INTRODUCTION
In common natural resources operation generally and in groundwater extraction
particularly, there are so many evidences that have proved possibility of tragedy
of commons (Clarke et al., 1997). As the optimization
models maximize one or more object to satisfy just one Decision Maker (DM),
practical results arising from applying optimization models in competitive situations,
have not met the expected success.
Game is called to the situation in which there is more than one DM and decisions
of each DM affect the rest of DMs’ payoff. Modern game theory may be said to
have begun with the work of and Von Neumann and Morgenstern
(1944). Next major development was John Nash’s modification of the Von
Neumann and Morgenstern’s (1994) approach. Nash (1950)
formally defined equilibrium of a noncooperative game to be a profile of strategies,
one for each player in the game.
Many researchers have applied noncooperative game to conflict resolution among
common natural operators. Transboundary fishery management (Cave,
1987; Fisher and Mirman, 1992), air pollution and
environment protection (Breton et al., 2006).
Nakao et al. (2002), analyzed potential gains
from cooperation in the withdrawal of water from the Hueco Bolson aquifer. Salazar
et al. (2007) applied game theory to a multiobjective conflict problem
for an aquifer in Mexico, where economic benefits from agricultural production
should be balanced with associated negative environmental impacts. Rubio
and Casino (2001) distinguished between cost and externalities by analyzing
two equilibrium concepts, open loop Nash and stationary Markov feedback in nonlinear
strategies to characterize private extraction. Msangi (2006)
studied asymmetric external effects on aquifer operators’ behaviors.
Coppola and Szidarovszky (2004) analyzed a twoperson conflict, where a
water company and a community are the players and water supply and health risk
constitute the payoff functions.
When players interact by playing a static game in finite stages, the game is called a stage dynamic game. In this study, a noncooperative dynamic game was developed to imply in conflict resolution among aquifer operators. In most of the mentioned study in groundwater extraction to reduce calculation content, it has been assumed that changes in the water level are transmitted instantaneously to all users. Most of the researches on conflict resolution among common natural resources operators have applied static game theory, but it seems that dynamic games are more compatible with the fact. In this study to simulate the exact effects of DMs’ decisions on groundwater table fluctuation, the wellknown Tiess Well equation was employed.
The proposed model is developed based on the earlier studies of Mangasarian
and Stone (1964) and Bellman (1957). Solution to static
games was represented by Mangasarian and Stone (1964).
Bellman (1957) represented dynamic programming for change
complex equation with many variables to many equations with little variable
(Denardo, 2003).
In this study, based on the amount of cooperation among the operators, common natural resources operation is modeled in 3 scenarios: (1) noncooperative static games, (2) noncooperative dynamic games and (3) cooperative games. Proposed scenarios are applied to conflict resolution among common aquifer operators in Bidestan area in Iran. In this area, two municipal and agricultural operators are simultaneously pumping common aquifer to provide potable water and irrigate wheat fields, respectively.
NONCOOPERATIVE STATIC GAME
Game theory is a mathematical method for analyzing strategic interaction. This theory would be useful when two or more DMs with conflicting objects try to decide on a common goal.
Here, we briefly review the equilibrium solution to the noncooperative static
game. Let I and II denote two players and M_{I} = {1...m_{I}}
and M_{II} = {1...m_{II}} be the sets of all pure strategies
available,
and
are strategy spaces and x and y are mixed strategies of players I and II, respectively,
e is the unit vector and T denotes the transposition of the vector. When player
I chooses a pure strategy iεM_{I} and player II chooses a pure
strategy jεM_{II}, the payoffs of players I and II are p^{I}
and p^{II}, respectively. The bimatrix game is defined by BG = {S_{I},
S_{II}, A, B}, where, A and B are the players I and II’s payoff
matrices, respectively.
Definition (equilibrium solution): (x*, y*) εS_{I}, x S_{II} is said to be a Nash equilibrium strategy of bimatrix game BG if x^{T} Ay* < x*^{T} Ay*, ∀ x ε S_{I} and x*^{T} By < x*^{T} By*, ∀ y ε S_{II}.
All the mixed strategies of player i that satisfy the relation above are called
the best response of player i to the opponent. Therefore, the mixed strategy
equilibrium point is a vector in which each player takes action the best response
to his/her opponents. No player intends to change the strategy in case that
all players play on this point and the vector above would be the equilibrium
point. Based on the Nash Existence Theorem every bimatrix game has at least
one equilibrium solution (Owen, 1995), which will be found
by following theorem:
Theorem (Mangasarian and Stone, 1964): A necessary
and sufficient condition that (x*, y*) be an equilibrium solution of BG is that
it is a solution of the following quadratic programming problem:
Further, if (x*, y*, α*, β*) is a solution to the problem above,
