Developing an integrated land allocation model based on linear programming and game theory

Land use configuration in any given landscape is the result of a multi-objective optimization process, which takes into account different ecological, economic, and social factors. In this process, coordinating stakeholders is a key factor to successful spatial land use optimization. Stakeholders need to be modeled as players who have the ability to interact with each other towards their best solution, while considering multiple goals and constraints at the same time. Game theory provides a tool for land use planners to model and analyze such interactions. In order to apply the spatial allocation model and address stakeholder conflicts, an integrated model based on linear programming and game theory was designed in this study. For implementing such model, we conducted an optimal land use allocation process through multi-objective land allocation (MOLA) and linear programming methods. Then, two groups of environmental and land development players were considered to implement the optimization model. The game algorithm was used to select the appropriate constraint so that the result would be acceptable to all stakeholders. The results showed that during the third round of the game, the decision-making process and the optimization of land uses reached the desired Nash Equilibrium state and the conflict between stakeholders was resolved. Ultimately, in order to localize the results, a suitable solution was presented in a GIS environment.


Introduction
Land use planning is a complex process in which all land use types are evaluated simultaneously to set a systematic framework to address the conflicting goals and constraints of landowners (Kaiser et al., 1995;Guoxin et al., 2004;Ligmann-Zielinska et al., 2008;Cao et al., 2011;Batty, 2018;Song & Chen, 2018;Maleki et al., 2020). Spatial optimization of land use is a complex decision-making problem with multiple antagonistic objectives from different parties and proper coordination between among stakeholders is a major key to solve land use conflicts and successful spatial optimization. The limited quantities of land resources and their suitability for different utilities are the main reasons for such conflicts (Chen, 2007;Maleki et al., 2020). From a spatial point of view, land use conflicts can be considered as competition of different land use types over a shared landscape to occupy land; however, in principle, such competitions are mainly due to conflicts of interest among several stakeholders (Zhang et al., 2012). Regarding land use management and optimization, selecting a suitable alternative from a set of available options is a challenging task (Gu et al., 2021;Lund & Palmer, 1997) since criteria are contradictory and hardly reconcilable, and any decision can cause objection from one or multiple parties, who think their interests are neglected (Lee & Chang, 2005) and it is often difficult to satisfy all stakeholders with different interests, values, and perspectives (Shields et al., 1999;Collins & Kumral, 2020). Considering land use conflicts, the contradiction between the economic benefits of land use development (e.g., timber cultivation, agricultural practices, and recreational activities) and ecological values (e.g., water and soil conservation and eutrophication reduction) has been well documented in the literature (Lund & Palmer, 1997;Jana et al., 2020). This contradiction in our study area is between the environmental approach of governmental decision makers in the Zagros basins, Iran, and the economic interests of local people living in these basins and it has led to the situation that majority of decision makers are trying to establish a balance between such conflicting goals. In this context, silo approaches and excluding influential parties in the region from decision making have only led to trade-off among different goals and negative feedbacks and feedback loops among players (Raquel et al., 2007).
Many researchers have conducted extensive researches on land use optimization and allocation. Existing optimization models can be roughly divided into the following three categories: linear programming models, cellular automation (CA) models and intelligent algorithm models (Liu et al., 2015). Linear programming models can quickly detect the structure of the optimal land use in response to specific goals and constraints (Arthur & Nalle, 1997;Chuvieco, 1993;Sadeghi et al., 2009). Automated cellular models are based on land use conversion principles and follow a bottom-up approach to create different land use patterns under different conditions (Li & Yeh, 2000. There are various machine learning and artificial intelligence algorithms in the literature amongst which simulated annealing (Aerts et al., 2003), particle swarm optimization (Liu et al., 2012a, b, c;Ye et al., 2021), ant colony (Li et al., 2011;Liu et al., 2012a, b, c;Tang et al., 2020) and genetic algorithm (Liu et al., 2015;Ozsari et al., 2021) are important representatives to mention. Geographic information systems (GIS) plays an important role in the application of intelligent algorithms and spatial optimization of different land use categories (Wu & Grubesic, 2010). GIS is used to process and visualize spatial data for these algorithms; however, these models do not take into account local land use conflicts and lack a coordinated game-based mechanism for solving local land use competitions (Liu et al., 2015).
