w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6

Avai lab le a t www.sc iencedi rec t .com

journa l homepage : www.e lsev ie r . com/ loca te /wat res

Game theory based models to analyze water conflicts in theMiddle Route of the South-to-North Water Transfer Projectin China

Shouke Wei a,b,*, Hong Yang a,1, Karim Abbaspour a,1, Jamshid Mousavi a,c,1,Albrecht Gnauck b

a Department System Analysis, Integrated Assessment and Modelling, the Swiss Federal Institute of Aquatic Science and Technology

(EAWAG), Ueberlandstrasse 133, CH-8600 Dubendorf, Switzerlandb Department of Ecosystem and Environmental Informatics, Brandenburg University of Technology, Konrad-Wachsman-Allee 1,

D – 03046 Cottbus, Germanyc Department of Civil Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, Tehran, Iran

a r t i c l e i n f o

Article history:

Received 13 August 2009

Received in revised form

10 December 2009

Accepted 24 January 2010

Available online 1 February 2010

Keywords:

Game theory

Water conflicts

Economic valuation

Scenario analysis

The water transfer

China

* Corresponding author at: Department SystScience and Technology (EAWAG), Ueberlan

E-mail addresses: shouke.wei@eawag.chjmosavi@aut.ac.ir (J. Mousavi), Albrecht.gna

1 Tel.: þ41 44 823 5568; fax: þ41 44 823 5370043-1354/$ – see front matter ª 2010 Elsevidoi:10.1016/j.watres.2010.01.021

a b s t r a c t

This study applied game theory based models to analyze and solve water conflicts con-

cerning water allocation and nitrogen reduction in the Middle Route of the South-to-North

Water Transfer Project in China. The game simulation comprised two levels, including one

main game with five players and four sub-games with each containing three sub-players. We

used statistical and econometric regression methods to formulate payoff functions of the

players, economic valuation methods (EVMs) to transform non-monetary value into

economic one, cost-benefit Analysis (CBA) to compare the game outcomes, and scenario

analysis to investigate the future uncertainties. The validity of game simulation was evalu-

ated by comparing predictions with observations. The main results proved that cooperation

would make the players collectively better off, though some player would face losses.

However, players were not willing to cooperate, which would result in a prisoners’ dilemma.

Scenarios simulation results displayed that players in water scare area could not solve its

severe water deficit problem without cooperation with other players even under an opti-

mistic scenario, while the uncertainty of cooperation would come from the main polluters.

The results suggest a need to design a mechanism to reduce the risk of losses of those players

by a side payment, which provides them with economic incentives to cooperate.

ª 2010 Elsevier Ltd. All rights reserved.

1. Introduction been an important cause of water scarcity in countries (Wang

From an economic perspective, water resources are composite

assets which provide a variety of services for consumptive and

productive activities. However, water quality degradation has

em Analysis, Integrated Adstrasse 133, CH-8600 Du

(S. Wei), hong.yang@eauck@ti-cottbus.de (A. Gna5.er Ltd. All rights reserved

et al., 2003; Wei and Gnauck, 2007a). Water resources

management on those problems is usually involved with

interactive and interdependent stakeholders with contradic-

tory or conflicting interests (Fang et al., 1998, 2002; Van der

ssessment and Modelling, the Swiss Federal Institute of Aquaticbendorf, Switzerland. Tel.: þ41 44 823 5568; fax: þ41 44 823 5375.wag.ch (H. Yang), karim.abbaspour@eawag.ch (K. Abbaspour),

uck).

.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62500

Veeren and Tol, 2003), goals and strategies (Wei and Gnauck,

2007a). Pollutant discharge is an essential but complex issue

in water resources management, and this complexity is not

only from intricate biochemical processes, but also from

different pollutant sources and multi-polluters with conflict-

ing aims. Water quality and quantity conflicts are usually

caused by (1) water scarcity due to uneven precipitation, (2)

multiple users and pollutant sources discharging waste into

water, (3) different degrees of upstream pollutions restricting

the water use in downstream catchment, and (4) interbasin

water transfer breaking the long-established balance of water

quality and quantity in a basin.

To solve water conflicts cause by water scarcity, Donevska

et al. (2009) proposed some engineering solutions in terms of

reducing water losses, increasing water use efficiency and

waste water recycling, water conservation, and water trans-

fer, and some other non-engineering measures. However,

methods of using water efficiency and waste water recycling

are not so sufficient to the regions facing extreme water

shortage. In addition, interbasin water diversion involves

a multidisciplinary problem (Yevjevich, 2001), which usually

brings about fundamental issues and conflicts concerning

socio-economical, environ-ecological, administrative and

legislative problems (Shao and Wang, 2003; Yang and

Zehnder, 2005). Besides, different economic and political

instruments have been widely used to solve water use

conflicts (Dinar and Howitt, 1997; Wang et al., 2003). Water

markets approach is one cited frequently in the literature

(Burness and Quirk, 1979; Howe et al., 1986; Colby, 1990; Green

and O’Connor, 2001; Bhaduri and Barbier, 2003). Water market

methods can provide water users with incentives to allocate

water and reduce pollutants discharge efficiently, and such

market really exists in some countries, such as Australia

(Pegram et al., 1992), California (Howe and Goodman, 1995),

Chile (Hearne and Easter, 1995), India (Saleth, 1996), and Spain

(Reidinger, 1994), etc. However, water market requires

defining the original water rights, creating institutional and

legal mechanisms, and establishing basic infrastructures for

water trade (Holden and Thobani, 1996; Wang et al., 2003)

before it can operate well. Waste discharge is a public bad, and

every polluter can free-ride others’ achievement of treatment

(Wei and Gnauck, 2007b). Free-riding problem will cause

market failure. In the absent of market and property right,

conflicts between multi-stakeholders competing for water

uses are unavoidable (Pethig, 1992; Wei and Gnauck, 2007a).

There are rare water markets in reality and they are not real

free market (Dellapenna, 2000). Those economic and political

based water conflict solving methods can be summarized into

two classes, direct regulations and economic instruments

(OECD, 1989; Markandya and Recharddson, 1992; Wei and

Gnauck, 2007a). Direct regulation is also known as the

‘‘command and control’’ strategies, which usually include

limitation quotas, standards, laws, etc. Economic and political

instruments make use of market mechanism, price incen-

tives, water rights, subsidies, compensation, tradable permits,

green taxations, etc. However, environmental resource prob-

lems and its interrelationships with economic activities and

the dynamic ecosystem are very complex and cannot be

solved with simple policy tools (Carraro and Filar, 1995).

Command and control strategies usually lack incentive,

because it is mainly in virtue of legislation, power or force. Wei

and Gnauck (2007a) stated that the existing economic and

regulation instruments do not work so well in solving these

conflicts. From a technical strategy point of view, multi-

objective optimization models have been used early to maxi-

mize the overall benefit in order to solve transboundary water

conflict ina riverbasin (Zeng et al., 2001; Yang andZeng, 2004). In

recent years, more advanced and popular multi-objective

evolutionary algorithms have been used to solve conflicts

objectives in watershed (Bekele and Nicklow, 2005; Muleta and

Nicklow, 2005). In general, however, those optimization

measures neglect the real interests and benefits of the stake-

holders in the basin, though they can capture the multiple

optimalsolutions, sometimescalled asParetooptimal solutions.

Game theory is an appropriate approach to model and

solve such water conflicts. It was launched by John von Neu-

mann, a great mathematician, and Oskar Morgenstern in

1944. Game theoretical modelling concepts and reasoning

have been widely applied in economic, commercial, social,

political, biological, and other sciences to help people analyze

social and behavioural phenomena. However, the applica-

tions of game theory to solve conflicts in water resources

management are comparatively few. As for this topic, it was

originally applied into the cost distribution in joint water

resource projects, i.e. waste water treatment and disposal

facilities (Giglio and Wrightington, 1972; Dinar and Howitt,

1997). Bogardi and Szidarovszky (1976) introduced possible

application of game theory, especially the oligopol game, in

four main areas of water management, and offered solutions

for the typical problems in decision analysis. Lewandowski

(1979) used a game-theoretic approach to model the behav-

iour of water users in a quality control problem, and he has

proposed a game-theoretic solution to different uses of

a water system. Coppola and Szidarovszky (2004) designed

a two-person conflicting game to analyze the optimal trade-

off between water supply and contamination risk for

a municipal wellfield. Salazar et al. (2007) described the

application of conflict resolution methods to a two-person

conflicts game in groundwater management. Besides, game

theory is regularly used to analyze equitable allocation of

waste loads to a common receiving medium (Kilgour et al.

1988; Okada and Mikami 1992; Wei and Gnauck, 2007b,c). It

was also applied to solve water allocation and pollution

problems in transboundary river, including inter-country river

(Frisvold and Caswell, 2000; Van der Veeren and Tol, 2003) and

intra-country river (Zeng and Yang, 2004; Yang and Zeng,

2004). In game theory, the development of conflict concepts

and methods has received increasing attention since the

pioneering work of Nash (1950). Based on the axiomatic

approaches of Nash, many modified and extended solutions

have been introduced. Among those solutions, four particular

methods have frequently citied and used in water manage-

ment in the literature (Coppola and Szidarovszky, 2004; Sala-

zar et al., 2007), including: (1) non-symmetric Nash solution of

Harsanyi and Selten (1972); (2) non-symmetric solution of

Kalai and Smorodinsky (1975); (3) non-symmetric area

monotonic solution of Anbarci (1993); and (4) non-symmetric

equal-loss solution of Chun (1988).

China possesses total water resources of 2812.4 billion m3,

ranking the 6th in the world (World Bank, 2002; Wei, 2007).

Fig. 1 – A sketch map of the South-to-North Water Transfer Project.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6 2501

However, due to spatially uneven distribution of precipitation,

water shortage has been a prolonged and widespread problem

in Northern regions of China (Yang and Zehnder, 2001; Wei,

2007). In order to mitigate the existing water crisis, the engi-

neers in China proposed the South-to-North Water Transfer

(SNWT) Projects, including the East Route Project, the Middle

Route Project and the West Route Project. The Middle Route

Project covers two municipalities and four provinces. It is

difficult to manage when water transfer involving such

different large regions. This study aimed to establish game

theoretical models to analyze water allocation and pollution

conflicts existing in the Middle Route of South-to-North Water

Transfer Project in China. The main goals of the study include:

� To forecast and analyze water supply, water demand and

water deficit in the different sectors in water scarce cities

(only Beijing municipality) in northern China,

� To predict and estimate pollutant production, discharge and

reduction from the different pollutant sources in the upper

basin of the Hanjiang River,

� To evaluate the economic benefits and losses of water diver-

sion and pollutant reduction to the cities in the study area,

� To analyze future uncertainty of the game simulation under

different scenarios.

Table 1 – Description of the cities, their main interests and pro

Province City ormunicipality

ID Interest

Beijing (BJ) Beijing R1 Obtaining sufficient water for so

and environmental protection

Shaanxi (SX) Hanzhong C1 Developing economy, improving

standard, reducing pollution bas

economic abilities

Ankang C2

Shangluo C3

Hubei (HUB) Shiyan C4

He’nan (HN) Xixia C5

Xichuan C6

2. Material and methods

2.1. Area description

The South-to-North Water Transfer (SNWT) projects in China

comprise the Western Route Project (WRP), the Middle Route

Project (MRP) and the Eastern Route Project (ERP) (Fig. 1). We

focus the study area on the MRP, including Beijing munici-

pality, and the 6 cities in the provinces of Shaanxi, He’nan and

Hubei in the upper basin of Hanjiang River (Table 1). The MRP

will divert water in 2010 from the Danjiangkou Reservoir in

the Hanjiang River Basin for 20 big cities and 100 counties in

Beijing and Tianjin Municipalities, He’nan and Hubei prov-

inces (CWRPI, 2005). It covers a total area of about 155,000 km2

and crosses about 200 river channels or canals with the total

cannel distance of 1246 km. Interbasin water transfer projects

to reduce water shortage are not new in China. However, WRP

covers two municipalities and four provinces. This project will

change the runoff and water level of the rivers, break the long-

established balances of benefits of different stakeholders, and

cause conflicts. Water transfer action within a region can be

effectively managed through the coordination of local

government and regional river administration, while water

blems in the study.

Main problems

cio-economic Developed region, severe water shortage, overusing

ground and ecological water, transferring water and

compensating to improve water quality in the basin

living

ed on their

Undeveloped regions, reducing pollutants for water

transfer, imposing extra cost

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62502

transfer involving different regions with such large areas is

usually difficult to manage.

The Hanjiang River basin lies in 30�080–40�110N latitude,

106�120–114�140E longitude. The river originates in the

southern part of Shaanxi Province, flows through Shaanxi and

Hubei provinces and joins the Yangtze River at Wuhan, the

capital city of Hubei province. It is about 1577 km long, the

longest tributary of Yangtze River; and the basin covers

a watershed area of 159,000 km2, the second largest river basin

in Yangtze River catchment. The Hanjiang River basin belongs

to subtropics monsoon area, and it is temperate and moist and

annual precipitation is 873 mm. According to the data series of

hydrology from 1956–1998, the river has total water resource

of 58.2 billion m3 and average annual natural runoff is

56.6 billion m3 (CWRPI, 2005). The river is traditionally divided

into three parts: upper, middle and lower rivers. The upper

river is from the river source to the Danjiangkou city, the

middle river from the Danjiangkou city to Zhongxiang City,

and the lower part from Zhongxiang City to the river mouth

(Zhang, et al., 2000). This paper only dealt with the upper basin

of the Hanjiang River. The upper river is 925 km long, and the

upper basin includes part of provinces of Shaanxi, He’nan and

Hubei. This part possesses surface water of 36.796 billion m3,

groundwater of 10.647 billion m3, and overlap amount is

10.387 billion m3. In the upper basin, the u-shaped Danjiang-

kou Reservoir is the water source of MRP, covering an area of

1050 km2 with a total storage capacity of 17.45 billion m3. The

average annual inflow of the Reservoir is 38.78 billion m3

approximately occupying 70% of water volume of the entire

basin.

Beijing municipality, the capital of China, is located in the

north-eastern part of China and covers an area of 16,808 km.

Beijing belongs to semi-humid climate region dominated by

the Pacific monsoon, and the annual average precipitation is

about 601 mm. Water scarcity is one of the serious problems of

this city due to uneven distribution of precipitation and water

pollution (Wei and Gnauck, 2007a; Wei et al., in press). Beijing

has a population of 15.38 million (BJSB, 2001–2009); but its

current available water resource per capita is only 247 m3 per

year, which is much less than the standard for water shortage

(1000 m3 per capita) defined by the United Nation (UN), and is

also below the minimum (300 m3 per capita) that the United

Nations Educational, Scientific and Cultural Organization

(UNESCO) defines to ensure a modern social life and produc-

tion. According to the internationally recognized standards of

extreme water deficit (�500 m3/person), Beijing belongs to the

areas of extreme water deficit (Li and Xiu, 2004; Wei, 2007).

2.2. Data sources

The data in this study includes climatological and hydrolog-

ical data (1986–2005), water quality data (1995–2004) environ-

ecological data (1994–2005) and socio-economic data

(1978–2008). Climatological and hydrological data include

precipitation, evaporation, surface water, groundwater, and

water flows. Water quality data comprise pollutants concen-

trations, point pollution sources (industrial waste discharge

and urban domestic waste water discharge), waste water

reclaim amount, and non-point pollution sources (agricultural

fertilizer consumptions, soil erosions, rural domestics and

animal husbandry). Environ-ecological data contains ecolog-

ical water use, urban water surface areas, public green areas,

and the numbers of newly planned trees. Those socio-

economic data mainly include urban and rural population,

water supply and water demand, gross industrial and agri-

cultural products, urban and rural per capita net incomes.

Data in the studied area were collected mainly from the

following sources: (1) different monitoring stations and

numerous controlling sections along the Hanjiang River and

its tributaries, (2) Database of the Changjiang Water Resource

Protection Institute (DB-CWRPI), (3) Online Database of

National Bureau of Statistics of China (DB-NBSC), (4) Chinese

statistic yearbooks in related fields at different administration

levels (BJSB, 2001–2009; BJWB, 2005; NBSC and SEPAC,

2001–2005; CWRA, 1998–2004; HBEPB, 2004–2005; HBSB,

1996–2005; HNSB, 1994–2007; NBSC, 1985–2007; NBSC and

SEPAC, 2000–2005; SXSB, 1991–2006), (5) Official reports and

planning documents (CWRPI, 2005; BDRC, 2006), and (6)

Previous studies (Hu and Zhang, 2003; Zhang, et al., 2000; Wu

and Zhang, 2005).

2.3. Analysis of water quality of the DanjiangkouReservoir

The Danjiangkou Reservoir has been deteriorated in recent

years, due to great amount of waste discharged into the River

without being treated. The water transfer project requires that

the reservoir water quality should conform to water class II of

Chinese Surface Water Standard II (GB 3838d2002) (SEPAC

and AQSIQC, 2002) before water transfer in 2010. We

selected annual average concentrations of BOD5 (Biology

Oxygen Demand after Five Days), DO (Dissolved Oxygen),

CODMn (Permanganate Index), NH3-N (Ammonia Nitrogen), TP

(Total Phosphorus) and TN (Total Nitrogen) during 1995–2004

from three water quality monitoring stations – Dam, Tai-

zishan (TZS), Taocha (TCA) for analysis of water quality in the

reservoir (Fig. 2).

The analyzing results illustrate that concentrations of

BOD5 (0.68–2.2 mg/L), DO (7.5–9.4 mg/L), CODMn (1.4–2.3 mg/L),

NH3-N (0.05–0.24 mg/L), all meet the Class II (4 mg/L) (Fig. 3a–

d). The concentrations of TP reach 0.6 mg/L and 0.06 mg/L,

which cannot conform to the standard of Class II (0.025 mg/L)

in 2001 and 2003 in Taocha, but they meet the standard in

other years (Fig. 3f). However, the concentration of TN cannot

conform to the Class II, and it belonged to Classes IV and V

(Fig. 3e). The analysis suggests that the water quality deteri-

oration of the Reservoir is mainly reflected by the increase of

concentration of TN, and thus this study focuses on TN

concentration reduction.

2.4. Game-theoretic approach

Games exist in the situations where the actions of actors

(individuals or groups) are interacting and interdependent and

the choices of all actors affect the outcome (Scharpf, 1997). A

game is a metaphor of the rational behaviors of multi-actors

in an interacting or interdependent situation, such as coop-

erating or coalition, conflicting, competing, coexisting, etc.

(Wei and Gnauck, 2007a). A country, a region, a group, an

individual, organism, abiotic and biotic constituents or even

Fig. 2 – The Danjiangkou Reservoir and the water monitoring stations.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6 2503

nature proper each can be an actor. The essence of this theory

is to study the interaction, strategies and equilibrium of

different actors. A game can be defined by set (1), but a normal

form game (or strategic game) can be generally described as

set (2).

GTbCN;A;V; I;O;ED (1)

GbCN;S;VD (2)

where GT – a general symbol for all kind of games, G – a normal

form game (or strategic game), N – set of players, A – actions

(moves), S – strategy profile, V – payoffs (or utility), I – infor-

mation set, O – outcomes and E – equilibrium or equilibria, i.e.

NAVI-OE. NAPI are collectively known as the rules of a game

and OE are the game results. The main aim of constructing

a game model is to define the rules (NAVI) in mathematical

language and get the solution from OE.

To analyze and solve the problems by means of non-

cooperative and cooperative games, we modelled and simu-

lated the conflicts of pollution (i.e. TN) reduction and water

allocation in the study area as a set of games with two levels,

including one main game with five players at the first level and

four sub-games with each containing three sub-players at the

second level. In the game model, we used subscripts ‘‘i’’ and

‘‘�i’’ to stand generally for every player and every other

‘‘n � 1’’ player (or player i’s opponent in some senses),

respectively. In game with sub-games, we also used ‘‘m’’ and

‘‘j’’ to refer to every main player and sub-player, and ‘‘mj’’ to

point out which main player a sub-player belongs to, like 11

means sub-player belongs to main player 1 (see Nomencla-

ture). Our game modelling and simulation process can be

generally expressed as defining a problem (or conflict) as

a game, analyzing the game, setting up game models,

analyzing the game models and solving the game. Non-

cooperative game modelling approach is used to find out

what the real payoff (or utility) of the players, and cooperative

game approach is to get the optimal solution. The main aim of

studying non-cooperative game is to find the solutions for

cooperation.

2.5. Regression model

In order to formulate functions of the input variables and

payoffs of the players, statistical and econometric regression

methods were used. The linear regression model can be

simply expressed by the following equations:

Yp ¼ b0 þXn

k¼1

Xm

p¼1

�bkXkp

�þ 3p (3)

where Yp – values of dependent variables in observation kp;

Xkp – independent (or explanatory) variables; bk – parameters

of the equation; 3p – disturb (or error) term. The equation

includes two components:

(1) b0 þPnk¼1

Pmp¼1ðbkXkpÞ, the non-random component,

(2) 3p, the random component.

To use ordinary least squares (OLS) estimation methods,

some nonlinear models were transformed into linear ones by

logarithm conversions at one side or both sides of the

modelling equations. Besides, polynomial regression and

vector auto-regression were also applied. Autoregressive (AR)

and/or Moving average (MA) terms were included in some

model equations to account for serial correlation. In addition,

balanced panel data and its related modelling approaches

were employed to establish model of the gross agricultural

products and nitrogen fertilizer consumptions. The validity of

the models was evaluated by comparing predictions with

observations.

2.6. Transport process of pollutants

The transporting process of pollutants (W) – total nitrogen

(TN) in this study – into the reservoir during a period of time

can be classified as (1) producing, (2) entering the rivers, (3)

reaching into the reservoir, (4) nitrification/denitrification

processing and forming the final concentration in reservoir.

Part of pollutants will be decayed due to biochemical and

0.0

0.5

1.0

1.5

2.0

2.5

3.0

95 96 97 98 99 00 01 02 03 04

Dam TaochaTaizishan ClassII

BO

D5 (m

g/L

)

6

7

8

9

10

95 96 97 98 99 00 01 02 03 04

Dam TaochaTaizishan ClassII

DO

(m

g/L

)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

Dam TaochaTaizishan ClassII

CO

DM

n (m

g/L

)

0.0

0.1

0.2

0.3

0.4

0.5

89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

Dam TaochaTaizishan ClassII

NH

3-N

(m

g/L

)

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

95 96 97 98 99 00 01 02 03 04

Dam TaochaTaizishan ClassIIIClassIV

TN

(m

g/L

)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

95 96 97 98 99 00 01 02 03 04

Dam TaochaTaizishan ClassI I

TP

(m

g/L

)

Time (year)

a b

c d

e f

Fig. 3 – Water quality of the Danjiangkou Reservoir in three monitoring stations: Dam, Taizishan and Taocha (a) BOD5, (b)

Do, (c) CODMn, (d) NH3-N, (e) TN, and (f) TP from 1995 to 2004.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62504

ecological processes. This process and the annual mean

concentration of TN reached in the reservoir can be expressed

by Eqs. (4) and (5), respectively:

Ltra ¼ €W

t

ra[ralrakrafra (4)

Ctra ¼ Lt

ra=Qtf (5)

where ra – subscript presenting human activity in a region; Ltra

– load of TN into the reservoir from a certain human activity in

a region during time t; €Wt

ra – pollutant TN production from

a certain human activity in a region during time t;

[ra; lra; kra;fra – generally called pollutant transport coeffi-

cients, i.e. rate of TN loss, coefficient of TN into the river, rate

of TN into the reservoir, as well as rate of TN finally main-

taining in the reservoir, respectively; Ctra – average TN

concentration into the reservoir during time t; Qtf – mean

water inflow into the reservoir during time t.

Based on the previous studies (Yang et al. 2006, Cheng et al.

2006, Song et al. 2006), the values of nitrogen transport coef-

ficients were defined in Table 2. Urban domestic sewage and

industry waste water are transported by pipelines, and they

are emitting directly into the local river surface. Therefore,

nearly 100% of nitrogen enters regional rivers, and thus rate of

loss ([) from production resource and rate of entering river (l)

are defined as 1. We also defined that the rate of TN main-

taining in the reservoir is 1, i.e. no loss in the reservoir. A unit

of Pig equivalence was used to measure the livestock and

poultry by pig unit based on their annual average nitrogen

production. According to the study on the spatial and

temporal change of nitrogen and phosphorus produced by

livestock and poultry in China (Wu, 2005), it defined that 1 pig

Table 2 – Different transportation coefficients of nitrogen.

Nitrogen source Shaanxi Hubei He’nan

[ l k 4 [ l k 4 [ l k 4

Nitrogen fertilizer 0.10 0.96 0.80 1.00 0.10 0.96 0.90 1.00 0.10 0.96 0.90 1.00

Soil erosion 0.21 0.81 0.80 1.00 0.21 0.81 0.90 1.00 0.21 0.81 0.90 1.00

Urban domestic sewage 1.00 1.00 0.80 1.00 1.00 1.00 0.90 1.00 1.00 1.00 0.90 1.00

Industry waste water 1.00 1.00 0.80 1.00 1.00 1.00 0.90 1.00 1.00 1.00 0.90 1.00

Animal husbandry 0.10 0.96 0.80 1.00 0.10 0.96 0.90 1.00 0.10 0.96 0.90 1.00

Rural domestic life 0.10 0.96 0.80 1.00 0.10 0.96 0.90 1.00 0.10 0.96 0.90 1.00

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6 2505

is equal to 1/5 of large animals, 2 goals or sheep, and 30

poultry, respectively, in nitrogen production. Those

researches also stated that the average annual nitrogen

amounts from a person’s manure and liquid, a pig’s manure

and liquid are 1.32 kg a�1, 3.07 kg a�1, 7.58 kg a�1 and

3.93 kg a�1, respectively.

2.7. Total nitrogen reduction

Total nitrogen (TN) concentration reduction in the Danjiang-

kou Reservoir was planned to follow a linear trend to reach the

Chinese water quality standard of Class II (0.2 mg/

L � TN � 0.5 mg/L) by 2010, and the two main reasons for this

consideration are: (1) a straight line is the shortest distance

between two points in geometric and mathematic principle;

(2) a straight line trend to reduce TN means time-cost saving.

The linear trend of upper threshold (Cmax) and lower threshold

(Cmin) of TN concentrations during different years (t), are

expressed by Eqs. (6) and (7).

Cmax ¼ �0:127tþ 255:1 (6)

Cmin ¼ �0:177tþ 355:3 (7)

2.8. Nominal and real values

In the case area, there is a clear time value included in the

benefits and losses of players, because pollution reduction

(cost) will be processed before water transfer (benefit).

Table 3 – Consumer Price Index of Beijing used for the value tr

t CPI t CPI t

1978 100.0 1988 187.6 199

1979 101.8 1989 219.9 199

1980 107.9 1990 231.8 200

1981 109.2 1991 259.4 200

1982 111.2 1992 285.1 200

1983 111.8 1993 339.3 200

1984 114.2 1994 423.8 200

1985 134.4 1995 497.1 200

1986 143.5 1996 554.8 200

1987 155.8 1997 584.2 200

aData from 1978 to 2008 (BJSB, 2001–2009).bValues from 2009 to 2015 are predictions.cThe values in parenthesis from 2006 to 2008 are predictions used to testdThe prediction model is: PI ¼ �6521.67 � 0.79 PI (�2)** þ 1.6 PI (�Prob(F-statistic) < 0.000001, *significant at p <0.01, **significant at p < 0.0

Therefore, the payoff values of the players are not at the same

time level. In details, the benefits of Beijing obtaining from

water diversion will be produced after 2010, while the losses of

the cities in the Hanjiang River basin due to reduction pollu-

tion for water transfer will be generated before 2010. In this

study, we start our pollution reduction from the base 2005,

and we only calculate the benefits of Beijing from 2010 to 2015

in order to compare those 6-year benefits to the 6-year losses.

In this sense, the future values should be discounted and

transformed into the current values. The future values are

termed as ‘‘nominal values’’ and the present values as

‘‘comparable or real values’’. In economics, Consumer Price

Index (CPI) is one of widely used deflator to kick out the price

inflation and change the nominal values into comparable

values. The CPI observation values of Beijing used for the

value discount in this study are listed in Table 3, and the

discount formula can be expressed as:

Dt0

R ¼ DtNpt0

I =ptI (8)

where Dt0R – comparable or real value of payoff (V0 and U0) in

year t0, DtN – nominal value of payoff (V and U ) in year t, pt0

I –

Consumer Price Index in year t0, ptI – Consumer Price Index in

year t.

2.9. Other methods

We used demand-supply principle (DSP), cost-benefit analysis

(CBA) and economic valuation methods (EVMs) to compare the

ansformation.

CPI t CPI

8 598.2 2008 703.4 (702.2)

9 601.8 2009 740.4

0 622.9 2010 783.8

1 642.2 2011 829.0

2 630.6 2012 873.4

3 631.9 2013 914.8

4 638.2 2014 952.1

5 647.8 2015 984.6

6 653.6 (666.9)

7 669.3 (672.3)

the accuracy of the prediction.

1)** þ 3.30t* with R2 ¼ 0.99, Adj-R2 ¼ 0.99, F-statistic ¼ 2148.56,

001.

Table 4 – Descriptions of the four scenarios.

Scenario Description

SN1 Demographic changes, economic growth, waster water discharge and reclaiming rate, environ-ecology protection,

water resource exploitation, hydro-climatology condition were just as usual; and water resource was in the condition

of normal years (P ¼ 50%).

SN2 Compared with SN1, demographic changes largely decreased, economic growth rate greatly increased, waste water

discharge significantly reduced, waste water reclaiming rate rapidly enhanced, environ-ecology and water resource

exploitation greatly protected, and it met wet years (P ¼ 20% in water resources and P ¼ 10% of water flow).

SN3 Compared with SN1, demographic changes decreased, economic growth rate increased, waster water discharge reduced,

waste water reclaiming rate enhanced, environ-ecology and water resource exploitation well protected, and it was in

moderate dry years (P ¼ 75% of both water resources and water inflow).

SN4 Compared with other 3 scenarios, demographic changes increased, economic growth rate decreased, waster water discharge

increased, waste water reclaiming rate declined, environ-ecology and water resource exploitation not well protected, and it

met high dry years (P ¼ 95% of both water resources and water inflow).

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62506

outcomes and results of the game modelling. EVMs were also

applied to estimate the benefit and loss in monetary term.

Various economic valuation methods can be used to quantify

economic value of water resources and losses of water

pollution, such as Direct market value method (DMVM),

Shadow engineering Method (SEM) or Replacement cost

approach (RCA), and Opportunity cost or benefits method

(OCM/OBM), Cost analysis method (CAM), Hedonic Pricing, etc.

(Feng and Wang, 2003; Li and Xiu, 2003, 2004; Wei, 2005). We

adopted DMVM to measure the benefits of water transfer,

CAM to calculate the losses of nitrogen pollutant reduction,

and OCM and OBM to evaluate the losses of Beijing and

benefits of cities in the upper river basin of Hanjiang without

water transfer.

2.10. Scenario design

Four scenarios were designed to analyze the uncertainties of

the simulation. The normal modelling and simulation were

established based on past data under an assumption of

‘‘business as usual’’, and they were regarded as the first

Table 5 – Assumption of the main scenarios for allplayers.

Main force (average annual changerate % on base of baseline SN1)

SN2 SN3 SN4

Demographic changes �1.0 �0.3 þ0.3

Gross industrial products þ3.0 �3.0 �6.0

Net income þ3.0 �3.0 �6.0

Gross agricultural products þ6.0 þ3.0 �3.0

Livestock and poultry þ6.0 þ3.0 �3.0

Fertilizer consumptions �6.0 �3.0 þ3.0

Soil erosion �6.0 �3.0 þ3.0

Industry waste water discharge �6.0 �3.0 þ3.0

Urban domestic sewage discharge �6.0 �3.0 þ3.0

Reclaim water þ3.0 þ2.0 þ1.0

Industry waste water treatment þ2.0 þ1.0 þ0.5

Urban and rural sewage treatment þ12 þ8.0 þ5.0

Ecological water demand þ5.0 þ2.0 �1.0

Ecological water use �4.0 �3.0 �0.5

Water resource (probability) 20 75 95

Water inflow (probability) 10 75 95

scenario (SN1). The other three main scenarios were designed

according to the possible changes of constrains and input

variables in SN1. The second scenario (SN2) is very optimistic,

in which the situation is much better than that in SN1 from an

economic and environmental perspective. By contrast, the

fourth scenario (SN4) is much more pessimistic. The third

scenario (SN3) is a coordinated one, whose situations

approximately lied between SN2 and SN4. The main descrip-

tions of those scenarios are showed in Table 4. Based on those

descriptions of the main scenarios, the assumptions of

scenarios were quantified in the Table 5.

3. Model

Water conflicts in the study can be briefly stated that R1 will

transfer water from the Danjiangkou Reservoir in the Han-

jiang River. Water transfer requires the cities (C1, C2, C3, C4, C5,

C6) reducing their pollutant (TN) discharge in order to improve

the water quality in the Reservoir, while TN reduction will

raise cost to those cities (Fig. 4). In this connection, the

conflicts concerning water allocation and water pollutant

Fig. 4 – Sketch of the regions and cities included in the

game simulation.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6 2507

reduction in this study area are unavoidable if the interests

and benefits are not balanced well.

3.1. Formulating the game model

The water conflicting problem in the study area is simulated

as normal (or strategic) form games with two levels, including

one big game and 4 sub-games. The mathematic definitions of

those games can be expressed generally as follows:

GTJCG1;G0DG0JCG01;G

02;.;G0mD

(9)

G1bCNm;Sm;VmD (10)

G0mbCNmj;Smj;VmjD (11)

m ¼ 1; 2;3;4; j ¼ 1;2;3; and m; j˛i; i ¼ 1;2; 3;.;n (12)

where i, m, j, and mj – subscripts refer to every player, main

player, sub-player, and which main player a sub-player belong

to, respectively; GT – a game set; G1 – the first level game; G0, G0m– set of sub-games, and a sub-game of main player m,

respectively; Nm, Nmj – set of main players m, and sub-players

mj, respectively; Sm, Smj – strategy profile of every main player

and sub-player, respectively; Vm, Vmj – payoff of main player m

and sub-player mj, respectively.

3.1.1. Definition of the playersIn the game at the first level, R1 is defined as main player 1 (P1),

C1, C2 and C3 main player 2 (P2), C6 main player 3 (P3), and C4

and C5 main player 4 (P4). In each sub-games, industry,

household and agriculture of every main player are defined as

sub-players 1, 2 and 3, denoted by (P11, P12, P13,.Pmj). They can

be expressed as follows:

Nm ¼ fP1;P2;.Pmg; m ¼ 1;2; 3; 4 (13)

Nmj ¼�

P11;P12;.Pmj

�; m ¼ 1;2;3; 4; j ¼ 1;2; 3 (14)

Among them:

P1 ¼ fR1g;P2 ¼ fC1;C2;C3g;P3 ¼ fC4g; and P4 ¼ fC5;C6g (15)

Pm1 ¼ fIndustryg;Pm2 ¼ fhouseholdg; and Pm3

¼ fagricultureg (16)

3.1.2. Definition of the strategiesTo simplify the problems, we defined that every player has

only two strategies. Rather, in the main game, P1 has either

strategies of transferring water from other players (SWT) or not

transferring water (SNT), i.e. solving water shortage internally;

and P2, P3 and P4 have similar two strategies: reducing TN

pollution for water transfer (SRP) and not doing so (SNR). In the

sub-games 1, the sub-players of main player 1 will choose

strategies of struggling for water without considering too

much of environ-ecology (SWS) and sharing their imitated

water resources considering environ-ecology and increasing

water use efficiency (SNS). In the sub-games 2, 3 and 4, the two

strategies of the sub-players are (1) to discharge pollutant into

reservoir freely (SPF); (2) to reduce the pollutant TN discharge

according to their economic abilities (SPA).

3.1.3. Definition of the payoff functionsThe payoffs of the main player 1 (P1) and his sub-players (P11,

P12 and P13) are the benefits obtained by using water, and

therefore their payoff functions can be formulated by their

available water and the economic values produced by water

use. For other 3 main players and their sub-players, their

payoffs are the cost to reduce TN discharge, and thus their

payoff functions can be formulated by the TN reduction and

the cost to reduce TN. Those payoff functions can be generally

expressed by Eq. (17).

Vti ;U

ti ¼

� f�Qt

i

�; m; j˛i;m ¼ 1; j ¼ 1;2;3

g�

€Wt

i

�; m; j˛i;m ¼ 2; 3;4; j ¼ 1;2; 3

(17)

where i, m, j – subscripts refer to every player, main player and

sub-player, respectively; t – superscript stands for time, V, U –

payoff of every player i non-cooperative and cooperative

game, respectively, Qti – available water of every player i

during time t, €Wt

i – pollutant TN reduction by every player.

3.2. Assumptions

- The games are static and finite with incomplete

information;

- All the players are rational, and their aims are to maximize

their welfares;

- There is no administrative intervention during game pro-

cessing, but the game processing is influenced by the

current policies;

- The cities in the same administrative regions is willing to

cooperate with each other due to the similar interests, i.e.

C1, C2, and C3 cooperation with each other; the same for C4

and C5;

- Cooperation or non-cooperation of other players excluded

from this example will depend on whether players 1, 2, 3

and 4 cooperate or non-cooperate;

- Water demand of each player will keep the same in

different hydrological conditions.

3.3. Game simulation processes

The game simulation flow can be illustrated by Fig. 5, which

includes 5 games. These five games are divided in two levels.

Game 1 is the main game at first level and games 2–5 are the

four sub-games at the second level. The main game is named

as a benefit-loss game, in which we will compare the benefits

and losses of different players under non-cooperation and

cooperation. The first sub-game (game 2) is a water obtaining

game, where sub-players make decision how to get their

water. The other three sub-games (games 2–5) are classed as

TN reduction game, where sub-players will decide if they should

reduce pollutant TN discharge into the river (Fig. 5). We start

the games from the main game, and then game 2, game 3, and

so on. In game one, P1 starts first, and he uses either strategy 1

(SWT) or strategy 2 (SNT). Then P2 moves and he knows P1 has

two strategies, but does not know which strategy P1 will use

due to incomplete information. Thus he will use his strategy 1

or strategy 2 according to his real situations. P3 moves next,

and then P4 moves. For each sub-game, we also start from sub-

player 1, then sub-player 2 and sub-player 3. In game 1, when

Fig. 5 – A sketch of game simulation process, (a) the main game at first level, (b) sub-game 1, (c) sub-game 2, (d) sub-game 3,

and (e) sub-game 4.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62508

P1 uses strategy of transferring water, and others players

reducing pollution for water transfer, the game enters

a cooperative game. In game 2, it also enters a cooperative

game if sub-players 11–13 all adopt the strategy 2 (SNS). Simi-

larly, games 3–5 each will also enter a cooperative game when

sub-players employ their strategy 2 (SPA). All problems in

those situations need cooperative methods to solve and get

the results. The results in those cooperative situations and

reverse situations have more practical values than those in

other mixed situations. Those results are the outcome (V1, V2,

V3, V4) and (U1, U2, U3, U4), (Q11, Q12, Q13) and ðQ 011;Q012;Q

013Þ,

ð €W21;€W22;

€W23Þ and ð €W021;

€W022;

€W023Þ, ð €W31;

€W32;€W33Þ and

ð €W031;

€W032;

€W033Þ, as well as ð €W41;

€W42;€W43Þ and ð €W

041;

€W042;

€W043Þ.

Therefore, we just calculate those results in the study.

3.4. Game solution

3.4.1. A non-cooperative game modelA non-cooperative game solution model for water resources

management presents that every player i maximizes the net

benefits, i.e. differences between benefits produced from

water usage and the costs charged for waste water reduction

or treatment. The model is expressed by Eq. (18).

MaxVti

Q; €W;t

¼Zn

t

Bt

iðQÞ � Kti

�€W�

e�dtdt (18)

where i – subscript general refers to every player, t – super-

script stands for time (year), Vti – payoff of every player i

during time t, Q – available water for player i, W – pollutant

TN reduction by every player i, e�dt – discount factor, BtiðQÞ –

benefit function of every player i to use available water

during time t, Ktið €WÞ – cost of every player i to abate pollutant

TN during time t. Unlike other study to use interest rate, we

used Eq. (8) to make discount or value transformation in this

study.

3.4.1.1. Water quantity optimization. Water quantity optimi-

zation means that per unit economic value will be produced

by consuming minimum unit of water. It also means that

consumption per unit of water will produce maximum unit of

economic values. It can be expressed by:

MaxBtiðQÞ ¼

Xn

t¼0

btiQ

ti (19)

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6 2509

subject to

Wts þWt

g þ Rt � Qted � Qt

i (20)

Qtws þ Qt

ga þ Qttr � Qt

ed (21)

Qty � Qt

y�1 þ aQti � Qt

eu � Etws � Qt

i (22)

0 � Rt � Qti (23)

Qti � Qt

i � Qt

i (24)

where i – subscript refer to every player including the main

player and sub-player, t – superscript stands for time (year),

BtiðQÞ – benefit function of available water of every player i, bt

i –

benefit coefficient of water use, i.e. value produced by every

player i using per unit of water during time t, Qti – available water

for every player i during time t, Qted – environ-ecological water

demand of player i during time t, Qteu – available water used for

environ-ecology of player i during time t, Wts – surface water

resource amount during time t, Wtg – W0 resource amount; Qt

ws –

water demand to keep certain amount of urban water surface

during time t, Qtga – water demand to maintain certain area of

public green area during time t, Qttr – water demand of newly

planed trees during time t, Rt – reclaimed waste water during

time t, a – coefficient of waste water and sewage back into water

during time t, Qty�1 – water inflow from upstream controlling (i.e.

player – i’s) section y � 1 in during time t, Qty – water flow in the

observed (player i’s) section y during time t, Etws – evaporation of

water surface during time t, Qti and Q

t

i – minimum and

maximum of water demand of player i during time t.

3.4.1.2. Water quality optimization. Water quality optimiza-

tion means that every player i minimizes the costs to reduce

pollutant (TN) discharge into the water body. It can be

expressed as follows:

Min Kti

�€W�¼ g €w

X€w ¼ 1

nXm

y¼1

h�L €w;yt�L €w;y� 1t

��1� h€w;y

�� L €w;yc

i

(25)subject to

L €w; y� 1t ¼ Qt

y�1C €w;y�1t(26)

L €w; yt ¼ Qt

yC €w;yt(27)

L €w; yc ¼ Qt

yC €w;yc(28)

Q > 0; L > 0; C > 0; h � 0 (29)

where Ktið €WÞ – cost of every player i to abate pollutant W (TN)

during time t, g €w – cost coefficient to reduce pollutant W (TN)

of every player i; L €w;yt – load of pollutant W in controlling (i.e.

player i’s) section y during time t, L €w;y�1t – load of pollutant W

from upstream controlling (i.e. player – i’s) section y� 1 during

time t, L €w;yc – controlling load of pollutant W in player i’s

controlling section y, h €w;y – decomposition (assimilation)

coefficient of pollutant W in player i’s controlling section y,

C €w;yt – concentration of pollutant W in player i’s controlling

section y during time t; C €w;y�1t – concentration of pollutant W

from player – i’s controlling section y � 1 in upstream during

time t; C €w;yc – controlling concentration (i.e. standard) of

pollutant W in player i’s controlling section y.

3.4.2. A cooperative game modelThe cooperative game model means that all the players

cooperate with each other to maximize the overall net bene-

fits. It is expressed by Eq. (30). Every player in cooperative

game is to maximize the net benefits which he can obtain

from cooperation. It is expressed by Eq. (31).

Max Ut

Q; €W;t¼Zn

t

BtðQÞ � Kt

�€W�

e�dtdt (30)

Max Uti ¼ Vt

i þmaxYn

i

�Ut

B=j�

i

(31)

subject to

Ut �Xn

i¼1

Vti þ Ut

B (32)

UtB � 0 (33)

where i – subscript refer to every player, t – superscript stands

for time, Ut – the total benefit obtained from cooperative game

during time t, BtðQÞ – the benefit function of water use in

cooperative game during time t, Ktð €WÞ – the cost to abate

pollutant TN in cooperative game, Uti – payoff of each player i

in cooperative game, Vti – payoff of every player i non-

cooperative game, UtB – total net benefit obtained from coop-

erative game, j – distribution factor of cooperative benefit.

4. Results and discussion

4.1. Evaluation of game simulation

The validity of the simulation was evaluated by comparing

predictions with observations (data) during 2000 and 2006. To

display the compared results of all the values at different sizes

in one diagram, scientific notion method was applied to

transform all numbers into the form a � 10b, the significant

a was defined as any real number between 1 and 10, and three

decimal places was set to a to keep all residuals (differences

between observations and predictions) from zero after the

transformation. The evaluation results illustrate that fore-

casts and observation are very close (Fig. 6a) with an average

error of 3.6% except one error is 16.51% and another 10.48%

(Fig. 6b). Through this evaluation, it is clear that the simula-

tion has a very good ability to reflect reality and can be used

for future prediction.

4.2. Water deficit

The simulation results of water deficits of sub-players 11, 12

and 13 in game 2 are displayed in Table 6. In the non-

cooperative game, available water for sub-player 11 (Q11) and

sub-player 13 (Q13) will decrease from 5.38 � 108 m3 to

4.10 � 108 m3, and 10.46� 108 m3 to 7.99 � 108 m3, respectively

Fig. 6 – Game simulation evaluation (a) comparison of predictions with observation, (b) forecasting errors.

Table 6 – Simulation results of available water (Q) and water deficit (Qd) of sub-players 11–13 in game 2 (3108 m3).

t Non-cooperation Cooperation Comparison (water deficit)

Q11 Q12 Q13 Q1 Q 011 Q 012 Q 013 Q 01 Q11d Q12d Q13d Q 01

2010 5.38 15.92 10.46 31.76 4.02 11.89 7.81 23.73 �1.36 �4.03 �2.65 �8.03

2011 5.10 16.31 9.94 31.35 3.82 12.21 7.44 23.46 �1.28 �4.1 �2.5 �7.89

2012 4.83 16.71 9.43 30.97 3.62 12.52 7.06 23.20 �1.21 �4.19 �2.37 �7.77

2013 4.58 17.1 8.93 30.61 3.43 12.8 6.69 22.92 �1.15 �4.3 �2.24 �7.69

2014 4.33 17.49 8.45 30.27 3.24 13.08 6.32 22.63 �1.09 �4.41 �2.13 �7.64

2015 4.10 17.88 7.99 29.97 3.06 13.33 5.96 22.34 �1.04 �4.55 �2.03 �7.63

Table 7 – Simulation results of total nitrogen production (W, W0) and reduction (WR) by players 21–23 in game 3 (tons/a).

t No-cooperation Cooperation Comparison (reduction)

W21 W22 W23 W2€W021

€W022

€W023

€W02 W21R W22R W23R W2R

2005 694.0 40,131.7 273,586.4 314,412.1 530.8 30,692.3 250,061.8 240,459.1 163.2 9439.4 64,350.3 73,953.0

2006 634.2 40,139.5 275,665.6 316,439.3 451.9 28,603.8 237,215.5 225,497.5 182.3 11,535.7 79,223.8 90,941.8

2007 601.7 40,195.2 279,338.8 320,135.8 315.2 21,055.7 187,125.1 167,699.1 286.5 19,139.5 133,010.7 152,436.7

2008 570.9 40,275.3 283,116.7 323,962.9 301.5 21,271.8 190,377.2 171,104.4 269.4 19,003.5 133,585.7 152,858.5

2009 541.7 40,367.7 286,994.6 32,7904.0 270.2 20,135.9 184,065.9 163,562.6 271.5 20,231.8 143,838.1 164,341.4

2010 514.0 40,466.3 290,772.4 331,752.7 157.8 12,426.3 130,270.4 101,874.3 356.2 28,040.0 201,482.3 229,878.4

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62510

from 2010 to 2015. In contract, available water for sub-player

12 showed an increasing trend, from 15.92 � 108 m3 to

17.88 � 108 m3 during the same period of time. In cooperation

game, the water amounts obtained by them are less than that

in non-cooperation, mainly because the players stop over-

using underground and ecological water and share their

limited water resources. By comparing the available water in

cooperation to that in non-cooperation, it was found that

those players will face serious water deficits in the condition

of cooperation.

4.3. Nitrogen reduction

The simulation results of nitrogen reductions in games 3–5 are

revealed from in Tables 7–9. In game 3, players 21, 22 and 23

can produce total nitrogen (TN) of 694.0–514 tons, 40,131.7–

40,466.3 tons, and 273,586.4–290,772.4 tons, respectively from

2005 to 2010 in non-cooperative game. In cooperative game,

they have to reduce the nitrogen production in order to

improve the water quality for water transfer. Comparing the

TN productions in non-cooperative to that in cooperative

games, it found that players 21, 22 and 23 will reduce,

respectively, TN of 163.2–356.2 tons, 9439.4–28,040.0 tons, and

64,350.3–201,482.3 tons from 2005 to 2010 (Table 7). Based on

the similar analysis, it revealed that in game 4 sub-players 31,

32 and 33 should reduce, respectively, nitrogen of 89.2–

506.2 tons, 3695.0–11,581.8 tons and 15,672.5–51,276.9 tons,

during the same period of time (Table 8). In game 5, the results

conformed that players 41, 42 and 43 should reduce, respec-

tively, TN of 45.6–165.8 tons, 1120.7–3247.4 tons and 13,553.3–

52,755.2 tons to meet the water quality standard from 2005 to

2010 (Table 9).

Table 8 – Simulation results of total nitrogen production (W, W0) and reduction (WR) by players 31–33 in game 4 (tons/a).

t No-cooperation Cooperation Comparison (Reduction)

W31 W32 W33 Total €W031

€W032

€W033

€W03 W31R W32R W33R W3R

2005 379.2 15,709.5 66,632.0 82,720.7 290.0 12,014.5 67,048.2 63,263.9 89.2 3695 15,672.5 19,456.8

2006 467.7 15,873.6 68,232.0 84,573.3 333.3 11,311.7 64,964.0 60,267.7 134.4 4561.9 19,609.3 24,305.6

2007 543.8 16,055.1 69,751.5 86,350.4 284.8 8410.3 53,137.3 45,233.5 259.0 7644.8 33,213.1 41,116.9

2008 606.1 16,255.1 71,198.2 88,059.4 320.1 8585.3 54,465.3 46,509.5 286.0 7669.8 33,594.1 41,549.9

2009 668.3 16,474.5 72,602.2 89,744.9 333.4 8217.7 53,357.6 44,765.9 334.9 8256.8 36,387.3 44,979.0

2010 730.6 16,714.4 74,001.1 91,446.0 224.4 5132.6 40,169.1 28,081.1 506.2 11,581.8 51,276.9 63,364.9

Table 9 – Simulation results of total nitrogen production (W, W0) and reduction (WR) by players 41–43 in game 5 (tons/a).

t No-cooperation Cooperation Comparison (reduction)

W41 W42 W43 W4€W041

€W042

€W043

€W04 W41R W42R W43R W4R

2005 193.7 4764.5 57,621.9 62,580.1 148.1 3643.8 49,026.8 47,860.6 45.6 1120.7 13,553.3 14,719.5

2006 203.8 4750.4 60,562.8 65,517.0 145.2 3385.2 48,111.8 46,688.0 58.6 1365.2 17,405.2 18,829.0

2007 213.7 4734.0 64,212.5 69,160.2 112.0 2479.8 38,584.6 36,228.7 101.7 2254.2 30,575.6 32,931.5

2008 223.5 4718.3 67,959.0 72,900.8 118.0 2492.0 40,835.1 38,503.3 105.5 2226.3 32,065.7 34,397.5

2009 233.4 4702.3 71,948.2 76,883.9 116.4 2345.6 40,824.4 38,350.7 117.0 2356.7 36,059.5 38,533.2

2010 243.2 4686.5 76,135.0 81,064.8 74.7 1439.1 28,309.3 24,893.3 168.5 3247.4 52,755.5 56,171.5

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6 2511

4.4. Payoffs

Results of payoffs at nominal prices in game 1 are presented in

the Matrix 1. In the matrix, the first column and the second

column refer to the payoffs resulting from the simulations of

non-cooperative and the cooperative games, respectively. In

each column, the first, second, third and fourth numbers refer

to the payoffs of players 1, 2, 3 and 4 respectively. The zeros

are used to (1) keep the matrix symmetric, (2) state no game

played there. These results proved that the non-cooperative

game will cost player 1 a total loss of 17.3 � 1011 yuan from

year 2010 to 2015, but it would yield players 2, 3 and 4 a benefit

of 1.1 � 1011 yuan. However, compared the overall costs to

benefits, there was an overall loss of 16.2 � 1011 yuan when

each player does not cooperate with the others. In contract,

the cooperative game results showed that there is an overall

benefit of 16.2 � 1011 yuan, though players 2–4 lose

1.1 � 1011 yuan. Those nominal values have been transformed

Matrix 1 – Payoff matrix of players 1–4 in game 5

(3108 yuan at nominal prices).

into comparative value (real value) (Matrix 2) based on Eq. (8)

to make reasonable comparison. Based on those results, the

payoffs of players 1 and his sub-players in the years of 2010,

2011, 2012, 2013, 2014 to 2015 have been transferred into the

values in years of 2005, 2006, 2007, 2008, 2009 and 2010,

respectively. These results calculated at real prices proved

that the non-cooperative game will cost player 1 a total loss of

13.6 � 1011 yuan during 2010–2015, but it yields players 2–4

a benefit of 1.1 � 1011 yuan from 2005 to 2010. However,

comparing the overall costs and benefits, there is an overall

loss of 12.5 � 1011 yuan when each player does not cooperate

with the others. In contract, the cooperative game result

showed that there is an overall benefit of 12.5 � 1011 yuan,

though players 2–4 lose 1.1 � 1011 yuan (Matrix 2).

Matrix 3 explained the real losses of all sub-players under

the game 1. In the matrix, the rows above the line present non-

cooperative results and down are cooperative results. From

those results, it showed that non-cooperation among players

1, 2, 3 and 4 will cost sub-players 11, 12 and 13 losses of

662.83 � 108–1222.25 � 108 yuan, 1230.70 � 108–2614.94 �108 yuan and 24.51� 108–27.74� 108 yuan, respectively, due to

water deficits during 2010–2015. However, sub-players of

Matrix 2 – Payoff matrix of players 1–4 in the non-

cooperative and cooperative game (3108 yuan at

comparable prices).

Matrix 3 – Payoff matrix of all the sub-players in the non-cooperative and cooperative game (3108 yuan at comparable

prices).

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62512

11–13 can avoid those losses if players 1–4 are cooperative, but

cooperation impose cost to sub-players of 21–23, 31–33, 41–43.

The losses of sub-players of 21, 22, 23, 31, 32, 33, 41, 42 and 43

are 0.15 � 108–0.32 � 108 yuan B, 39.07 � 108–40.18 � 108 yuan,

38.89 � 108–98.50 � 108 yuan, 0.59 � 108–3.36 � 108 yuan,

11.46 � 108–16.98 � 108 yuan, 6.96 � 108–26.04 � 108 yuan,

0.08 � 108–0.29 � 108 yuan, 38.21 � 108–38.41 � 108 yuan and

3.47� 108–17.83� 108 yuan, respectively from 2005 to 2010. On

Wa

te

r d

em

an

d (×

10

8

m3

)

39.0

39.5

40.0

40.5

41.0

41.5

42.0

2010 2011 2012 2013 2014 2015

SN1 SN2 SN3 Sn4

-24

-20

-16

-12

-8

-4

0

2010 2011 2012 2013 2014 2015

SN1 SN2 SN3 SN4

Tim

Wa

te

r d

efic

it (×

10

8

m3

)

a

c d

Fig. 7 – Scenarios (SN) of (a) water demand, (b) available water re

sub-players 11, 12 and 13 (P11, P12, P13) without cooperation wi

the contrary, the sub-players of 11–13 will have no such losses

if players 1–4 are cooperative, but cooperation also imposes

cost to the sub-players of 21–23, 31–33, 41–43. For example, the

sub-players 21, 22 and 23 will lose 0.15 � 108–0.32 � 108 yuan,

39.07 � 108–40.18 � 108 yuan and 38.89 � 108–98.50 � 108 yuan,

respectively from 2005 to 2010. Therefore, all the players will

be better off if a side payment is made between them at the

end of the cooperative game.

16

20

24

28

32

36

40

2010 2011 2012 2013 2014 2015

SN1 SN2 SN3 SN4

-10

-8

-6

-4

-2

0

2010 2011 2012 2013 2014 2015

SN1P11 SN1P12 SN1P13SN2P11 SN2P12 SN2P13SN3P11 SN3P12 SN3P13SN4P11 SN4P12 SN4P13

e (year)

Availab

le w

ater reso

urces (×

10

8

m3

)W

ate

r d

efic

it (×

10

8

m3

)

b

sources, (c) water deficit of player 1 (P1), (d) water deficits of

th outside players.

0

5,000

10,000

15,000

20,000

25,000

30,000

2005 2006 2007 2008 2009 2010SN1P2 SN1P3 SN1P4 SN2P2SN2P3 SN2P4 SN3P2 SN3P3SN3P4 SN4P2 SN4P3 SN4P4

0

4,000

8,000

12,000

16,000

20,000

24,000

2005 2006 2007 2008 2009 2010SN1P21 SN1P22 SN1P23 SN2P21SN2P22 SN2P23 SN3P21 SN3P22SN3P23 SN4P21 SNP22 SN4P23

0

4,000

8,000

12,000

16,000

20,000

2005 2006 2007 2008 2009 2010

SN1P31 SN1P32 SN1P33SN2P31 SN2P32 SN2P33SN3P31 SN3P32 SN3P33SN4P31 SN4P32 SN4P33

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

2005 2006 2007 2008 2009 2010

SN1P41 SN1P42 SN1P43SN2P41 SN2P42 SN2P43SN3P41 SN3P42 SN3P43SN4P41 SN4P42 SN4P43

Time /year

To

ta

l n

itro

gen

red

uc

tio

n (to

ns

)

To

ta

l n

itro

gen

d

isch

arg

e(to

ns

)

To

ta

l n

itro

gen

d

isch

arg

e(to

ns

)

To

ta

l n

itro

gen

d

isch

arg

e(to

ns

)

a b

c d

Fig. 8 – Scenarios (SN) of (a) nitrogen reduction by players 2–4 (P2–P4), (b) nitrogen discharged into the reservoir, by sub-

players 21–23 (P21–P23), (c) nitrogen discharged into the reservoir by sub-players 31–33 (P31–P33), and (d), nitrogen discharged

into the reservoir by sub-players 41–43 (P41–P43).

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6 2513

Form those comparing results, it is clear that the players

should cooperate with each other so as to maximize the

overall benefits. However, every player is usually afraid of

cooperation, because they face risks and uncertainties of

losses when they are not sure if others really want to coop-

erate. Furthermore, every player can be better off by free

riding in non-cooperation. In water scarce area, players can

get their water by free riding in terms of overusing ground-

water and ecological water. In the Hanjiang River basin, every

player can also be better off by free riding others’ achievement

of pollution reduction. In a long run, non-cooperation will

deteriorate the environ-ecology and water quality. Therefore,

non-cooperation results in a game of ‘‘Prisoners’ dilemma’’.

The methods to solve the dilemma are usually to design

a mechanism to change the rules and drive the players to

reach collective rationality. The driving forces usually refer to

something like laws, regulations, contracts and other binding

agreement. In contract with those legislation methods,

economic methods such as tax, fine, compensation and so on,

are also such kinds of driving forces. In this study, reducing

waste water and increasing water quality will impose cost to

players in the reservoir catchment, but they can create a large

benefit to the players in water receiving area. In this sense, all

the players will have incentives to cooperate if a mechanism

could guarantee to transfer part of the benefits obtained from

cooperation to cover the losses of players.

4.5. Scenario simulation

The main comparison results from the water demand game

simulation under the four scenarios are illustrated in Fig. 7.

The scenario results revealed that, from 2008 to 2015, water

demand of player 1 (Fig. 7a) will increase under each of those

four scenarios, though the efficiency of water consumption

will be highly increased in those scenarios. Player 1 and his

sub-players would face shortage problems (Fig. 7c and d) in

each of the four scenarios (Fig. 7b), mainly due to increase of

ecological water demand. Comparing to other sub-players,

player 12 will face most serious water deficits in each of the

four scenarios (Fig. 7d). It also found that, due to extremely

severe water scarce situation, those players cannot solve their

water deficits without cooperation with other players, even

under the optimistic scenario 4 (S(4)), where it is in the wet

years (P ¼ 20%), high waste water recycling amount, etc.

The comparing simulation results of TN reduction under

the four scenarios are illustrated in Fig. 8. The results showed

that, in each of the scenarios, player 2 should take more

responsibility to reduce nitrogen production (Fig. 8a), because

he is the main polluter discharging more pollutant TN into the

reservoir than that of other players. The uncertainty of non-

cooperation probably come from this player and his sub-

players because they face big loss to reduce their TN

discharge based on the payoffs results in scenario 1 (Matrix 2).

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62514

From the scenario results of sub-games, it is also clear that

sub-players 23, 33 and 43 are the main polluters in games 3, 4

and 5, respectively, because they discharge more nitrogen

than that other players do in each of those gamed under the

four scenarios (Fig. 8c and d).Those sub-players will be the

uncertainty sources of non-cooperation in those games.

5. Conclusions

This study established game-theoretic simulation models to

analyze the problems of water scarcity and nitrogen reduction

in the South-to-North Water Transfer Project. The simulation

is consisted of two levels, 1 main game with 4 players and 4 sub-

games with 12 sub-players. Beijing municipality, Shaanxi

(Hanzhong, Ankang and Shangluo cities), Hubei (Shiyan city)

and He’nan (Xixia and Xichuan cities) were definedas players1,

2, 3 and 4; and industry, household and agriculture of those

four players as the sub-players 11, 12 and 13, 21, 22 and 23, 31, 32

and 33, and 41, 42 and 43, respectively. The main results

revealed that player 1 and its sub-players cannot solve their

water deficit problem without cooperation with other players

even under an optimistic scenario. Sub-player 12 will face most

serious water deficit based on the simulation results of four

scenarios. Cooperation with other players is the dominant

strategy of those players. Players 2–4 and their sub-players will

face costs to reduce pollutant total nitrogen for the water

diversion. The uncertainty of non-cooperation might come

from player 2 in game 1, and sub-players 3 in the games 2–5.

This study also proofed that non-cooperation will cause whole

society a loss although some players can get benefits. In

contract, cooperation brings some players losses, but it will

produce much more collective benefits. However, players

usually are not willing to cooperate, because they will face risks

of losses. This usually results in a game of ‘‘Prisoners’

dilemma’’. The players are willing to cooperate if a mechanism

can guarantee to transfer part of the benefits obtained from

cooperation to cover the losses of players. Suggestions on the

mechanisms can include: (1) to sign a binding agreement on

the beneficial players funding losers to build necessary pollu-

tion treatment plan; (2) to transfer water using and controlling

right to the losing players; and (3) to include the losses of losers

into the water prices for winners. These game simulation

results will not only benefit the water users to be better off, but

also benefit water administration for decision support on water

distribution, water pricing and ecological compensation, etc.

r e f e r e n c e s

Anbarci, N., 1993. Noncooperative foundations of the areamonotonicsolution.Quarterly Journal of Economics108,245–258.

Bekele, E.G., Nicklow, J.W., 2005. Multiobjective management ofecosystem services by integrative watershed modeling andevolutionary algorithms. Water Resources Research 41,W10406. doi:10.1029/2005WR004090.

BDRC (Beijing Development and Reform Committee), 2006. WaterResources Protection and Use Planning for No. 11 Five-yearPlan of Beijing. The Beijing Development and ReformCommittee, Beijing (in Chinese).

Bhaduri, A., Barbier, E.B., 2003. Water Transfer and InternationalRiver Basin Cooperative Management: The Case of the Ganges.Department of Economics and Finance, University ofWyoming, PO Box 3985, Laramie, WY 82071, USA.

BJSB (Beijing Statistic Bureau), 2001–2009. Statistic Yearbooks ofBeijing (2001–2009). China Statistics Press, Beijing (in Chinese).

BJWB (Beijing Water Resources Bureau), 2005. Beijing WaterResources Bulletin. Beijing Water Resources Bureau, Beijing (inChinese).

Bogardi, L., Szidarovszky, F., 1976. Application of game theory inwater management. Applied Mathematical Modelling 1, 16–20.

Burness, H.S., Quirk, J.P., 1979. Appropriative water rights and theefficient allocation of resources. American Economic Review69 (1), 25–37.

Carraro, C., Filar, J.A. (Eds.), 1995. Control and Game-TheoreticModels of the Environment. Birkhauser, Boston.

Cheng, H.G., Hao, F.H., Ren, X.Y., Yang, S.T., Xiong, W., Lei, S.P.,2006. The study of the rate loss of nitrogenous non-pointsource pollution loads in different precipitation levels. ActaScientiae Circumstantiae 26 (3), 392–397 (in Chinese).

Chun, Y., 1988. The equal-loss principle for bargaining problems.Economics Letters 26, 103–106.

Coppola, E., Szidarovszky, F., 2004. Conflict between water supplyand environmental health risk: a computational neural networkapproach. International Game Theory Review 6 (4), 475–492.

Colby, B.G., 1990. Transaction costs and efficiency in westernwater allocation. American Journal of Agricultural Economics72 (5), 92–1182.

CWRA (China Water Resources Administration), 1998–2004. ChinaWater Resource Bulletin (1998–2004). China Water ResourcesAdministration, Beijing (in Chinese).

CWRPI (Changjiang Water Resources Protection Institute), 2005.Report on the Environmental Impacts of Middle Route of South-to-North Water Transfer Project. Wuhan, China (in Chinese).

Dellapenna, J.W., 2000. The importance of getting names right:the myth of markets for water. William and MaryEnvironmental Law Policy Review 25, 317–377.

Dinar, A., Howitt, R.E., 1997. Mechanisms for allocation ofenvironmental control cost: empirical test of acceptability andstability. Journal of Environmental Management 49, 183–203.

Donevska, K., Dodeva, S., Tadeva, J., 2003. Urban and agriculturalcompetition for water in the Republic of Macedonia. Internetresources. Available at: http://afeid.montpellier.cemagref.fr/mpl2003/Conf/Donevska.pdf (accessed 27.11.09).

Fang, L., Hipel, K.W., Kilgour, D.M., 1998. The graph modelapproach to environmental conflict resolution. Journal ofEnvironmental Management 27 (2), 195–212.

Fang, L.,Hipel, K.W., Wang,L.Z., 2002. Gisborne waterexport conflictstudy. In: Schmitz, G.H. (Ed.), Proceedings of the ThirdInternational Conference on Water Resources andEnvironmental Research, vol. 1. Dresden, Germany, pp. 432–436.

Feng, Y.L., Wang, H.J., 2003. Study on resources value of water.Journal of Hydraulic Engineering 34 (8), 111–117 (in Chinese).

Frisvold, G.B., Caswell, M.F., 2000. Transboundary watermanagement: game theoretic lessons for projects on US-Mexico. Agricultural Economics 24, 101–111.

Giglio, R.J., Wrightington, R., 1972. Methods for apportioning thecosts of a water resource project. Water Resources Research 8,1133–1144.

Green, G.P., O’Connor, J.P., 2001. Water banking and restoration ofendangered species habitat: an application to the Snake River.Contemporary Economic Policy 19 (2), 225–237.

Harsanyi, J.C., Selten, R., 1972. A generalized Nash solution fortwo-person bargaining games with incomplete information.Management Science 18, 80–106.

HBEPB (Hubei Environmental Protection Bureau), 2004–2005. TheHanjiang River Bulletin (2004–2005). Hubei EnvironmentalProtection Bureau, Wuhan (in Chinese).

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 6 2515

HBSB (Hubei Statistic Bureau), 1996–2005. Statistic Yearbooks ofHubei Province (1996–2005). China Statistics Press, Beijing (inChinese).

Hearne, R.R., Easter, K.W., 1995. Water allocation and watermarkets: an analysis of gains-from-trade in Chile. In: TechnicalPaper Series Number 315. World Bank, Washington, DC, USA.

HNSB (He’nan Statistic Bureau), 1994–2007. He’nan StatisticYearbooks (1994–2007). China Statistics Press, Beijing (inChinese).

Holden, P., Thobani, M., 1996. Tradable Water Rights: A PropertyRights Approach to Resolving Water Shortages and PromotingInvestment. World Bank Policy Research Working Paper No.1627. http://www-wds.worldbank.org/servlet/WDSContentServer/WDSP/IB/1996/07/01/000009265_3961214131318/Rendered/PDF/multi_page.pdf Available at:(accessed 3.12.09).

Howe, C.W., Goodman, D.J., 1995. Resolving water transferconflicts through changes in water market process. In:Dinar, A., Loehman, E.T. (Eds.), Water Quantity/QualityManagement and Conflict Resolution: Institutions, Processedand Economic Analysis. Praeger Publishers, Westport,Connecticut, pp. 119–129.

Howe, C.W., Schurmeier, D.R., Douglas Shaw, W., 1986.Innovative approaches to water allocation: the potential forwater markets. Water Resources Research 22 (4), 439–448.

Hu, J.J., Zhang, Y.H., 2003. The current soil erosion situations inthe areas of the Danjiangkou Reservoir and its upstream andthe countermeasures. China Water Resources A, 47–49 (inChinese).

Kalai, E., Smorodinsky, M., 1975. Other solutions to Nash’sBargaining problem. Econometrica 43, 513–518.

Kilgour, D.M., Okada, N., Nishikori, A., 1988. Load controlregulation of water pollution: an analysis using game theory.Journal of Environmental Management 27, 179–194.

Lewandowski, A., 1979. A game-theoretical approach tomodelling the behaviour of water users in a quality controlproblem. In: Findeisen, W. (Ed.), Studies in ControlMethodology for Water resource Systems. Techn. Dept., Inst.Automat. Control, Techn. Uni. Warsaw, Warsaw, pp. 83–97.

Li, J.X., Xiu, S.L., 2003. Calculation model of water pollutioninduced economic loss for river basin. Journal of HydraulicEngineering 10, 68–74 (in Chinese).

Li, S.T., Xiu, X.Y. (Eds.), 2004. South-to-North Water Transfer andChina Development. Economic Science Press, Beijing (inChinese).

Markandya, A., Recharddson, J. (Eds.), 1992. EnvironmentEconomics: A Reader. St. Martin’s Press. Inc., New York.

Muleta, M.K., Nicklow, J.W., 2005. Decision support for watershedmanagement using evolutionary algorithms. Journal of WaterResources Planning and Management 131 (1), 35–44.

Nash, J.F., 1950. The bargaining problem. Econometrica 18,155–162.

NBSC (National Bureau of Statistics of China), 1985–2007. ChinaStatistic Yearbooks (1985–2007). China Statistics Press, Beijing.

NBSC (National Bureau of Statistics of China), SEPAC (StateEnvironmental Protection Administration of China), 2000–2005. China Environmental Statistic Yearbook. China StatisticsPress, Beijing (in Chinese).

OECD (Organisation for Economic Co-operation andDevelopment), 1989. Economic Instruments for EnvironmentalProtection, Paris.

Okada, N., Mikami, Y., 1992. A game-theoretic approach to acidrain abatement: conflict analysis of environmental loadallocation. Water Resources Bulletin 28, 155–162.

Pegram, J.J., Delforce, R.J., Cowell, M.L., Norris, V., Anthony, G.,Anderson, R.L., Musgrave, W.F., 1992. Transferable WaterEntitlements in Austria. Centre for Water Policy Research,University of Armidate, Austrila.

Pethig, R. (Ed.), 1992. Conflicts and Cooperation in ManagingEnvironmental Resources. Springer-Verlag, Berlin, Heidelberg,New York, Lodon, Paris, etc.

Reidinger, R., 1994. Observations on water markets for irrigationsystems. In: Le Moigne, G., Easter, K.W., Ochs, W.J., Giltner, S.(Eds.), Water Policy and Water Markets, Technical Paper 249.World Bank, Washington, DC, USA.

Salazar, R., Szidarovszky, F., Coppola, E., Rojano, A., 2007.Application of game theory for a groundwater conflict inMexico. Journal of Environmental Management 84, 560–571.

Saleth, R.M., 1996. Water Institutions in India. CommonwealthPublishers, New Delhi.

Scharpf, F.W., 1997. Games Real Actors Play: Actor-CenteredInstitutionalism in Policy Research. Westview Press,Boulder, CO.

SEPAC (State Environmental Protection Administration), AQSIQC(Administration of Quality Supervision, Inspection andQuarantine of China), 2002. Environmental Quality Standardsfor Surface Water of the People’s Republic of China (GB3838d2002) (in Chinese).

Shao, X.J., Wang, H., 2003. Interbasin transfer projects and theirimplications: a China case study. International Journal of RiverBasin Management 1 (1), 5–14.

Song, G.Q., Yan, M., Zhang, W.M., Zhang, Y., 2006. Preliminaryestimation of TN load of Shiyan to TN concentration ofDanjiangkou Reservoir. Environmental Science andTechnology Supplement(SUM 26), 164–165 (in Chinese).

SXSB (Shaanxi Statistic Bureau), 1991–2006. Shaanxi StatisticYearbooks (1991–2006). China Statistics Press, Beijing.(in Chinese).

Van der Veeren, R.J.H.M., Tol, R.S.J., 2003. Game theoreticanalyses of nitrate emission reduction strategies in the Rhineriver basin. International Journal of Global EnvironmentalIssues 3 (1), 74–103.

Wang, L.Z., Fang, L., Hipel, K.W., 2003. Water resource allocation:a cooperative game approach. Journal of EnvironmentalInformatics 2 (1), 11–22.

Wei, S.K., 2005. Managing World Heritage Sites in China from anEconomic Perspective. Dissertation. de-Verlag, Berlin.

Wei, S.K., 2007. A survey on the situation of water resource andwater management in China. In: Gnauck, A. (Ed.), Modelingand Simulation of Ecosystems. Shaker Verlag, Aachen, pp.171–186.

Wei, S.K., Gnauck, A., 2007a. Water supply and water demand ofBeijing – a game theoretic approach for modeling. In:Gomez, J.M., Sonnenschein, M., Muller, M., Welsch, H.,Rautenstrauch, C. (Eds.), Information Technologies inEnvironmental Engineering. Springer Verlag, BerlinHeidelberg, pp. 525–536.

Wei, S.K., Gnauck, A., 2007b. Game theoretic approaches to modelwater conflicts on a river basin scale. In: Gnauck, A. (Ed.),Modellierung und Simulation von Okosystemen. ShakerVerlag, Aachen, pp. 22–40.

Wei, S.K., Gnauck, A., 2007c. Simulating water conflicts usinggame theoretical models for water resources management. In:Tiezzi, E., Marques, J.C., Brebbia, C.A., Jørgensen, S.E. (Eds.),Ecosystems and Sustainable Development VI. WIT Press,Southampton, Boston, pp. 3–12.

Wei, S.K., Lei, A.L., Islam, S.N. Modelling and simulation ofindustrial water demand in Beijing. Frontiers ofEnvironmental Science & Engineering in China, in press. doi:10.1007/s11783-010-0007-6.

World Bank, 2002. World Development Indicators 2002. Availablevia World Bank website www.worldbank.org. Accessed on 28Jan 2007.

Wu, S.X., 2005. The Spatial and Temporal Change of Nitrogen andPhosphorus Produced by Livestock and Poultry and theirEffects on Agricultural Non-Point Pollution in China. Ph.D.

w a t e r r e s e a r c h 4 4 ( 2 0 1 0 ) 2 4 9 9 – 2 5 1 62516

dissertation. The Graduate School of Chinese Academy ofAgricultural Sciences, Agricultural Resources and RegionalPlanning Institute of Chinese Academy of AgriculturalSciences, Beijing (in Chinese).

Wu, P.L., Zhang, W., 2005. Water crisis and sustainable waterresource utilization in Beijing. Journal of Liaoning TechnicalUniversity 24 (3), 436–439 (in Chinese).

Yang, H., Zehnder, A., 2001. China’s regional water scarcity andimplications for grain supply and trade. Environment andPlanning A 33, 79–95.

Yang, H., Zehnder, A., 2005. The South-North Water TransferProject in China: an analysis of water demand uncertainty andenvironmental objective in decision making. WaterInternational 30 (3), 339–349.

Yang, S.T., Cheng, H.G., Bu, Q.S., Zhang, J.Y., Shi, X.X., 2006.Estimation of soil erosion and nitrogen and phosphorus loadin its application in assessment of the absorbed China. ActaScientiae Circumstantiae 26 (3), 366–374 (in Chinese).

Yang, Z.F., Zeng, Y., 2004. Mathematical model for water conflictand coordination in transboundary regions. Acta ScientiaeCircumstantiae 24 (1), 71–76.

Yevjevich, V., 2001. Water diversions and interbasin transfers.Water International 26 (3), 342–348.

Zeng, W.H., Cheng, S.T., Yang, Z.F., 2001. Integrated managementof river basin. China Environmental Science 21 (2), 173–176 (inChinese).

Zeng, Y., Yang, Z.F., 2004. Policy conflict analysis of waterquality improving for the transboundary regions of Guantingreservoir. Advances in Water Science 15 (1), 40–44.(in Chinese).

Zhang, J.Y., Luo, L., Li, C.S., Ma, R.H., Wang, D.B., Tan, Y.Y.,Kuang, Q.J., 2000. Study on the ecological impacts of middleroute of south-to-north water transfer project on middle-lower streams of Hanjiang River. Environmental Science andTechnology Supplement SUM 90, 1–89 (in Chinese).

Nomenclature

A: action (or moves) in a gameB(Q): water benefit function in cooperative gameBi(Q): water benefit function of player i in non-cooperative gameBOD5: biology oxygen demand after 5 days (mg/L)Cra: TN concentration into reservoir from one human activity in

a region (mg/L)C €w;yc : limiting concentration of pollutant (TN) in controlling

section y (mg/L)C €w;y: concentration of pollutant (TN) in the controlling section y

(mg/L)C €w=;y�1: concentration of pollutant (TN) from upstream controlling

section y � 1 (mg/L)CODMn: permanganate index (mg/L)Cmax: upper threshold of TN concentrations (mg/L)Cmin: lower threshold of TN concentrations (mg/L)DN: nominal value of payoff (V and U ) (108 yuan)DR: real value of payoff (V0 and U0) (108 yuan)DO: dissolved oxygen (mg/L)e�dt: discount factorE: game equilibriumEws: evaporation of water surface (mm)GT, G: game, normal form (or strategic)gameG1, G0: first level game, sub-game respectivelyG0m: main player m’s sub-gameI: information set of a gameK(W): cost function to abate pollutant (TN) in cooperative game

(108 yuan)

Ki(W): cost function of player i to abate pollutant in non-cooperative game (108 yuan)

L €w;yc : limiting TN load in section y (tons)Lra: load of TN into the reservoir from one human activity in

a region (mg/L)LW,y: load of pollutant (TN) in section y (tons)LW,y�1: load of pollutant W from upstream controlling section y� 1

(tons)N: set of playersNm, Nmj: set of main players, sub-playersNH3-N: ammonia nitrogen (mg/L)O: game outcomePm, Pmj: main player, sub-playerpt

I ; pt0I ,: consumer price index (CPI) in time t, t0

Qeu: water used for environ-ecology (108 m3)Qf: water inflow into the reservoir (108 m3)Qi,Q 0i : available water of every player in non-cooperative, cooper-

ative game (108 m3)Qed: environ-ecological water demand (108 m3)Qid: water deficit of player i (108 m3)Qi: maximum water demand of player i (108 m3)Qi: minimum water demand of player i (108 m3)Qy�1: water flow from upstream section y � 1 (108 m3)Qy: water flow in the section y (108 m3)Qws: water demand to keep certain water surface (108 m3)Qga: water demand of public green area (108 m3)Qtr: water demand of newly planed trees (108 m3)R: reclaimed water (108 m3)SN: scenariosS: strategy profile of a gameSm,Smj: strategy profile of main player, sub-playerTN: total nitrogen (mg/L)UB: net benefit in cooperative game (108 yuan)Ui, U0i: payoff, real payoff of player i in cooperative game (108 yuan)V: payoff (or utility) in a gameVi, V0i : payoff, real payoff of player i in non-cooperative game

(108 yuan)Vm,Vmj: payoff of main player, sub-player (108 yuan)Wg: groundwater resources (108 m3)Ws: surface water resources (108 m3)Wi, €W

0m: pollutant TN production of player i in non-cooperative,

cooperative game (tons)WiR: pollutant TN reduction of player i (tons)Wra: pollutant TN produced from a certain activity in a region

(tons)Xkp: independent (or explanatory) variablesYp: dependent variablesGreek symbolsa: coefficient of waste water back into waterb: parameter in linear equationb: benefit coefficientg: cost coefficient to reduce pollutant (TN)j: distribution factor of cooperative benefit[: loss coefficient of TN from production sourcel: coefficient of TN into riverk: coefficient of TN into reservoir4: coefficient TN finally maintaining in reservoirh: assimilation coefficient of pollutant3p: disturb (or error) termSubscripts and superscriptsra: one certain human activity in a regionk, p: observation numberst, t0: time (year)i, �i: every player, other n � 1 playerm, j: every main player, sub-player in sub-gamemj: which main player a sub-player belongs toW: referring to pollutant (TN in this study)y, y � 1: lower, upper stream controlling sections