Volume 35, Issue 2. Allocation of costs to clean up a polluted river: an axiomatic approach. Wilson da C. Vieira Federal University of Vicosa, Brazil

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1 Volume 35, Issue 2 Alloto of osts to le up polluted rver: xomt pproh Wlso d C. Ver Federl Uversty of Vos, Brzl Abstrt Ths pper proposes method to shre the osts of leg up polluted rver mog the gets loted log t. Ths method s bsed o model of pollutt trsport d the polluter pys prple. We provde xomt hrterzto for ths method d vestgte ts reltoshp wth both the (weghted) Shpley vlue d the tuvlue of the orrespodg ost gme geerted from the problem. We show tht the soluto of the proposed method odes wth both the weghted Shpley vlue d the tu-vlue. We would le to th the Assote Edtor (Dr. Joh P. Coley) d two oymous referees for ther vluble ommets d suggestos whh helped mprove ths pper. The usul dslmer pples. Ctto: Wlso d C. Ver, (205) ''Alloto of osts to le up polluted rver: xomt pproh'', Eooms Bullet, Volume 35, Issue 2, pges Cott: Wlso d C. Ver - wver@ufv.br. Submtted: August 30, 203. Publshed: Jue 0, 205.

2 . Itroduto Wter polluto s serous problem fed by my outres, espelly developg outres. I ths pper we vestgte the followg questo: how to dvde frly the osts of redug polluto rvers mog the gets loted log t? The swer to ths questo s ot trvl se wter pollutts re trsported wth the wter d some of them re bodegrdble. To swer ths questo we resort to model of trsport of pollutt d the polluter pys prple. Ths prple s wdely epted d ssgs the resposblty for the polluto to the polluters. Thus, they would be resposble, for exmple, the otext of our lyss, for the osts of leg up polluted rver. The polluto pys prple hs reeved strog support from the Orgzto for Eoom Co-operto d Developmet (OECD) d the Europe Uo (EU). Se the 972 Reommedto by the OECD Coul o Gudg Prples oerg Itertol Eoom Aspets of Evrometl Poles ths prple hs bee reommed s the prple to be used for llotg osts relted to polluto. The sope of ths prple hs evolved over tme to lude lso detl polluto d le-up osts d, ts ew verso, s ofte referred to s exteded Polluter Pys prple. There re vrous deftos, terprettos or versos of the polluter pys prple the lterture. The defto tht best fts the method proposed ths pper s gve the Glossry of Sttstl Terms of the OECD (202): The polluter pys prple s the prple ordg to whh the polluter should ber the ost of mesures to redue polluto ordg to the extet of ether the dmge doe to soety or the exeedg of eptble level (stdrd) of polluto. Bsed ths prple, we propose method to dvde frly the osts of leg up rver pollutts mog the polluters d provde xomt hrterzto for ths method. Ths pper s losely relted to the pper by N d Wg (2007). These uthors propose two methods to splt the osts of leg up polluted rver mog the gets loted log t. They bsed ther methods, lled the Lol Resposblty Shrg (LRS) method d the Upstrem Equl Shrg (UES) method, respetvely, o the two m dvoted dotres tertol dsputes: the Absolute Terrtorl Soveregty (ATS) d the Ulmted Terrtory Itegrty (UTI). 2 These two dotres re terpreted terms of resposbltes the polluto ost lloto problem. Uder the ATS dotre, for exmple, the osts to le up pollutts gve segmet of the rver should be ssged oly to polluters loted tht segmet. However, uder the UTI dotre, pollutgleg osts gve segmet should be ssged to polluters loted tht segmet s well s ll upstrem polluters. Our pproh dffers from the pproh of N d Wg (2007) o two ey pots. Frst, the method we propose s bsed o model of pollutt trsport d the polluter pys prple. Seod, to shre dowstrem osts, we see to estblsh the resposblty of eh upstrem polluter ordg to ts otrbuto to the polluto level eh segmet of the rver, tht s, upstrem polluters re ot treted symmetrlly s do N d Wg (2007). As these uthors, we lso vestgte the reltoshp of the proposed method d ts soluto wth the (weghted) Shpley vlue of the ssoted ost gme geerted from the problem. I ddto, we lso vestgte the relto betwee the τ- vlue of the orrespodg ost gme d the soluto of the method proposed ths pper. For ltertve versos or terprettos of the polluter pys prple see, for exmple, Bugge (996). 2 For more detls o the ATS d the UTI dotres see, for exmple, Klgour d Dr (996).