Game theory, which was first introduced with the pioneering work of Bočková et al. (2015), is the study of mathematical models of conflict and cooperation between decision makers (Bočková et al., 2015). It is also a powerful tool in determining the equilibrium among decision makers and it is used to analyze situations where stakeholders' decision making affect each other's decision. Game theory has been used in various fields such as economics (Camerer, 1997) and social sciences (Myerson, 1992), water resource management (Parrachino et al., 2006a, b;Carraro et al., 2007;Homayounfar et al., 2010;Sobuhi & Mojarad, 2010;Liu et al., 2021;Mohammadifar et al., 2021), optimal groundwater consumption (Mazandaran Zadeh et al., 2010;Pourzand & Zibaei, 2010;Nazari et al., 2020;Yazdian et al., 2021), wood market (Mohammadi Limaei, 2006, paper market (Mohammadi Limaei, 2010), forest management (Rodrigues et al., 2009;Shahi & Kant, 2007;Ikonen et al., 2020), and watershed management (Lee, 2012;Moradi & Mohammadi Limaei, 2018;Adhami et al., 2020). The game theory can simulate the decision-making behavior of different stakeholders with conflicting interests and facilitate reaching a consensus among them (Maleki et al., 2020;Rasmusen, 2001;Zhang, 2004). The application of game theory in the context of land use change can be categorized into monitoring (Wu et al., 2005) multi-objective optimization (Lee, 2012), and resolving land use conflicts (Hui & Bao, 2013;Maleki et al., 2020) studies. However, game theory methods are rarely associated with land use allocation models and in this research, we have attempted to make a connection between land use optimization and conflict resolution to develop an informed spatial allocation model. In this study, we first optimize different land use categories using the multi-objective land allocation (MOLA) method and linear programming. Then, a contradiction is simulated during land allocation process, which will ultimately be resolved using the game theory algorithm. Despite much research on separate application of game theory and linear programming in environmental studies, their integrated application for developing land allocation models is a less noted attempt in the literature. Therefore, the main objective of this study is to develop an integrated land allocation model to explore the potential of different land use optimization methods such as MOLA, game theory, and linear programming when combined together. In addition, this study provides a participatory basis to include the views of different stakeholders during the decision-making process which supports application of the results. It also provides a spatial decision support system, which helps decision makers to simulate and quantify the probable effects of strategies adopted by influential parties in the region.

Materials and methods
The study site covers Gorgan and Kordkoy cities spanning over an area of 243,921 hectares, located in Golestan Province, northeastern Iran. Gorgan is located at 48° 28′ 54″ northern longitudes and 48° 49′ 36″ eastern latitudes and Kordkuy is located at 47.88° 6′ 54″ northern longitudes and 30.12° 47′ 36″ eastern latitudes. The population size in Gorgan and Korkoy is 462,455 and 70,244, respectively (Census Yearbook of Golestan Province, 2013). Figure 1 shows the geographical location of the study area.
The trend of land use change in the study area shows that in 1984, forest cover accounted for more than 50% of the area and by 2036, the area of forest cover is anticipated to be to less than 20% of its current total area (Asadollahi & SalmanMahiny, 2017). Forest lands in this region have mainly been transformed into land use categories such as agriculture and urban areas. Such land use conversions have been occurring mostly based on economic interests with no respect to land suitability and ecologically valuable ecosystems in the region. Therefore, in order to protect forest and protected lands, and simultaneously, regulate land development plans in accordance with land capability of the area, governmental authorities need to find a balance between economic players and environmental stakeholders whose interests are conflicting in the region. The environmental stakeholders include pro-environmental organizations and bodies, as well as pro-environmental people. The economic players include government, land developers, and private companies. Thus, in order to resolve conflicts between pro-environmental stakeholders and land developers, a solution based on game theory and linear programming can be achieved, so that the results of land management would be acceptable to all.
The general framework of the study is given in Fig. 2. First, we examined land suitability using the Multi Criteria Evaluation (MCE) method for seven selected land uses. In the next step, we used the MOLA method for land allocation and land use planning. Then, we used the multi-objective linear programming method to better optimize land uses and improve MOLA outputs. At this stage, to meet the demands of different stakeholders in the study area, two objectives of minimizing runoff depth and maximizing profits from each land use were selected for land use optimization. Also, restrictions were designed to implement the objectives according to the needs of each stakeholder group. Due to various limitations, it was not possible to continue optimization process, and therefore, the game theory was implemented to resolve the conflict. Finally, by resolving the conflict, based on the result of the game theory, one of the constraints was selected as the constraint accepted by all stakeholders, and the multi-objective linear programming was performed based on these constraints to optimize and upgrade MOLA outputs.
This study was conducted through three main steps to allocate land uses. In the first step, an initial multiobjective land allocation model is presented using the MOLA method. The purpose of this step is to provide a model of primary land use allocation based on the potential and natural characteristics of the region. In the second step, the initial land use allocation model is updated according to higher goals such as the interests of pro-environmental stakeholders and land developers (i.e., secondary land use allocation). Finally, in the third step, in order to implement the secondary land use allocation, it is necessary to resolve the existing conflicts between the stakeholders with their conflicting interests, so that the results of land use allocation are acceptable to all of them. To achieve this goal, we will use a designed game algorithm.

