected), ad f the acto s a, the H s rejected (ad H accepted). We ow have the bass for a statstcal test to decde f the dsperso of a group ca be cosdered larger tha usual,.e., that the probablty of obtag the value of the sample geometrc dsperso of the group s greater tha a pre-

7 specfed sgfcace level (e.g., 5 percet) the dstrbuto of the group geometrc dsperso. Hece, to reject H at a gve sgfcat level α, the followg must hold: X PS [ s( x,..., x)] = P s υ = P[ υ ] α = X = <. [ : ] For example, for a group of sze 6, whose judgmets o a gve ssue are equal to {, 3, 7, 9,, }, the geometrc dsperso of the group s equal to The average geometrc dsperso s estmated to be equal to exp( /6) = Tag the usual sgfcace level of 5 percet, we observe that PS [ (6) <.9569] =.4347 <.5. Thus, the p-value correspodg to the sample geometrc dsperso dcates that t seems rare to observe values of the geometrc dsperso smaller tha the sample geometrc dsperso, ad hece, the geometrc dsperso of the group s ot uusually large, whch tur mples that the geometrc mea ca be used as the represetatve preferece judgmet for the etre group. 5. roup Member Classfcato by the eometrc Dsperso Let us assume that { x,,,..., } = s a group of judgmets ad let { x[ : ],,,..., } = be ther order statstcs. If FD[ s ( x,..., x )] P[ S ( X,..., X ) s ( x,..., x )] < α (where α s usually tae to be equal to.5) the the geometrc mea ca be used as a represetatve of the group judgmet. O the other had, f FD[ s( x,..., x)] P[ S( X,..., X) s( x,..., x)] > α the the group eeds to dscuss the pared comparsos further a attempt to reach cosesus. To determe whch members of a group dsagree the most ad hece mae the geometrc dsperso large, we fd the p-values correspodg to the geometrc dspersos of the groups of judgmets gve by: { x[: ], x [: ] },,{ x[: ], x[: ],, x[ : ] },,{ x[: ], x[: ],, x[ : ] }. Let s s x[: ] x[ : ] ( ) = (,..., ), =,,. Lemma : s( ) = s( x[: ],..., x[ : ] ) s a o-decreasg fucto of,.e., s( ) s( ). s ( ) x [ : ] Proof: Let s( ) = s( x[: ],..., x[ : ] ) = where s s the dex for whch x [:] s x[ s: ] x[ : ] x[ s+ : ]. Because x[ : ] x[ : ], the geometrc mea of the frst order statstcs wll x x x + or x [ s+ : ] x[ : ] x[ s+ : ]. If x [:] s x[:] x [ s + :], the satsfy [:] s [:] [ s :] s x [ : ] x[ : ] ( ) = ( [: ],..., [ : ] ) = ( ) = x[:] s [:] ( ) ( ) s s x x s ( x s ) s s x [ : ] x [ : ] we have ( ) ( ) s ( ) ( x[:] s ) ( x[:] s ) s ad sce x[:] s x[:] s ( ) = s ( ). Lewse, f x x x, the [ s+ : ] [ : ] [ s+ : ]

8 ( s+ ) x [ : ] ( ) = ( [: ],..., [ : ] ) = = ( ) x [ s+ : ] ( s ( ) ) ( s+ )( ) s ( s+ ) ( x[ : ]) ( x[ : ]) ( s+ ) ( x[:] s ) ( x[ s+ : ) ] ( ) s s x x s = ( s+ )( ) s ( s+ ) ( x[ : ]) ( x[ s : ]) s+ ( x[:] s ) ( x[ s+ :] ) ( ) ad sce x[ s+ : ] x[ : ] ad x[:] s x[:] we have s ( ) ( ) s ( ) ( ) s ( ) ( ) s ( ) ( ) ( + ) ( + ) ( s+ )( ) x [ : ] x[ : ] x[ : ] x[ : ] s ( + ) ( + ) x[ s : ] x[ s+ : ] x[ s : ] x[ s+ : ] s ( ) = ( s ( ) ). s ( ) Theorem : ve a set of judgmets { x,,,..., [ : ] } dspersos { s ( ),,,..., } = wth correspodg ordered geometrc =, f for ay, PS [ ( ) s( )] α the PS [ ( ) s( )] α. x, =,,...,, we have Proof: By Lemma, gve a set of judgmets { } s () s (3) s ( ) s ( ). If PS [ ( ) s ( )] α, the α PS [ ( ) s( )] = PS [ ( ) s( )] because PS [ ( ) S( )] =. I addto, sce for the gve set of judgmets, s ( ) s ( ), the PS [ ( ) s ( )] PS [ ( ) s ( )], ad the result follows. Defto: A group of judgmets { x,,,..., } = s sad to be α-cohesve f PS [ ( ) s ( )] α. Defto: A member of a group of α-cohesve judgmets s sad to be a laso of the group f the group s ot α-cohesve after the elmato of the correspodg judgmet from the set of judgmets. The Laso Theorem: ve a group of α-cohesve judgmets, a laso does ot exst f ad oly f all subgroups of cardalty (-) are α-cohesve. The exstece of a laso meas that we may be able to dvde a group to two subgroups whose prefereces dffer, ad for whch the geometrc mea caot be used as the represetatve group judgmet. Ths s the subject of further study. 6. eometrc Dsperso ad Prorty Varato To study the relatoshp that exsts betwee the geometrc dsperso of a group ad the dsperso of the correspodg egevectors, we fd the rage of varablty of each compoet of the egevector for gve sets of group judgmets. Ths s doe by frst fdg the dstrbuto of the egevector compoets for radom recprocal matrces whose etres are dstrbuted accordg to recprocal uform dstrbutos RU[ l, u ]. It ca be emprcally show usg smulato for ay recprocal matrx A whose etres are dstrbuted accordg to a recprocal uform dstrbuto RU[ 9,9], the average prcpal egevector s gve by Ew [ ] =, =,,, where w Beta( α, β ). The beta dstrbuto has a desty fucto gve by Γ ( α+ β) α β z ( ) t Γ( α) Γ( β) f( x) = x ( x) where Γ z = t e dt. It s also ow that

