Communication on the Paper A Reference-Dependent Regret Model for. Deterministic Tradeoff Studies
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1 Councaton on the Paper A Reference-Dependent Regret Model for Deterntc Tradeoff tude Xaotng Wang, Evangelo Trantaphyllou 2,, and Edouard Kuawk 3 Progra of Engneerng cence College of Engneerng Louana tate Unverty Baton Rouge, LA E-al: xwang8@lu.edu 2 Departent of Coputer cence Louana tate Unverty Baton Rouge, LA E-al: tranta@lu.edu 3 Departent of yte Engneerng Naval Potgraduate chool Monterey, CA E-al: ekuawk@np.edu ABTRACT Th councaton focue on a fundaental proble related to the recently ntroduced Reference-Dependent Regret Model (RDRM) [Kuawk, 2005] for deterntc ultcrtera decon akng. In [Kuawk, 2005] t wa aerted that the RDRM odel atfe three properte. The frt of thee properte, referred to a the ndependence of donated alternatve, ee to be an ntutve one. Accordng to th property, the RDRM odel preerve the rankng of two alternatve A and A wth rankng A f A when a new alternatve donated by A ntroduced or an old alternatve donated by A dropped. In th councaton t deontrated algebracally and alo by ean of a nuercal exaple that the RDRM odel ay fal to atfy th property. The plcaton that when the concept of regret and/or reocng are condered and defned n ter of all the avalable alternatve n accordance wth the RDRM, addng or droppng a donated alternatve can change the rankng of the alternatve and volate the ndependence of donated alternatve property. Key word: Regret theory, rank reveral, Pareto-optal alternatve, utlty theory, ultcrtera decon analy. * Author to who all correpondence hould be addreed (tranta@lu.edu)
2 . INTRODUCTION Incorporatng behavoral apect, uch a feelng of regret and reocng, nto Mult-Crtera Decon Analy (MCDA) ha been an ntrgung area. Aong all the eoton, regret the one that ha receved ot of the attenton. avage [95] frt propoed the noton of regret ore than 50 year ago. Later on, Looe and ugden [982] and alo Bell [982] ultaneouly propoed the regret theory for ratonal decon-akng under uncertanty. In ther regret theory (to be referred to a RT-B/L), t aued that regret depend on the dfference of the utlte of the choen alternatve and the forgone alternatve that were condered but were not choen. Recently, Kuawk [2005] argued that a peron level of regret often depend explctly on the abolute value of the utlte of the choen and forgone alternatve rather than ply the dfference. He propoed the Reference-Dependent Regret Model (RDRM) to account for th behavor. More pecfcally, accordng to the RDRM the aocated regret when choong alternatve A and forgong alternatve A under crteron C k, denoted a R(u k, u k ), gven by G( uk ) G( uk ), uk < uk Ru ( k, uk ) () 0, otherwe. In Eq. (), u k the utlty of alternatve A for crteron C k. The notaton G(.) denote the regret-buldng functon. It eaure the level of regret referenced to the axu poble utlty noralzed to. The G-functon ued n the RDRM odel gven a follow:, x > 0, 2* *( B+ x) Gx ( ) + ( B/ x) 0, otherwe. (2) In Eq. (2), B and are two paraeter that depend on the regret atttude of the ndvdual decon aker. The total level of regret for choong A fro a et of n (where n 2) alternatve wth crtera gven by R w R u u ) (3) n ( ) k ( k, k n k The fnal utlty of alternatve A gven the et, calculated a follow: U w u R w u w ( ) R( u, u (4) n k k k k k k k k k n l lk)
3 In Eq. (4), the frt ter the clacal utlty of alternatve A and the econd ter the antcpated regret for choong alternatve A. Fnally, the alternatve are ranked by ther overall utlte. Furtherore, n [Kuawk, 2005], three properte were ntroduced whch were argued to be neceary condton that any effectve MCDA odel hould atfy. The frt property tered the ndependence of donated alternatve. Accordng to th property, f a odel rank two alternatve A and A a A f A, then when addng a new alternatve whch donated by A or droppng an extng alternatve whch donated by A hould preerve the ntal rankng A f A. Th property appear to be ntutve and t wa thought that t had been proved to hold for the RDRM odel [Kuawk, 2005]. However, a llutrated and proved n the next ecton, the RDRM odel doe not alway atfy th property. 2. RANK REVERAL WITH THE RDRM MODEL In th ecton, a randoly generated decon proble ued to llutrate how the RDRM odel ay fal to atfy the property of ndependence of donated alternatve tated n [Kuawk, 2005]. Th further nvetgated by analyzng the RDRM odel algebracally. 2. An exaple of rank reveral wth the RDRM odel Th nuercal exaple defned for two alternatve wth three decon crtera. The followng atrx D repreent the utlte of the two alternatve n ter of the three crtera whle the row vector W repreent the weght of the three crtera. D W [ ] Furtherore, the two paraeter ued n the G-functon are aued to be B 0.60 and When Eq. (), (2), and (3) are appled to copute the overall regret value aocated for choong each alternatve and forgong all the other the reult are: U P R U 2 P 2 R Where P the clacal utlty value of alternatve A, R the overall regret value for choong alternatve A and forgong all the other alternatve, and U the fnal utlty value of alternatve A after ncorporatng the effect of regret. nce U > U 2, alternatve A ranked hgher than A 2 ;.e., A f A 2. Next, a new alternatve A 3 whch donated by alternatve A ntroduced. Pleae note that n th exaple t a concdence that A 2 alo donate A 3. The new decon atrx gven by
4 D The weght vector W rean the ae a before. The ae forula are appled a before to rank the alternatve. The reult are U P R U 2 P 2 R U 3 P 3 R Now U 2 > U, o A 2 f A. Th reult dfferent than the prevou reult of A f A 2. A rank reveral ha ut occurred. Thu, the RDRM odel faled the property of ndependence of donated alternatve. 2.2 Matheatcal Analy Th ecton nvetgate atheatcally why the RDRM odel doe not alway follow the frt property. Gven a et of n alternatve and crtera, uppoe that two alternatve, ay alternatve A and A, are ranked a A f A. A decrbed prevouly, the RDRM utlty for alternatve A and A are calculated a follow: U w u R w u w ( ) R( u, u n k k k k k k k k k n l lk ) U w u R w u w ( ) R( u, u n k k k k k k k k k n l lk) For convenence of dcuon, let R w R( u, u ). k k lk k l n Then U w ( ) R. n kuk k larly, let n R w R( u, u ). k k lk k l Then
5 The dfference between U and U U ( ) n R. wkuk k U U w u w u R R ) ( k k k k) ( )( k k n. (5) Gven that the two alternatve are ranked a A f A, U U > 0 (6) When ntroducng a new alternatve A k whch donated by A, the value of R rean unchanged whle the value of R ay ncreae f A k donate A n ter of one or ore crtera. Thu, n Eq. (5), the part ( R R ) ay becoe le than before. Meanwhle, the nuber of alternatve n the et ncreaed by. Along wth the above change, f the orgnal value of ( R R ) potve, the ter ( )( R R ) n Eq. (5) ay becoe n aller than before. Then the nequalty n Eq. (4) tll hold. However, f the orgnal value of ( R R ) negatve, the ter ( )( R R ) ay becoe larger than before. The n nequalty relaton n Eq. (6) can be revered and hence the rankng between A and A ay be altered. Th how the RDRM odel can fal to atfy the property of ndependence of donated alternatve. 3. ANOTHER WAY FOR MODELING REGRET It ay be quetoned whether there a way to avod the above tuaton of rank reveral. Quggn [994] decrbe the proble where anpulaton of a et of the alternatve ay yeld rratonal choce a the rankng of the alternatve ght be oney puped ;.e., the rankng of the alternatve nfluenced by the ntroducton of donated or non-pareto optal alternatve. In order to avod beng oney puped, Quggn [994] propoed that the eaure of regret hould atfy a property called the Irrelevance of tatewe Donated Alternatve (IDA). Th property lar to the ndependence of donated alternatve property propoed n [Kuawk, 2005]. In order to atfy the IDA property, Quggn [994] proved that regret ut be deterned olely by the bet attanable outcoe n each tate of the world or equvalently the bet perforance value of each decon crteron n MCDA proble. Th n contrat wth deternng the regret aocated wth an alternatve by conderng the entre et of alternatve, lke averagng the regret contrbuton that are fored when conderng all avalable choce par. When th dea appled to the tudy of how to odel regret n MCDA proble, the regret aocated wth