thus:
DYNAMIC GAME AND SOLUTION
When players interact by playing a static game in finite stages, the game is called dynamic game. In every stage of such a game, players simultaneously move knowing the moves in the earlier stages.
Development of dynamic game: Let us consider noncooperative dynamic
game for two players. The decisions of players I and II at stage t are expressed
as Assume
that the initial system state at stage t be R_{t} and the maximum and
minimum possible values of it be R^{max} and R^{min}, respectively.
The players’ payoffs at stage t, denoted by
for players I and II, respectively, depend on the current system state R_{t},
player I and player II’s decisions, so that:
where f and g are desired utility functions for players I and II, respectively.
Each player chooses optimal policies to maximize his/her utilities all over
the stages with regard to his/her opponent’s probable action. Players’
optimal decision at stage t will be a decision that leads to the maximum payoff
to the end of the planning horizon for him/her. In other words, at each stage
both Eq. 3 and 4, should be satisfied simultaneously:
where,
are the maximum expected cumulative payoffs for players I and II from stage
t to the end of the planning horizon, respectively.
are the optimal decisions of players I and II, respectively at stage t as well.
Attention to the right hand side of equations above shows that players’
payoffs consist of two components: the first one is the current payoff and the
second is the future payoff.
Solution to the dynamic game: To solve the stage dynamic game, the combination
of dynamic programming and solution to the static games are employed. Dynamic
programming is a theory extensively adopted by Bellman (1957).
Programming starts with the final stage t T_{f}. For all players’ decisions,
the players’ payoff matrices are created. By solving each of static game by
Eq. 1, the optimal decision of players I and II,
respectively, are obtained for each possible system state. At stage t = T_{f}1
for all players’ decisions, the current players’ payoffs are obtained. Since,
the system state at the end of stage t = T_{f}1 is equal to the beginning
of the next stage t = T_{f} and players’ optimal decisions at stage
t = T_{f}, have been defined previously, so, by adding the current and
future payoffs, the cumulative payoffs are created at stage t = T_{f}1.
By solving this game, the players make a decision which leads to the maximum
payoff from stage t = T_{f}1 to the final stage. The mentioned process
continues till it reaches the first stage t = 1.
SCENARIOS OF AQUIFER OPERATION
Scenario 1: Noncooperative static game: In this scenario, it is assumed
that the players monthly decision is defined by myopic policy. In other words
players are not long sighted and their monthly decision is just to reach monthly
compromise. Therefore during the operation stages, players should solve T_{f}
independent static games. If denotes
the payoff of player i at stage t, Eq. 5 shows the interaction
of operators with each other:
Scenario 2: Noncooperative dynamic game: In this scenario the goal of each operator is getting the maximum and possible benefit at the whole stages of operation and at the same time watching the probable moves of the opponent.
Scenario 3: Cooperative game: To better evaluation of before conflict resolution scenarios, another scenario with the aim of optimization of aquifer operation was prepared. The objective function of this model is maximizing the total extraction of players during the whole stages of the game. In other word, it is supposed that instead of two DMs, one DM is the owner of the wells and try to maximize the whole extraction of them:
CASE STUDY AND DECLINE EQUATIONS
Figure 1 indicates the position of Bidestan and the wheat
field around it. This city is located at 150 km to the Northwest of Tehran,
the capital of Iran. Some of the potable water of this city, with 10000 inhabitants,
is provided with M1 and M2 wells. The water for wheat fields is drawn from pumping
A1 to A7 wells. To make the calculation easier, one municipal equivalent well
(ME) instead of M1 and M2 and one equivalent agricultural well (AE) instead
of A1 to A7 are used. The equivalent of the municipal pumping rate is Q_{ME}
= 1500 m^{3} day^{1} and the equivalent of the agricultural
pumping rate is Q_{A} = 2000 m^{3} day^{1}.
First and second rows of Table 1 shows monthly value of municipal and agricultural operators. Third row indicates the monthly increasing level of groundwater table fed by precipitation.
The planning period is based on a oneyear scale. The set of pure strategies for operator i is a discrete one which including ten elements, M_{i} = {0, 10%,...100%} i ε {I, II}, where symbols I and II stand for municipal and agricultural operators, respectively. Groundwater level at the beginning of each month determines the system state at the beginning of each stage.
Operators payoff at stage t, depends on the current system state, operators’
decisions and the amount of monthly demand, so that:
Q_{i} is the pumping rate for player I,
is the demand of operator i at month t. To avoid the aquifer overdraft if
thus punishment is enforced on the operator i. The amount of drawdown in the
groundwater table at the end of each month is calculated by Tiess Well equation
(Maidment, 1993):

Fig. 1: 
Position of municipal and agricultural wells 
Table 1: 
Monthly demand of operators (m^{3} day^{1})
and groundwater table incensement at Bidestan aquifer (m) 

where,
is the amount of water decline in well i at month t, r_{i} is the distance
from the point at which the decline is measured to well i is 1000 m, T is the
aquifer transmissivity equal to 432 m^{2} day^{1}, D is the
duration of operation with the fixed pumping rate equal to 30 days and S_{S}
is the specific storage coefficient equal to 4x10^{4}.
RESULTS AND ANALYSIS OF EQUILIBRIUM POINTS
Here, we analyze the results arising from different scenarios. Table
2 and 3 explain the amount of extracted water, Fig.
2 and 3 show the equilibrium points and Fig.
4 and 5 shows the changes on ground water basin, respectively
in the locations of municipal and agricultural operating wells.
Comparison and analysis of tables and figures above shows as the following:
• 
According to Table 2 and 3
the average of obtained water by players in scenario 3 is more than scenario
1 and 2. The reason of this matter is because of the cooperation between
the two operators. As in scenario 3 extracting the maximum amount of water
by two operators are assumed as the goal function, so the players cooperate
with each other to achieve a collective aim. In this scenario the collective
aims of operators is prior to individual aims 
• 
In scenarios 1 and 2 to resolve the conflict among the operators, individual
interests are considered. According to Table 2 and 3
the amount of obtained water arising from scenario 2 is more than scenario
1. In scenario 1 the operators on the basis of monthly needs and also the
probable movements of the opponent choose a decision that leads into the
individuals benefits at the mentioned month. But in scenario 2 the players
in addition to watching the probable moves of opponents analyze the effects
of current decisions on acquired water on the oncoming months and take decisions
that require more extraction at the sum of current stages and future stages 
• 
Comparing Fig. 25 show that in scenario
1 municipal and agricultural operators without considering the effects of
current decision on future payoff tries to extract water from aquifer. The
process of aquifer extraction in scenario 1, emphasis on this fact that
as long as possible the operators tried to extract water from the aquifer.
In other word, in scenario 1, players obey myopic policy. In this scenario
obtaining water at the first half of the year is more than the second half
of the year. But in scenarios 2 and 3 water extraction is done gradually
throughout the year with the monthly need of operators 
• 
Generally, the results show that cooperative model has more profits for
the group of players. But we should emphasize that the operation on the
base of optimization is not applicable. Because this method is assumed on
the basis of cooperation among the operators, but it is possible that the
operators don’t keep their words on the agreement or by optimization
model be spoiled the interest of one of the operators on behalf of other
operators. 

Fig. 2: 
Monthly decision of municipal operator in different scenarios 
Table 2: 
The amount of extracted water by municipal operator in different
scenarios 

Table 3: 
The amount of extracted water by agricultural operator in
different scenarios 


Fig. 3: 
Monthly decision of agricultural operator in different scenarios 

Fig. 4: 
Monthly fluctuation of groundwater table in municipal well
in different scenarios 
As it was said in introduction, despite having less benefit for the players,
using noncooperative conflict resolution model is a necessity. There is no
need to the agreement of operators with each other but logical operators follow
the operation rules which arises from noncooperative conflict resolution models.

Fig. 5: 
Monthly fluctuation of groundwater table in agricultural well
in different scenarios 
Results show that among the noncooperative conflict resolution models, the
results of the proposed model (scenario 2) on providence of operators and in
addition to conflict resolution among the operators, there is a little difference
with the results of cooperative model in a way that the differences of suggested
conflict resolution and cooperative model in the case study is 3%
CONCLUSION
In this research, the dynamic conflict resolution is presented on the basis of compilation of static games study and the dynamic programming. The theory of static games for conflict resolution among operators and dynamic programming for transferring players payoff from one stage to another stage is used. This model application is in long term and mid term programming of conflict resolution for common natural resources. The proposed model is applied for conflict resolution among municipal and agricultural operators from Bidestan aquifer, which is located in Iran. To analyze the effectiveness of the proposed model, the static noncooperative and cooperative model on the Bidestan aquifer were applied also. Results showed that among the above scenarios, cooperative model has more benefit for the players but in practice the possibility of using cooperative model is less. The results show that in noncooperative conflict resolution the results of the proposed model on the basis of operators’ providence has a little difference with the results of cooperative model.