3 Gómez-Rú (203) lso vestgtes the lloto of osts to le up polluted rver d tes s strtg pot the wor of N d Wg (2007). She reples the xom tht trets upstrem polluters symmetrlly by three ltertve xoms d ordgly osders three dfferet systems of weghts to hrge polluters. Regrdg the method proposed ths pper, the m dfferee s tht our system of weghts s dfferet from those osdered by Gómez-Rú (203). We bse our system of weghts o trsfer oeffets of model of pollutt trsport whle she spefes how the weghts re determed oly oe of her systems of weghts d, ths se, they deped o exogeous ftors, suh s ty populto, wter osumpto, et. Her lyss lso does ot lude omprso wth the τ- vlue of the ost gme geerted from the problem. Gómez-Rú (202) exteds her prevous wor to the otext of rver etwor. The rest of ths pper s strutured s follows. The ext seto presets the model used to formlze the Polluter Pys method of llotg the totl ost to le up pollutts rver. The thrd seto provdes xomt hrterzto of the Polluter Pys method. We used four xoms to provde ths hrterzto. I ths seto we lso show tht the Polluter Pys method s the oly method tht stsfes the four xoms preseted d tht ts soluto odes wth both the weghted Shpley vlue d the τ- vlue d s the ore of the orrespodg ost gme geerted from the problem. Flly, the fourth d fl seto, we preset some oludg ommets. 2. The model As N d Wg (2007), osder rver dvded to segmets tht re dexed gve order =,..., from upstrem to dowstrem. We ssume tht eh segmet s hbted by people who my be dversely ffeted by wter polluto d tht t the begg of eh segmet, ordg to the bove order, there s frm or dustry tht dshrges pollutts of some d to the rver. I every segmet, evrometl uthorty sets the mxmum level of polluto tht s llowed for tht segmet. If the polluto level exeeds tht lmt, the polluters re requred to sped so tht the qulty of the wter body meets the evrometl stdrd set for the segmet. The totl osts to le pollutts ross the rver s equl to = = We ssume tht these osts re t ther lowest levels d tht the tehology used leg the rver s the most effet. We wt to fd method to llote to totl ost of leg up rver pollutts frly mog the pollutg frms loted log the rver. To estblsh the method to llote those osts we eed to trodue some ddtol ottos. We rell tht ey feture of wter polluto s tht the pollutt s trsported wth the wter from upstrem pot (soure) to dowstrem pot (reeptor). Some rver pollutts re bodegrdble d the oetrto of polluto they use s redued log the rver. To desrbe the trsport of the wter pollutt d the level of polluto t uses log the rver we use trsfer oeffets. 3 To defe these oeffets, suppose there s le rver d tht ert mout of pollutt e s dshrged t upstrem pot (soure) d tht, fter some tme, we hve the level (oetrto) of polluto p t the dowstrem pot (reeptor). To estblsh the reltoshp betwee the emsso level t 3 Severl uthors hve used trsfer oeffets to lyze the sptl spets ssoted wth polluto. Hug d Shw (2005), for exmple, use trsfer oeffets the lyss of system for trdg wter polluto dshrge permts.

4 the pot (segmet) d the polluto oetrto t the pot (segmet), we use the followg expresso (see, for exmple, Kolstd, 2000, p ): p = e () where s the trsfer oeffet. Typlly these oeffets te strtly postve vlues d, >. Note tht f the emssos t the pot hge by lttle ( e), ordg to the expresso (), the level of polluto t the pot wll hge by e, so we defe the trsfer oeffet s the rto of the hge polluto oetrto t pot to the hge emssos t pot,.e., = p / e. Bslly, ths oeffet gves the overso rte for emssos to polluto level. If, for exmple, =. 2, ths mes tht ut of emssos t the pot geertes. 2 ut of polluto oetrto t pot. The expresso () be geerlzed to osder, for exmple, emssos t vrous upstrem pots (segmets), e, e2,..., ei (where I represets the umber of soures), d the rver s ot ompletely le. I ths se, the expresso () hges to I = = p e + B where p represets the level of polluto t the dowstrem pot ( I) d B s the bgroud level of polluto t the pot (segmet). 4 Note tht emssos from frm (loted segmet of the rver) rese polluto levels ll dowstrem segmets. However, polluto levels the segmet 2 re due oly to the emssos of frm (loted segmet ) d frm 2 (loted segmet 2). We ssume tht ll the segmets of the rver represet both soures d reeptors of pollutts. Thus, the mtrx of trsfer oeffets of the segmets s represeted by the followg upper trgulr mtrx I ths pper we lso ssume tht the trsfer oeffets re determed expermetlly for eh rver d type of wter pollutt. For gve mout of pollutt, the vlues of these oeffets my vry ordg, for exmple, to the volume of wter flowg rver d/or wter temperture. Oe determed the trsfer oeffets, the evrometl uthorty eed oly mesure the level of polluto eh rver segmet. Bsed o the levels of polluto eh segmet d trsfer oeffets, the levels of emssos eh segmet be lulted dretly. Therefore, t s ot eessry to mesure dretly the levels of emssos of eh frm eh segmet. Now we defe the four xoms tht support the ost lloto method proposed ths pper. The formlzto of these xoms s mde the ext seto. The ft tht dowstrem polluters re ot resposble (or do ot hve otrol) for the osts urred upstrem segmets s trslted to xom lled Idepedee of Upstrem Costs. 4 Note tht we re ssumg tht the reltoshp betwee emssos d level of polluto s ler. Ths ssumpto s very mportt the otext of our lyss.

5 Bslly ths xom sd tht the osts urred the segmet shll be lloted oly to the polluters loted ths segmet s well s ll the upstrem polluters. The ext xom, lled Upstrem Asymmetry, s bsed o the ssumpto tht the otrbutos of upstrem polluters to the level of polluto gve dowstrem segmet re ot eessrly equl. I ths se, ordg to the polluter pys prple, the resposblty of eh upstrem pollutg frm must be orde wth the proporto of ts otrbuto to the oetrto of polluto eh segmet of the rver. The thrd xom s lled Effey. It smply sttes tht the totl osts of leg up pollutts should be fully lloted mog the polluters. The lst xom s lled Addtvty. It s used s requremet of osstey order to obt uque vlue lloto. N d Wg (2007) estblsh d use the xoms Idepedee of Upstrem Costs, Upstrem Symmetry, Effey d Addtvty to hrterze ther UES method. I ths pper, we reple the xom Upstrem Symmetry by the xom Upstrem Asymmetry. Ths hge reflets the wy upstrem polluters re treted the ost lloto problem to redue polluto gve dowstrem segmet. Whle N d Wg (2007) tret upstrem polluters symmetrlly, we tret them symmetrlly. Ths hge s possble wth the help of the polluter pys prple d the prevously defed trsfer oeffets d exteds the sope of the UES method. Gómez-Rú (203) uses lso the sme set of xoms suggested by N d Wg (2007), exept Upstrem Symmetry. Ths xom s repled by three ltertve xoms: Upstrem Moototy, δ- Bodegrdto Rte, d Proportol Tx. For the frst, t s ssumed tht, for bodegrdble pollutts, the further wy the segmet of the rver s from polluter, the smller the prt of the ost ths polluter should py for leg tht segmet. For the seod, t s ssumed tht the bodegrdto rte of pollutt s ow d equl to δ. For the lst ltertve xom, t s ssumed tht the ost ssged to polluter for leg gve segmet should be lulted bsed o exogeous ftors, suh s ty populto, wter osumpto, polluto lod, et. Now suppose tht the totl pollutg-leg ost = s represeted by the vetor = (,..., ) R+, where deotes the osts requred to redue polluto the th segmet, d tht the polluto ost lloto problem s represeted by the pr C = ( N, ), where N = {,..., } s fte umber of gets (polluters). A soluto to ths problem s vetor x = ( x,..., x ) R+ suh tht x = = =, where x s the ost shre ssged to the th polluter. A method s rule tht ssgs to eh problem ( N, ) soluto x ( N, ). Applyg the polluter pys prple d bsed o the model desrbed bove, we propose for y R+ the followg method, lled Polluter Pys method. PP x ( ), =,..., (2) = = = where re the trsfer oeffets. To llustrte the use of ths method, osder the followg smple exmple: polluted rver s dvded to three segmets,.e., N ={,2,3}, d the ost to redue polluto set by the evrometl uthorty the three segmets s gve by the vetor = (, 2, 3). Aordg to the expresso (2), the ost shre ssged to polluter N s gve by 2 3 x PP ( ) =

6 Note tht lerly, gve by = 3, = x PP 2 ( ) = x PP 3 ( ) = x = = = = 33. If we ssume tht the trsfer oeffets re d 3, the the ost shre ssged to the three polluters the thrd segmet re x PP 2 ( ) = 6 3, x PP 3 2 ( ) = 6 3, d x PP 3 ( ) = 6 3,.e., the upstrem polluters re treted symmetrlly. Note lso tht f we ssume tht ll trsfer oeffets re equl, _.e., f =, for ; =,..., ; =,...,, expresso (2), the we obt the UES method proposed by N d Wg (2007). I ths sese, the Polluter Pys method s more geerl d ludes the UES method s spel se. Regrdg the systems of weghts, Gómez-Rú (203) shows tht method x ( N, ) stsfes the xoms Idepedee of Upstrem Costs, Effey d Addtvty f d oly f for every segmet =,...,, there exsts weght system ( ) R suh tht s = 0 for s N + polluter > d = =. If we dd Upstrem Moototy to these three xoms, the s weght system must stsfy ddtolly s s for every <, or s =δ s for every <, f, sted, we hd dded the δ- Bodegrdto Rte xom. I both ses, she does ot spefy how the weghts re obted. Oly whe the Proportol Tx s dded to the three prevous xoms, she ssumes tht the weghts re gve by s = w/ l = w l for ll, where w s weght vetor bsed o exogeous ftors. I our pproh, the weghts re determed edogeously d ome from pollutt trsport model. 3. Results I ths seto we preset four propostos. The frst proposto shows tht the Polluter Pys method s the oly method tht stsfes the four xoms metoed erler. The seod d thrd propostos show, respetvely, tht the lloto of the totl osts of leg rver pollutts suggested by the Polluter Pys method odes wth both the weghted Shpley vlue d the τ- vlue of the ost gme geerted from the problem. The fourth proposto shows tht both the weghted Shpley vlue d the τ- vlue, d therefore the soluto of the Polluter Pys method, re the ore of the orrespodg ost gme geerted from the problem. We ow formlze the four xoms tht hrterze the Polluter Pys method. Idepedee of Upstrem Costs. For y 2 for ll >, we hve x ( ) = x ( ). 2 R+, d N suh tht =, l >, 2 l l Ths formlzto of the xom follows N d Wg (2007) d smply sttes tht the ost shre of gve frm orrespods to ts ow polluto ost s well s ll dowstrem osts, but does ot lude upstrem osts.

7 Upstrem Asymmetry. For y R+ d N, f 0, the we hve the followg proporto of ths ost to be pd by frm x ( ) =, =,...,, = where ( =,..., ; =,..., ) re trsfer oeffets s defed the prevous seto. Effey. x = = =. Addtvty. For (,..., = ) R+ 2 2 x ( + ) = x ( ) + x ( ) for ll =,...,. d (,..., = ) R+, we hve Proposto. The Polluter Pys method s the oly method stsfyg Idepedee of Upstrem Costs, Upstrem Asymmetry, Effey d Addtvty. Proof. It s esy to verfy tht the Polluter Pys method stsfes the four xoms bove. Now we show tht the Polluter Py method s the oly method tht stsfes these xoms. Cosder the -dmesol ost vetor =(0,...,0,,0,...,0), =,2,...,, where s the th ompoet of Now suppose tht for ll,. By Idepedee of Upstrem Costs, ( ) = 0 x ( x for ll >. ) = β, where β s some oegtve rel umber. By Upstrem Asymmetry d by Effey, we must hve x ( ) = f d x ( ) = 0 f >. l = l Note tht se the -dmesol ost vetors,, =,2,...,, form bss of y R be wrtte s = = =,,..., ). Addtvty mples x ( ) = x ( ) = = = x ( ) = = = x PP (). ( 2 l= for ll N. The proposto s proved. l se x ( ) = 0 f >. Ths result lerly dtes tht the Polluter Pys method trets ll frms frly, tht s, the ost ssged to eh frm s lulted orde wth the proporto of ts otrbuto to the oetrto of polluto ts ow segmet d ll ts dowstrem segmets. To lyze the reltoshp betwee the soluto of the Polluter Pys method wth vlues, suh s the (weghted) Shpley vlue d theτ- vlue, we eed to model the polluto ost lloto problem s oltol gme wth trsferble pyoff. 5 Ths type of gme s R, 5 For more detls o oltol gmes see, for exmple, Osbore d Rubste (994).

8 desrbed by pr G = ( N, v), where N s fte set of plyers d v (S) s rel umber tht represets the worth of the olto S ( S N). I the otext of ost gme, worth represets the osts urred by the olto S. Now, lettg N = {,2,..., } be the set of gets (pollutg frms) d lettg S N be y olto from the polluters, ll we eed s to defe the hrterst futo v (S). To do ths, we deote by m S the most upstrem pollutg frm the olto S. Uder the polluter pys prple, eh member of S N s resposble for the osts to redue polluto ts ow segmet d ll dowstrem segmets orde wth ts otrbuto to the level of polluto. Therefore, the ost of the olto S, for y gve R, should be Now, for y gve R+ + v ( S) =. (3) = m S, we hve oltol gme, ( N, v ), tht stsfes v ( ) = 0. Ths gme s detl to tht proposed by N & Wg (2007) uder the UTI dotre. Se both the (weghted) Shpley vlue d theτ- vlue detfy sgle result oltol gme, they re exmples of vlues. A vlue s futo φ defed o the spe of gmes N G, : G R v ), for N φ, tht detfes fesble lloto, φ = N ( ) v( N every oltol gme. The weghted Shpley vlue be hrterzed by the xoms of Dummy Plyer, Asymmetry, Effey d Addtvty d be defed usg umty gmes by the followg expresso (see, for exmple, Herger, 2006) φ ( v, ω) = α φ( u ( S), ω) = T T where φ ( v, ω) s the lloto to plyer t the result φ( v, ω), u T (S) s the umty gme defed by u T ( S) = f S T d u T ( S) = 0 otherwse, d ω s weght vetor (ω ) N s t beg ω R+ the weght ssoted to plyer N; α T = ( ) v ( T), beg s d t T S the umber of plyers oltos S d T, respetvely. I the ext proposto, we show tht the Polluter Pys method s osstet wth the weghted Shpley vlue of the gme ( N, v ), tht s, the soluto of the Polluter Pys method (2) odes wth the weghted Shpley vlue φ of the gme ( N, v ) for ll T T α T ω S ω R+. Proposto 2. For ll pollutg frms N. R+ d x ( = φ ω for ll PP v defed by (3), we hve ) ( v, ) Proof. Cosder g the -dmesol ost vetor =(0,...,0,,0,...,0), =,2,...,, where s the th ompoet of. The oltol gmes ssoted wth these ost vetors, =,2,...,, re gve by v ( S) = 0 f m S > d ( S) = otherwse. v Note tht for the gmes ( N, v ), ll pollutg frms > re dummes d ll pollutg frms re ot treted symmetrlly, tht s, dfferet weghts re ssged to them ordg to ther otrbuto to the oetrto of polluto ther ow segmets s well s dowstrem segmets. If these weghts re defed orde wth the trsfer oeffets preseted prevously, the weghted Shpley vlues of the gmes ( N, v ) re

9 φ ( v, ω) = 0 f > d φ (, ω) otherwse. (4) v = l = l Se y R+ be obted by = = S N of the gme ( N, v ), for ll v ( S) = = m S = = m S = v = m S By Addtvty d expressos (4), we get = m S ( S), φ ( v, ω) = φ ( v, ω) = = = x PP () =,2,...,. The proposto s proved. l= l, we hve, for ll oltos Ts (987) uses three xoms to hrterze the τ- vlue, beg Effey oe of them. The others two xoms re lled Mml Rght Property d Restrted Proportolty Property. The Mml Rght Property s mpled by Addtvty together wth Idvdul Rtolty ( x v({}) the se of ost gme) d Effey whle the Restrted Proportolty Property mples tht the gs or osts ssged to gve plyer s proportol to hs(her) mrgl otrbuto to the grd olto,.e., the olto formed by ll plyers. For the ext proposto, tht estblshes the reltoshp betwee the Polluter Pys