Water Distribution as a Noncooperative Game

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1 Water Ditribution a a Noncooperative Game Ardehir Ahmadi, IHU Univerity,ehran, Iran Ardehir79@yahoo.com ABRAC he water ditribution problem of the Mexican Valley i modeled a a three-peron noncooperative ame in which ariculture, indutry, dometic water uer are the player, the total water amount upplied to the uer are the payoff function. he equilibrium i detered by olvin a nonlinear optimization problem, which can be derived baed on the Kuhn-ucer neceary condition. All contraint are linear the objective function i quadratic, o tard olution alorithm oftware can be ued.. Introduction he limited amount of natural reource create conflict between the uer, ince any uer can increae it upply only in the expene of the other. A it i well nown, water hortae i one of the mot worryin problem of our ociety. Accordin to recent prediction, by 00 we will need about 7% more water than available to feed the world. herefore efficient uae of water optimal water ditribution cheme become neceitie. he Mexican Valley i one of the mot critical area, ince Maxico City with it 9 million inhabitant i the mot populated city in the world, ariculture i the main economic activity in the reion which require lare amount of irriation water. In addition there are indutrial uer, the development of indutrial employer i crucial for the welfare of the population. he water upply i divided into urface water, roundwater treated water. he roundwater upply ha the bet quality, treated water ha the wort. o, treated water can have only limited uae. he roundwater reource are overexploited at a rate of 00% or more, which ha the direct conequence of dryin prin inin round up to 0.4m/year in ome area. he current water hortae ituation raie the neceity of importin urface roundwater from the neihborin waterhed, which alo miht raie dipute over water reource. he water ditribution problem of the Maxican Valley i modeled a a three peron ame. where the three uer are the player the total water amount upplied to the uer are the payoff function. here are many earlier wor dealin with imilar problem. he urvey paper of Hipel (99) the more recent paper of Kapelan et al. (005), Doneva et al. (00), Coppola zidarovzy (004), alazar et al. (00) can be mentioned amon other. A the ame theoretical concept method are concerned the boo of zidarovzy et al. (986) Foro et al. (999) ive comprehenive ummarie.. Mathematical Model he three player are ariculture (=), indutry (=) dometic uer (=). For each player the deciion variable are =urface water upply from local ource =round water upply from local ource t =treated water upply =imported urface water upply =imported round water upply

2 o, the tratey of player i the 5-dimenional vector x (,, t,, ). he tratey et of the player are detered by feaibility contraint. he three uer have two common contraint: t D () t D (=,, ), () where D i the imum neceary amount, D i the total dem of player. Contraint () i requeted to avoid watin water. In addition to thee contraint each uer ha it own condition. Ariculture (=) ha two pecial water quality related contraint. ome crop can ue only roundwater, ince they are very enitive to the quality of the water they are irriated with. If i the ratio of the water need of thee enitive crop to the total water amount ued for irriation, then it i required that. () t ome le enitive crop can tolerate treated water, which ha the wort quality. If i ratio of the water need of thee le enitive crop to the total water amount ued for irriation, then we have to aume that t t (4) ince treated water cannot be ued to irriate crop which cannot tolerate treated water. Indutry (=) ha very imilar contraint, ince there are certain uae which can be upplied only by roundwater a well a ome other uae can tolerate treated water: (5). (6) Dometic uer (=) have limitation only on the amount of treated water, ince it can be ued only for limited purpoe (uch a irriatin par, olf coure, etc). Hence thi contraint can be iven a t t (7) Notice that all contraint ()-(7) can be rewritten into linear form. he total water availably in both local imported urface roundwater reource can be repreented by the additional contraint: (8) (9) (0) () where the riht h ide are the maximum available amount upplied from the different reource. We require equality in contraint (8) (9) to be ure that all local reource have to be ued before water i imported from other waterhed. he payoff function for each player i the water upply: Maximize t. () Hence we have a three-player noncooperative ame with linear payoff linear contraint definin the tratey et of the player.