6 choong one alternatve and forgong all the other alternatve deterned only by coparng the choen crtera value wth the bet crtera value. Addton or deleton of donated alternatve then cannot affect the regret level of the other alternatve becaue the bet crtera value wll be the ae a before under thee change. In order to keep the IDA property, t ee reaonable that regret hould be eaured only by the bet poble value under each crteron. However, th ay not ake uch ene a t llutrated n the followng hypothetcal tuaton. uppoe that the followng are the core acheved by four tudent n oe exa: tudent core A 30 A2 32 A3 3 A4 00 In th exaple there ut one tudent who earned a hgh core (.e., 00 pont). Therefore, t reaonable to expect that the tudent who earned only 30 pont feel oe but lted regret for not havng acheved a hgher core. Th reacton ay be ratonalzed becaue h/her perforance not a bad when t copared to that of ot of the other tudent. However, the ae tudent ay feel uch tronger regret for corng only 30 pont f the core of thee tudent were a follow: tudent core A 30 A2 98 A3 97 A4 00 Th undertandable becaue n the new hypothetcal exaple he/he the only tudent who ha acheved a very low core. Therefore, ntutvely n the prevou exaple, t ake ore ene to copute regret n ter of the entre et of alternatve. Thu, the concept of regret and reocng ay be ore realtcally expreed n ter of the crtera value of the entre et of alternatve, a gven by Eq. (3), than n ter of only the bet crtera value. However, on the other hand, f regret coputed by conderng all the alternatve then addng or deletng a donated alternatve ay alter the ntal rankng and thu leave the rankng of the alternatve open to anpulaton lke the Money Pup. 4. CONCLUDING REMARK Fro the above dcuon, one can ee that f the concept of regret and/or reocng are condered n the decon-akng proce and are defned n ter of all the avalable alternatve, the rankng of the alternatve when only the Pareto-optal alternatve are ued ay be dfferent than the rankng of the alternatve when oe or all of the donated alternatve are condered bede the Pareto-optal one. Addng or deletng a donated
7 alternatve can change the rankng of any alternatve becaue of the nterdependence aong the. Thu the RDRM doe not atfy the property of ndependence of donated alternatve that propoed n [Kuawk, 2005]. The followng queton tll left anwered and deerve further reearch: I the property of ndependence of donated alternatve a neceary condton that any effectve MCDA odel hould atfy or hould t unveralty be condered upect? Proble wth ncorporatng regret/reocng wthn the fraework of clacal utlty are not unexpected becaue clacal utlty theory doe not pert coparng dfference n utlty [Luce and Raffa, 957: 32]. The proble of forally odfyng clacal utlty theory to ncorporate ratonal pychologcal nfluence challengng; but worth purung. An alternatve approach propoed by Kuawk [2005] to ncorporate regret a an eleent of a cot-utlty-regret analy. REFERENCE D.E. Bell, Regret n decon akng under uncertanty, Operaton Reearch, 30 (982), E. Kuawk, A reference-dependent regret odel for deterntc tradeoff tude, yte Engneerng, 8 (2) (2005), J. Quggn, Regret theory wth general choce et, Journal of Rk and Uncertanty, 8 (994), G. Looe and R. ugden, Regret theory: An alternatve theory of ratonal choce under uncertanty, Econoc Journal, 92 (982), R.D. Luce and H. Raffa, Gae and decon: Introducton and crtcal urvey, Dover Publcaton, Boton, MA, UA, 957. L.J. avage, The theory of tattcal decon, Journal of the Aercan tattcal Aocaton, 46 (95),
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