A model of decision with lingustic knowledge

Ma th w are&Sof tC om puti ng 3( 19 94)25 3-26 3 AM od el of Decis ionw ithL inguis ti c K no wle dge 3 M.T.La ma ta Dept.Ci enci as de la Com putaci  oneIn tel igencia Arti ci al U niv ersidaddeGranada.1 80 71-Granada.Espa~ na e-mailmtl@r obinson.ugr.es Ab stract Theaimofthispape ri stodevelopanewaggregating met hodforthedecisionprobl eminw hic hthepossiblev al ue sofrew ardsareknow ninlin gu istic terms. W esho wnewopera torsforsolvingthisproblem,asw ellasthew a yin whic hO WA operatorspro videusw ithana dequateframew orkforrepresen tingtheoptimismdegreeofthed ecisionmak erincasew eha v en oinformation aboutthereal state. Keyw ords :Deci sion-making,linguisticlab el,uncertain t y ,operat ors. 1In troduc tio n Adeci sion-ma kingprobl emi n v ol v esse le ctingthebestal ternati ve in viewo fthe resul tsa nd inf orm ati ontha tw eha v ebeen abletoo bta inrega rdi ngthes tatesof na ture. In many cases,thei nf orm ationa vail abl eregardingthesta te so fn atureisobtained in an auxil ia rym a nner .N ormaly ,h o w ev er,inthec lass ic al mo delthisi nform atio nisav ai la bleinterm so faprobabil it ydistri bution,ori tm a yso ha ppenthat we are faced with total ig noranc e. On the othe r ha nd , thepa ym en ts that the de cisionmaker p e rce i ves a s a conseque nce of his/her c hoice whi ch sh a ll b e n um e rical , are a funct i on of the s ta te of na t ure that is pre sen t ed. If w e think that in e v e ryda yl ife the w ayo f e xpres sing a ni de a i smore l ingui sti c than numerical , this wil lm ean that b oth the i nf orm ati on a vail able ab out the states ofna ture and the p ossible re w ards are al lkno wn b yusinli ng uistic te rm s. That is wh yw esha ll rs t disc uss the problem of dec i si on-m aking unde r risk, 3 This work is supported by the DGICYT under pro ject PB92-0959 253 254 M. T.La m ata wher etheinf orm atio naboutt he st atesofnatureisk no wn in term sofaprobabil it y distri buti on,whereasthe po ss iblerew ardsarev a luese xpre ss edinli nguisticterms. Fi nall y ,w eshalldea lwith thec aseofdeci si on-m akingu nder ignorance,where no inform ationi sa v ail abl ea bo ut thestatesofna ture,a nd ther eforei no rderto beabletoactthedecis ion-ma k er' sattitudewhenface dw it hriskm ustbekno wn. W eshallsolv etheproblemb yi n troduci ngtheO W Aoperators (OrderedW ei gh ted Av erag ing),whi c hoe ru samoreg enera lf ra m ew orkt hant he onekno wntil lno w. 2 T he dec i si onprobl em Letusconsider adecisionm aki ngproblem withthefol lo wi ngelem en ts: f  ;D; r ;I; g whe re : 1. isthesetofstatesofna tu re whic hw esupposeareni te. 2. D i sthesetofalterna tiv esa v aila bletothedeci si on-m ak er, am o ngwhic hthe decisio n-m ak erm ustc ho oseone. 3. r : D 2  ! R i safunctionwherearealn um berco rrespondsto ev erydecision d i andtoev erystate w j . ( d i ;! j ) 0 ! r ( d i ;! j )  r ij Wi th r i j bei ngtherew ardg iv entothedeci si on-m ak er forha vi ngc hosenthe decisio n d i whenthestate w j hasbeenpres en ted r canberepresen tedb ya m atri xca lle dt he Matr ixofr ewar ds . D n  w 1 w 2 11 1 w j 11 1 w m A 1 r 11 r 12 111 r 1 j 111 r 1 m A 2 r 21 r 22 111 r 2 j 111 r 2 m . . . A i r i 1 r i 2 111 r ij 11 1 r im . . . A n r n 1 r n 2 111 r nj 11 1 r nm In thes e c ondi tions t he r ew ards whi ch can b e obtained b ythe deci si on-m aker for m ak ing a de cisio nwi ll b e an yv al ue of the n-di mens i onal v e ctor asso c i ate d with the chosen al te rnativ e. A i 0! [ r i 1 ;r i 2 ;:::; r ij ;:::;r im ]  ~r i 2 R m Am odelofdecision with lingui sti ckno wledge 255 4. Ar elati onofpreference s  by thedecisio nm ak er.W eshal la ssum eacoheren t de cisionm ak er,sow ewill tr yt om axi m i zehi sprotsore ls em ini m izehi s losses. 5.Certaini nform a ti on ab outt herealstateo fnature .D ependingonthedeci si onm ak er'skindo fkno wledge ab outtherealstate,three en vironm en tsappear intheli terature. (a) De cisionmaking unde rc er tainty .Thedeci sion-mak erkno wswhi chstate i sg oingtobepresen te d. I npa rti cul ar,i tisthec asei nwhic h P ( s j )=1 or0fo rall j =1 ;::m .Inthi scase,thedecision-mak ershouldc hoosethe al ternati v ewhic hm eansthem axim umpa y o/m inim umlo sstothi s stateofnature. (b) De cision-making unde rr isk .Inthedecisio nunde rr iskapro ba bil it y di stributi on ab outthestateso fna turei sa ss um edtobek no wn.Inthi s case,thestandard procedurei stousethe ex pe ct edv al ue. (c) De cisionmakingunderignor anc e .Thethird en vironm en ti n v o lv esthe caseo fi gnorance .Sincew eha v enoi nform a ti on a v ai labl eaboutthe statesofnature, thedecisio ncri teri atak ein toaccoun tthedeci si onm ak er's attitudeto w ardstherisk. Am ongtheseattitudesthe m ostrepresen tativ ea rethef ol lo wi ng:  Pessi misti cA ttit ude .By m ea nso fthisstrategy ,thedeci si on-m a k er hasthew orstpossible,point ofview.Hechoosesthew o rs to utcom e forev erya lternati v eandthenheselectstheonew hi cha chi ev esthe best/ w orst.This st rateg yisal sokno wna sMa ximin.  Optimist icatti tude .Underthi ss tr a tegythedeci si on-m akera ssum es the best resultpossi ble,select in gt hebestoutco m ef orea cha lter - nativ ea nd c hoosingthealternativ ew hi c hac hiev esthebest/b es t. Thisstrategyi sa lsoknowna sM axi m ax.  Hur wi cz st rate gy .W iththi sapproach thed ec ision-m ak ersel ectsa v al ue  2 [0 ; 1]whi c hr epresen tshis degreeofoptim i sm .F oreac ha lt ernativ ewew ei gh tm axi mum and m ini mum v alues b y  and ( 1 0  ) r esp ec tively . H i =  2 Max i r ij +(1 0  ) 2 Min i r ij Final ly ,theal t ernativ eofa chieving th e m axim um i scho se n.  