3 . olution Methodoloy In our cae the number of player i n=, the tratey of player i the five-dimenional vector x, the tratey et of thi player i defined by the individual contraint A x a () obtained from () throuh (7) by the joint contraint (8) throuh (): B x B x B x b. (4) he payoff of player can be written in eneral a maximize c x. (5) ince the contraint are linear, the Kuhn-ucer reularity condition are atified, o there are nonneative vector v w uch that A x a x B x B x b B v A w B v ( a A x ) w ( b B x B x B x) 0. c 0 (6) Becaue of the other contraint the left h ide of the lat condition i alway nonneative, o it imal value i zero. Conider now the followin quadratic proram problem with linear contraint: imize v ( a A x ) w b B x B x B x ubject to c v v, w A A x w a 0 B 0 (,, ) B x B x B x b. he Niaido-Ioda theorem implie the exitence of at leat one equilibrium, o there i at leat one olution of ytem (6). o the optimal objective function value i zero, all optimal olution atify the Kuhn-ucer neceary condition (6). Becaue of the linearity of the ame, condition (6) are alo ufficient, implyin that all optimal olution of problem (7) are Nah-equilibria of the three-peron ame. For the numerical olution of (7) tard methodoloy oftware i available. 4. Numerical Reult he data for our cae tudy were iven by a reearch roup of the Univeridad Autonoma Chapino, who invetiated the ame problem by uin different olution concept methodoloy (alazar et al., 007). he input data are iven in able. In addition, the available water upplie from the able = = = D D (7)

4 different ource are limited a iven in contraint (8) throuh () with =69. hee quantitie =58, =70, =45 D D are iven in mill m /year, the contant are ratio, unitle quantitie. he equilibrium wa computed by olvin the optimization problem (7). he reult are preented in able. he objective function at the optimum wa zero howin able. Numerical Reult = = = otal t otal that lobal optimum wa reached. All dem of ariculture indutry can be atified, ince the received amount are their total dem. he dometic dem can be atified only partially, on 59.4% level. All urface roundwater reource are ued. he retrictive contraint on treated water uae mae the ue of more treated water impoible. Importin the lare amount of urface roundwater into the Maxican Valley will raie evere conflict with the neihborin reion. In addition, a larer amount of treated water ha to be ued which raie the important iue of water quality, ince under the iven contraint the only way to increae water upply i to increae the treated water uae. In order to have larer water upply avoid eriou ocial conflict further development are needed in combination with more efficient water uae by the three ector. Maybe a maret-driven water pricin policy would ive incentive to the uer. 5. Concluion he water ditribution problem in the Maxican Valley wa modeled a a three-peron noncooperative ame. he three uer were the player, the upplied water amount the payoff, the tratey et were detered by upply water quality contraint. he Nah equilibrium of thi ame wa detered by olvin a pecial quadratic optimization problem with linear contraint, which wa derived baed on the Kuhn-ucer condition. he numerical reult indicate that a combination of invetment for further development in the infratructure i needed in combination with more efficient water uae in order to avoid eriou ocial conflict water hortae. 6. Reference zidarovzy, F., M. Gerhon, L. Ductein (986) echnique of Multiobjective Deciion Main in ytem Manaement. Elevier, Amterdam. Hipel, K. W. (99) Multiple objective deciion main in water reource. Water Reour Bull 8(): -. Kapelan, Z.., D. A. avic, G. A. Walter (005) Multiobjective dein of water ditribution ytem under uncertainty. Water Reour Re 4: W407. Doneva, K.,. Dodeva, J. adeva (00) Urban aricultural competition for water in the Republic of Macedonia. Coppola, E. F. zidarovzy (004) Conflict between water upply environmental health ri: A computational neural networ approach. International Game heory Review 6(4):

5 Foro F., J. zep, F. zidarovzy (999) Introduction to the heory of Game. Kluwer, Dordrecht. alazar, R., F. zidarovzy, A. Rojano (00) Water ditribution cenario in the Mexican Valley. Water Reource Manaement, DOI 0.007/ alazar, R., F. zidarovzy, E. Coppola A. Rojano (007) Application of ame theory for a roundwater conflict in Mexico. Journal of Environmental Manaement 84: