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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: [email protected]@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@[email protected] (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), [email protected] (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.
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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