Land use optimization
The MOLA method was used to optimize different land use categories in the study area. In doing so, based on ecological characteristics of the region, research objectives, experts' opinions and past studies, land use categories in the area were defined as follows: warm and cold water aquaculture, agriculture, forestry, urban and rural developments, conservation, and rangelands. Then, the relevant data and map layers were collected and standardized. In the next step, using MCE for each land use type, a suitability map was prepared, which then fed into MOLA. The data used and the steps of MCE and MOLA methods are fully described below.

Database preparation and multi-criteria evaluation
During the MOLA process, first a land suitability analysis for different land use categories was conducted. In this regard, based on objectives of the study, local knowledge of the research location, data availability and former studies in the region, hydrothermal aquaculture, cold-water aquaculture, agriculture, forestry, urban and rural development, conservation, and rangeland were selected as land use categories. The required factors for each land use were determined and a suitability map for each land use was prepared using the MCE method. In this case, important factors including elevation, slope, geographical aspect, geological characteristics, soil, and Fig. 1 The geographic location of the study area erosion risk were obtained mainly from digital elevation model (DEM) of the study area. The DEM of the area was acquired from National Cartographic Center (NCC) of Iran. In addition, hydrographic map, plant type and vegetation, climate of the region, bedrock, and percentage of vegetation density, soil type and texture (including drainage conditions, soil depth, and structure as well as fertility), distance from the road, distance from residential areas, and hydrological map of the study area were also prepared. The set of factor layers implemented in this study were obtained from Goragn University of Agricultural Sciences and Natural resources, which is the leading institute in conducting land use planning studies in the region.
The MCE method evaluates land suitability in response to several environmental criteria. During the MCE process, several map layers are standardized (i.e., fuzzified), weighted, and combined to evaluate the suitability of the land for a specific utility. In this regard, in the first step, the above-mentioned set of meaningful criteria for land suitability analysis were identified. The set of criteria is divided into constraints and factors. Constraints include those Boolean criteria that shows the suitability of the land in a 0 and 1 fashion. In other words, zero indicates absolute lack of potential and one indicates suitability of the land for the desired land use. In contrast, factors represent the degree of the land suitability in a continuous way, and therefore, layers with continuous values are standardized using fuzzy sets theory. It is because each layer has a different measurement scale and it is necessary to standardize the layers to be able to combine them into one single suitability map. The standardization process can be performed by fuzzy membership functions on a scale of 0 to 1 or 0 to 255 (Drobne & Lisec, 2009). In addition, the relative weight of each factor layer was determined based on analytical hierarchy analysis (AHP) and finally layers were combined into one final suitability map based on Weighted Linear Combination (WLC) method (Hasti et al., 2016a, b). The weights are calculated by a series of pairwise comparisons of the relative importance of factors in response to the suitability of the land use being evaluated. These pairwise comparisons are then analyzed to produce a set of weights that sum to 1.
The WLC equation is presented as follows: (1) where Wi is the AHP-derived relative weight of the factor i, Xi is the standardized layer i, Ci is constraint i, and ∏ is he multiplication operator.
MOLA is an iterative allocation process through which a specific area threshold for each land use category is achieved using the corresponding weight of the land use type. In the present study, the aim was to optimize land use categories considering ecological conservation perspectives and economic interests. In case of ecological conservation, the objective was minimizing the probability of runoff depth, and the economic goal was maximizing the profit from each land use. In order to achieve the targeted goals and improve MOLA allocation output, the multi-objective linear programming method was employed. A multi-objective linear programming model can be defined as follows (Eq. 2): where Z( x ) is an objective function and [Z1( x ), Z2( x ), Z3 (x ), …, Zp ( x )] is a set of objective functions G i (x), j is the most important constraint and x k , k is the decision variable. The goal of the optimization is to find the best acceptable answer under given limitations. There could be different answers to a problem and objective function compares them and selects the optimal solution. The choice of this function depends on the nature of the problem. For example, the objective function can be selected as profit maximization, employment, runoff minimization, erosion, environmental pollution, etc. Decision variables are the variables that are used to write the objective function, such as different types of land use. Constraints indicate what constraints exist or should be applied to the execution of the objective function.
Land use optimization in watersheds using linear planning and GIS is one of the appropriate management methods to achieve optimal landscape configuration and maximum profit (Riedel, 2003); however, the economic interests must be secured considering ecological and social sustainability aspects (Pfaff & Sanchez-Azofeifa, 2004;Ducourtieux et al., 2005). There are also other studies in the literature applying linear programming for land allocation (Shabani, 2010;Chuvieco, 2004;Nikkami et al., 2002).