method d the τ- vlue of the orrespodg ost gme, we eed some ddtol otos. These otos re used to estblsh the expresso of the τ- vlue for the ost gme. 6 The frst of them s the mrgl vetor ( m v ) of the gme ( N, v ) tht s defed s the vetor wth th oordte m( v ) = v ( N) v ( N {}). I ost lloto problem, the rel umber m ( v ) mesures the smllest otrbuto of plyer to the worth of the grd olto v (N) f he(she) os the olto of ll the others N {} plyers. I dvso of the totl ost of leg up polluted rver, for exmple, plyer must ot expet to otrbute less th hs(her) mrgl vlue m ( v ). Thus, we see m ( v ) s lower boud for plyer the lloto of the totl ost v (N) of the gme ( N, v ). It s esy to show tht, ll possble oltos 0 S N tht plyer o, m( v ) v ( S) g ( S) = v ( S) m( v ) plys mportt role the S. The gp S 6 For more detls o the τ - vlue see, for exmple, Ts (987).

10 defto of the τ- vlue of the gme ( N, v ). I ft, for oltol gmes wth g ( S) 0, we must ostrut pyoff vetor x suh tht m( v ) x m( v ) + λ( v ), where λ ( v ) = mx g( S) d the expresso m( v ) + λ( v ) represets upper boud for plyer S; S the lloto of the totl ost v (N) of the gme ( N, v ). For gmes ( N, v ) wth g ( N) 0, the τ- vlue s defed s follows. m( v ) f g( N) = 0 τ ( v ) = (5) m( v ) + α( v ) g( N) f g( N) > 0 where α ( v ) [0,] s hose suh tht τ ( v ) = v ( N). Tg to out ths Effey xom of the τ- vlue d the hrterst futo of the oltol gme defed by (3), we defe the weghts α ( v ) smlr wy s we dd the Proposto 2. N Proposto 3. For ll pollutg frms N. R+ d = τ for ll PP v defed by (3), we hve x ( ) ( v ) Proof. For y oltol gme ( N, v ) wth ost vetor = = = (, 2,..., ) R+, we hve g( N). I ths gme, the mrgl otrbuto of the pollutg frms re m 2 ( v ) = f = d,, d m ( v ) = 0, f = 2,3,...,. Note tht, f we wt to hrge frms,, 2, ordg to ther otrbuto to the oetrto of polluto ther ow segmets s well s ll dowstrem segmets, the prmeter α ( v ) of the equto (5) ot be ostt, but should vry ordg to eh frm d eh segmet of the rver. So, for y olto S N, v (S), we should hve α ( v ) = 0 f m S > d α ( v ) f( ) (0,] otherwse, (6) = where =,..., refers to the th pollutg frm, =,..., refers to the th segmet of the rver, d ( =,..., ; =,..., ) re the oeffets of the pollutts trsport model defed prevously. Tg ths to out, we lulte the τ- vlue for th pollutg frm the of the oltol gme ( N, v ) wth g ( N) > 0 s follows + = ( v ) τ ( v ) = m ( v ) α. (7) Equto (7) s equvlet to the followg equto se t stsfes expressos (6), m ( v ) = f =, d m ( v ) = 0, f = 2,3,...,, d the xom of Effey, tht s, ( v ) = for ll τ = = x PP () =,2,...,. The proposto s proved. Flly, we show the ext proposto tht both the weghted Shpley d the τ- vlue, d therefore the Polluto Pys method soluto, re the ore of the orrespodg ost gme ( N, v ). We show tht dretly, usg the oto of ovexty gmes. I ovex gmes the ore s lwys oempty d the Shpley vlue s ore lloto (see, for exmple, Moul, 988). Moderer et l. (992) show lso tht the set of ll weghted 2 l= l