9 EX [ ] α( α β ) = +. To approxmate the parameters of the beta dstrbutos for dfferet sze matrces, we geerated radom recprocal matrces of sze 3, 4, 5, 6, 7 ad 8 wth etres dstrbuted accordg to a recprocal uform RU[/9,9], ad computed ther prcpal rght egevector. Sce the average of each compoet would theory be equal to /, we estmated the parameters of each beta dstrbuto for each compoet ad averaged them to obta the followg parameter values gve Table, ad ftted a regresso equato to βα because βα = + from EX [ ] = αα ( + β) = /. Fgure 3 summarzes ths result. Table. Estmates of Egevector Parameters alpha beta beta/alpha Plot of Ftted Model E[ βα ] = * (R-squared =.) Fgure 3. Estmate of the Prcpal Rght Egevector of a Radom Recprocal Matrx wth Etres Dstrbuted accordg to RU [ 9,9] We exted ths result to a arbtrary recprocal matrx the followg theorem. Theorem : For a radom recprocal matrx X = ( x ) wth etres dstrbuted accordg to a recprocal uform dstrbuto, x ~ RU[ l, u ], the compoets of the radom varable w= ( w,..., w ) T correspodg to the prcpal rght egevector are dstrbuted accordg to a beta, w w Beta( α, β), where w = m{ w} ad w = max{ w}, ad the prcpal rght egevector of w w the recprocal matrx whose etres are gve by the geometrc mea of ts etres, E [ x ], s gve by: α ( α α + β α + β ) T T ( Ew [ ],..., Ew [ ]) = ( w w) + w,, ( w w) + w. Let x = µ where µ = lu s the geometrc mea ad σ s the geometrc dsperso of x ~ RU[ l, u ]. By defto, µ j = / µ ad σ j = σ. Thus, we have wj = / w. Let us assume that the recprocal matrx of geometrc meas s cosstet,.e., µµ j = µ. The the prcpal rght (pr-) egevector of the matrx ( x = µ w σ ) s gve by the Hadamard product of the pr-egevector of the matrx ( µ ), µ w, ad the pr-egevector of the matrx ( w σ ). The etres of ths matrx are radom recprocal uform varables RU[ l µ, u µ ] whose geometrc dsperso s gve by ( u l ) /4. w σ Sce the geometrc dsperso of the varables x ad that of the varables w σ s the same, because

10 x µ =, we have ( u l ) /4 w σ σ =. Thus, boudg the dsperso of the etres of the matrx ( ) bouds the dsperso of the etres of the matrx ( x = µ ). w σ w σ 6.. Example Cosder the followg matrx whose etres are dstrbuted accordg to recprocal uform dstrbutos, alog wth the correspodg geometrc expected value ad pr-egevector: A E[ w] E { } { } { } [ ] RU[,9] E RU[,] E RU[,4] Ew E { [,9]} { [,8]} [ ] RU E RU Ew = 5 E { [3, ] 3 } Ew [ 3] RU Ew [ 4 ].83 Usg smulato, we geerated a sample of 5 radomly geerated recprocal matrces, whose etres follow the recprocal uform dstrbutos descrbed above. The radom pr-egevector whose compoets are beta dstrbuted wth parameters gve Table Coclusos α β α α+ β Ew [ ] Table 3. Dstrbutos of the Egevectors w w w3 w I ths paper we put forth a framewor to study group decso-mag the cotext of the AHP. A prcpal compoet of ths framewor s the study of the homogeety of judgmets provded by the group. We developed a ew measure of the dsperso of a set of judgmets from a group for a sgle pared comparso, ad llustrated the mpact that ths dsperso has o the group prortes. A subject of future research s the study of the relatoshp betwee dspersos o the dvdual pared comparsos the etre matrx, the cosstecy of judgmets, the compatblty of the prorty vectors ad the measuremet of the volato of Pareto Optmalty. Refereces EN.REFLIST