Normative app r oach .Inthisa ppr o ach, t he de cisionmak er a dds al l val ue s re l ating to e v ery al t ernativ eino r der to c hoose that al te rnative whose sum is the highe st. 256 M. T.La m ata Fr om thism om en tletusassume that therew ards r ij ca nb eexpressedinl inguisticterm s.Thi swi llin vo lv es ev eralprobl em ssucha st he needtoa tta chl abel s with n um ericalv al ues. Therefo re,w eshallha v eto in trod ucenewoperatorssimilart ot hosewhi c happeari nthel iterature .I na ddi tion,theseoperatorss houl dbecapableofso lvi ng the pro bl emw eindi ca te. To startwith,letusdenethe fo llo wing : 3Se tofgenerali z edl abel s Let U = f u 0 =0 ;u 1 ;u 2 ;: :;u n g betheordina teset ofl ingui sti cv a luesthatrep resen t thesetofpossi blere w ards,i .e., U = f u i = i 2 I  Z;I =0 ; 1 ;: :;n g andsotha t u i  u j if i  j .Let P = f 0= p 0 ;p 1 ;: :;p m g wi th p j 2 [ 0 ; 1 ]and6 p j =1.Inthi s case, p j repre se ntsthepro babil it ydistri butionabouta ndaccordinglyt he setof po ssibl ew ei gh ting so fthedi eren tl abels. Bym eans ofthesesets,whic hcanbee itherli nguistico rn um erical, w edene these to fgeneral izedl abel s[4] . U = U 2 P wherethepai r( u i ;p j ) 2U represen tstheproducto f p j  u i .  If p j =1 ! ( u i ; 1)  u i :  If p j =0 ! ( u i ; 0)  0 : Asaresultofthi s, thel abelwhi chrepresen ts then ullv al uehastobel ongto theseto fl abel s U . 3.1Addition Le t( u i 1 ;p j 1 )a nd ( u i 2 ;p j 2 )b et wo ge nera li zedlabe ls;w ed ene th eiraddi tionas : 8 : U 2 0!U ( u i 1 ;p j 1 ) 8 ( u i 2 ;p j 2 )=( u s ;p i 1 + p i 2 )(1) whe re s = ( i 1 2 p j 1 )+( i 2 2 p j 2 ) p j 1 + p j 2 Ho w ev e r, thi sv alue s is not nec es sari ly an i nte ger n um ber, and u s may not b e alabel of U .We can use a rounding meth od as the mos t s traightforward wayof achieving it. Am odelofdecision with lingui sti ckno wledge 257 As we ha v etoaddsev eralgener al izedla belsw eshouldextend( 1) inthefo llo wi ngw a y: ( u i 1 ;p j 1 ) 8 ( u i 2 ;p j 2 ) 8 :: 8 ( u in ;p jn )=( u s ; n 6 k =1 p jk )=( u s ; 1) wi th s =round n 6 k =1 ( ik 2 p jk )(2) P roper ti es : 1.C om mutativi t y:Theorderofthea dditiono ft w og enera li zed labe ls doesnot ae cttheresul toftheaddition. 2. Associ ativi t y:Duetothew ay inwhic htheoperatio nhasbee ndenedi n(2). 3.NeutralEl em en t:An yl ab el sucha s: ( u i ; 0) 8 i =0 ; 1 ;: :;n 4D eci sionunde rri sk W henw eha v eap roba bil it ydistri buti onabo utthestatesofnaturewhic hi nform sus a bo uttheuncerta in t ya ssoci atedtotheexperience,them athem ati ca lexp ecta tion i stheclassico peratorwhi c hforeac hdecisio nw ei gh tsthedi eren tpa y os. F o rth eprobl emw eaddress,inwhi ch therew ardsa rel ingui sti c,w eproposethe f ollo wi ngopera tor. 4. 1 Thesetofprobabil itiesi sn um eri calandt hepa y osl inguisti c E :( U 2 P ) m 0 ! U 6 j ( u j 2 p j ) 0 ! u si Co ns i de ring U = f u i =i =0 ; 1 ;:: ;n g t o b e the ord i nate n i te set o f the p ossibl e lingui st i c lab els, and since eac h util it y u ij is a label u j 2 U ,we dene,asi nthe abo v e, i ts o r der as O ( u ij )= O ( u j )= j . Then E 2 [ A i ]= u si where s i =r ound ( U i ) 258 M. T.La m ata and U i =6 j ( O ( u j ) 2 p j ) Asthes et U i sni teandnorm a ll yitsg ra nulari t yislo w,thi sm ea ns that sev eral de cisionsac hiev ethesam eexpect ation.Inthiscase,i nor dert ot a k eadeci si on, th ea ctionforwhic hthe Max i U i is obtai nedshouldbec hosen. U 0 ! R u s i 0 ! U i Algo ri th m 1.F oreac hal ter nativ ew ecalcula tei tsexpec tatio n E 2 [ A i ] 2.W ec hooseasoptimumt hatdeci si onf orwhic hthem axi m umisrea ched. E 2 [ A 3 ]= Max i u si 3.Ifthe de cisio nisnotuni que,c hoos et he A 3 optim umsotha t E 2 [ A 3 ]= Max i U i 4.2Ex am ple Letussuppose thattheinfo rm atio naboutth ev estateso fnaturei sproba bili sti c andg iv enb y p ( w i )= i= 1 5f ori =1,2, 3, 4,5.Letusal sow econsiderthefol lo wing m atrixofrew ardsexpressedb ym eansofl ingui sti cterm s: w 1 w 2 w 3 w 4 w 5 a 1 u 7 u 4 u 4 u 6 u 5 a 2 u 2 u 5 u 7 u 4 u 7 a 3 u 5 u 5 u 7 u 4 u 3 a 4 u 6 u 4 u 3 u 5 u 2 where U = f 0 ;u 1 =v erylo w ;u 2 =lo w ;u 3 =so m ethingl es st hanfai r ;u 4 =fai r ;u 5 = m orethanf ai r ;u 6 =hi gh ;u 7 =v eryhigh g Thesoluti onisv erysim pl e E [ a 1 ]=7 2 1 = 15 + 4 2 2 = 15+ 4 2 3 = 15 + 6 2 4 = 15+5 2 5 = 15 = 5 : 60 s = round(5 : 60) = 6 = ) E [ a 1 ]= u 6 in the s a mew ay E [ a 2 ]= u 6 E [ a 3 ]= u 5 E [ a 4 ]= u 4 the best alte rnativebeing a 2 . Am odelofdecision with lingui sti ckno wledge 259 5 D eci si onunde ri gnorance Notha vingany inform a ti ona bo utthestates of nature,andsupposingthat the criteri aofdomi nanc edosnotpro vi dean yre su lt,i ti sthedecisio n-m ake rw ho must deci de,takingi nt oa ccoun tasubjecti v eru le . W esupposethatt he util iti esareling uistic. In thiscase,theclassical strate g ies o fdecision maxi m axa ndm ini m axa renot them os tsuitabl e,si ncewith thes et U beingnite ando fli ttl egran ul arit y ,it is lik elythatt w oal ternativ es reac hthesam e m axi m umand/orm ini m um .Duetothi s,w eal so re m o v etheHurwi czapproach a ndw econsideri tm oresuitabl etoinsert theO W Aoperators. 5. 1Op erator s D en i ti on (Y ag er)Ano rderedw ei gh teda v erag ingop erator(O W A)ofdim ension n isaf un ctio n F : R n ! R thathas as sociatedwithi taw eig h tingv ecto r W W =[ w 1 ;w 2 ;:: :;w n ] W = 2 6 6 6 4 w 1 w 2 . . . w n 3 7 7 7 5 suc hthat 1. w i 2 [ 0 ; 1] 2.6 i w i =1 wheref oran ysetofv alues a 1 ;: :; a n F ( a 1 ;: :;a n )=6 i w i ?b i where b i is the i-l arge st of the a 1 ;: :; a n . Den iti on .A ssum etha t X is a s et of elem en ts. A bag (o rm ulti -s et) d ra wn f rom X is a ny coll e ction of e l emen ts, eac h of whic hiscon tai ne d in X . Abagisdi eren t fro m a su bset i n that i tal lows m ulti pl e copi es o f the sam eel emen t. A bag is s i milar to a se t in that the order i ng of elem en ts do snotm at ter. If A is a bag consisting of a; b; c; d ,we denote this A = <a;b;c;d > . 260 M. T.La m ata Properti es 1. F sa ti s es ageneral izedcom m utati vity (sym m etry) F ( a 1 ;a 2 ;: :; a m )= F ( b 1 ;b 2 ;: :; b m ) ifthebag s <a 1 ;a 2 ;:: ;a m > and <b 1 ;b 2 ;:: ;b m > areeq ua l. 2. F ism ono to nic F ( a 1 ;a 2 ;: :;a m )  F ( b 1 ;b 2 ;: :;b m ) if a i  b i f oralli=1, 2, .. ,m 3. F is idem poten t F ( a;a; : :;a )= a forev ery a 2 R . Ifw et ak ei n toaccoun ttheprevious denitio ns,andsi ncew eha ve no an yinfo rm ationa boutthestatesofnatureandw ea renotcapabl eofconsideri ngsubjecti v e pro ba bil ities,w eshallsuggestthef ol lo wing: Cri te ri onofp ref eren ce .Gi v ent w odecisio ns, A 1 and A 2 ,w eshal lsa ytha t thersti spreferredtothese co ndifiti sv eriedthat F ( u 11 ;u 1 2 ;:: ;u 1 m )  F ( u 2 1 ;u 2 2 ;: :;u 2 m ) If <u 1 1 ;u 12 ;::;u 1 m > and <u 21 ;u 2 2 ;:: ;u 2 m > ar eequa lthenw esa ytheyare indieren t. Onthecon trary ,if thed ec i si on-m a k erw an tstow ei gh ts om eresultsm o rethan others,a ccordingto thei rdeg reeofo ptim i smaboutt he resul t,w esha lltak einto accoun tthatforan yd eg re eofoptim ism ,theoperator F whi chrepresen tsi t,i s alw a ysbetw ee nt he extrem eoperato rsm axim um andm i ni m um . Min ( a 1 ;a 2 ;: :;a m )  F ( a 1 ;a 2 ;:: ;a m )  Max ( a 1 ;a 2 ;:: ;a m ) Taki ng i nto acco un tthedeci si on-m ak er ' s attitude , w e repr o duce t he classical c riteria of de cisi on under ig noranc e when the rew ards are li nguistic, and w eop - e rate with the c oncep to f the ge nerali zed l ab e l pr evi ous l yi ns er ted and the O WA op e rators.  Pe ssim ist i cA tti tud e: If we selec t w wh ere w 3 =[ 0 ; 0 ;: ::; 1] T the n F ( a 1 ;a 2 ;::;a m )= Min j [ a j ]= u 3 j 2 U , whichistheaggre gation r ule us ed in t he p e ssimistic s trategy. Am odelofdecision with lingui sti ckno wledge 261  Opti m isticA ttitude:Ifw es el ec t w 3 wher e w 3 =[ 1 ; 0 ;::: ; 0 ] T then F ( a 1 ;a 2 ;:: ;a m )= Ma x j [ a j ]= l 3 j 2 U ,whi chiswhati su se dinthe optim i sti cstrategy.  Hurwic zS tra tegy:Ifw eselect w H =[ ; 0 ;: :: ; 0 ; (1 0  )] T then F ( a 1 ;a 2 ;:: ;a m )=  2 Max [ a j ]+(1 0  ) 2 Min [ a j ]=  2 l 3 j +(1 0  ) 2 u 3 j = u j 2 U .Thisopera ti onis thesam easinSubsectio n4.2. Thisistheform ul atio nuse dintheHurwiczstrategy .  Hu rwic zStra tegyGenerali zed: If we select w HG =[  1 ; 2 ;: :: ; m ] T with6  i =1andwhose va lues  i aretheresu lt ofthem athem aticalprogram m ingprobl emproposedb yO` Hagan. Maximize 6 j w j ln ( w j )(en trop y) Subje ctt o: 6 j ( hn ( j ) 2 w j )=  6 j w j =1 w j  0f oral l j =1 ;: :;m where  isthedeg reeofo pti m i smand hn ( j )= ( n 0 j ) ( n 0 1) . The n F ( a 1 ;a 2 ;:: ;a m )=  1 2 Ma x [ a j ]+  m 2 Mi n [ a j ]= u j 2 U . With the s i mila ro p eratio nasinth e Hur wi c z Strate g y.  No rm ativ e Approac h: If w e selec t w N =[ 1 =n; 1 =n; :: :; 1 =n ] T the n F ( a 1 ;a 2 ;: :;a m )=1 =n 6 j a j = u j 2 U .Thi sfunctio nisessen tia ll ythe nor mative strate gy.