11 Shpley vlues ots the ore of oltol gme. For ost shrg gmes, the ovexty property s equvlet to the ovty of the gme (N d Wg, 2007). A gme G = ( N, v) s lled ove f d oly f v( S {}) v( S) v( T {}) v( T) (8) for every N d for every S T N\ { }. We rell tht the ore of ost lloto gme, C (v), s defed by C( v) = { x X: x } S v( S) for every S N where X = { x R : x v( N) }. = N Proposto 4. For ll R+, both the weghted Shpley vlue d the τ - vlue of the ost PP gme ( N, v ) re the ore,.e., φ ( v, ) ( v ) x ( ) v ( S) S ω = τ S = S for every S N. Proof. We eed oly to show tht the gme ( N, v ) s ove,.e., for every N d for every S, T N\{ }, f S T, the (rerrgg(8)) v ( T {}) v ( S {}) v ( T) v ( S) (9) There re three ses to osder: Cse. If < m T = m S, m T =m S < or m T <m S <, the (9) s stsfed trvlly s equlty. Cse 2. If < m T < m S, the (9) s stsfed s strt equlty se v ( T {}) = v ( S {}) d v ( T) > v ( S). Cse 3. If m T < < m S, the (9) s stsfed s strt equlty se v ( T {}) v ( S {}) < v ( T) v ( S). These three ses over ll possbltes d show tht the gme ( N, v ) s ove. The proposto s proved. 4. Coludg remrs I ths pper we vestgte the lloto ost problem of leg up polluted rver mog the gets loted log t. We propose method bsed o the polluter pys prple d pollutt trsport model to solve t. Ths method s hrterzed by four xoms. We lso show tht the soluto of the proposed method odes wth the (weghted) Shpley vlue d the τ- vlue to the orrespodg ost gme tht s dued ordg to the polluter pys prple. Aother e property of ths soluto s tht t belogs to the ore of the ost gme lyzed d, therefore, s stble lloto. The formto requred to use the proposed method (Polluter Pys method) lude the type of wter pollutt (whether bodegrdble or ot), the trsfer oeffets lulted for type of wter pollutt d segmet of the rver, d the oetrto of polluto eh segmet. Wth ths formto, t s possble to lulte dretly the levels of emssos of the frms eh segmet of the rver. The ost lloto suggested by ths method s bsed o these oeffets d the estmted ost to le pollutts from the rver eh segmet. Note tht the method s bsed oly o tehl formto tht s spef for eh polluted rver. The more urte ths tehl formto s, the more relble the pplto of ths method wll be.

12 Referees Bugge, H. C. (996) The prples of polluter pys eooms d lw, Ede E. d v ser Bergh R. (eds) Lw d Eooms of the Evromet, Oslo: Jurds Forlg, 996. pp Gómez-Rú, M. (202) Shrg polluted rver etwor through evrometl txes Eooms Bullet 32, Gómez-Rú, M. (203) Shrg polluted rver through evrometl txes SERIEs 4: Herger, G. (2006) A ew weght sheme for the Shpley vlue Mthemtl Sol Sees 52, Hug, M-F. d D. Shw (2005) A trdg-rto system for trdg wter polluto dshrge permts Jourl of Evrometl Eooms d Mgemet 49, Klgour, M. d A. Dr (996) Are stble greemets for shrg tertol rver wters ow possble? Worg Pper 474. World B, Wshgto. Kolstd, C. D. (2000) Evrometl eooms, Oxford: Oxford Uversty Press. Moderer, D., Smet, D. d L. S. Shpley (992) Weghted vlues d the ore Itertol Jourl of Gme Theory 2, Moul, H. (988) Axoms of oopertve deso mg, Cmbrdge: Cmbrdge Uversty Press. N, D. d Y. Wg (2007) Shrg polluted rver. Gmes d Eoom Behvor 60, Orgzto for Eoom Co-operto d Developmet (OECD). (202) - Glossry of Sttstl Terms. Polluter-Pys-Prple. Avlble t essed Jury 202. Osbore, M. J. d A. Rubste (994) A ourse gme theory, Cmbrdge: The MIT Press. Ts, S. H. (987) A xomtzto of the τ- vlue Mthemtl Sol Sees 3, 77-8.