In order to obtain the runoff depth probability of each land use type, the long-term hydrological impact assessment (L-THIA) model was used in Arc View software environment and the runoff depth probability was extracted. This model provides an estimate of runoff changes, recharge and pollution of non-point sources due to past or predicted land use change to provide a measure of the long-term effects of development on hydrological conditions (Bahadori et al., 2000;Ma, 2004;Weng, 2001). Since one of our goals in land use optimization by linear programming is to reduce runoff depth, one of the fastest and most reliable methods for calculating runoff depth in the study area is the L-THIA method. This model provides a synoptic overview of the runoff situation throughout region.
This model was first developed for natural resource managers because they are familiar with land use change in a particular area and have access to land use information and are often interested in studying environmental impacts (Engle et al., 2003). In addition, economic data such as land use profit was retrieved from land use planning database of Golestan Province, which was establish in 2014.
The purpose of this study is to integrate game theory into the land allocation process. At this stage, in order to simulate contradictive perspectives during the land use allocation, two target categories can be considered for the implementation of multi-objective linear programming. In other words, there are two categories of constraints for the implementation of linear programming, which include constraints designed by stakeholders seeking environmental protection and land developers trying to maximize economic interests. Therefore, with two sets of constraints, it is not possible to implement land use allocation by linear programming. In this regard, stakeholders want to implement their goals, and at this stage, game theory can be used to resolve the simulated conflict.

Game algorithm
At this phase, in order to resolve the simulated conflict, we can implement the general framework and the game algorithm (Fig. 3). Figure 2 shows the game algorithm designed for this study. As shown in the figure, after the initial (MOLA) and the second (linear programming) land allocation, we have two sets of constraints according to the different needs of the stakeholders (environmental and economic), and therefore, the optimization process will stop (conflict). According to the various designed scenarios and strategies, which we will explain in the following, we will use game theory to resolve this conflict and select one of the limitations to continue the optimization process. This algorithm shows that in each stage of the game, if the result is Nash equilibrium, which is acceptable to all players, the created conflict will be resolved. The result of resolving the conflict is selecting one of the limitations for implementing linear programming and final land allocation.
Once the game algorithm has been designed, it is necessary to devise strategies for each step of the algorithm to simulate decision conditions and selection of players. Specifically, there are different categorizations of games. Games, in terms of format, can be divided into two forms of strategy or normal and wide or tree. Strategy games are a compact form of a game in which players simultaneously coordinate their strategy. Extensive games can also be referred to as a set of normal games (Hasti et al., 2016a, b). Each game contains three main elements including (i) players, which is the factor that makes decisions in the game, (ii) actions, which is a set of actions defined for each player, and (iii) final result, function or preferences, which is the result of each player from his decision according to the rules and scenarios of the game (Hasti et al., 2016a, b). In the present study, the idea of a prisoner puzzle game (non-zero outcome), which is a well-known example of normal games, was undertaken and the wide form of the game was used to implement it. The conflict is created in such a way that each player accepts his own restrictions and has designed restrictions according to his goals. The designed goal for players in the game is to meet their designed constraints. In other words, the choice of constraints designed by them is more important. In the next stage, game scenarios for each player were determined according to experts' knowledge and the designed goal. The game scenario in the first iteration is presented in Table 1.
According to Table 1, the first strategy shows that if the environmental player chooses his constraints (En constraints), he will get 5 points. Also, if the economic player selects his designed constraints (Ec constraints), he gets 5 points. Therefore, Table 1 shows the strategies that each player can choose and the points that he will receive from each choice. The logic applies to Tables 1, 2 and 3.
Then, the game model was designed in Gambit 13.1 software and the game results were determined, which included the balance of the game and the points obtained for each player as a result of solving the game. The results show that players have selected their own restrictions and the desired conflict is still present, and consequently, the game enters the second iteration. In the second phase of the game, the objective for each player was to achieve the environmental and land development interests as well as respecting constraints designed by other players. In other words, each of the constraints designed by each group achieves the maximum value of the objective function (minimizing the probability of runoff depth and maximizing the profit from each land use). To do this, the environmental objective function was first implemented based on constraints that are separately designed in response to environmental conservation and economic benefits. Then, the same procedure was repeated for the economic function. In the next step, based on the experts' opinion and according to the results of the multi-objective linear programming for objective functions (based on the constraints designed by each player separately), game scenarios were introduced for each decision of the players. Table 2 shows the game scenario in the second iteration.
The land use allocation game model was implemented with the second iteration in Gambit 13.1 software. The results of the game including the desirability of the players in each decision and the game's balance were obtained. Given the nature of the game theory (Nash Equilibrium concept) in predicting player's decisions and game resolution results, it shows that each player selects his or her own constraints as the final decision. Due to outcome of the game, the conflict has not yet been resolved and it is not possible to continue the process of optimization and land use allocation. Therefore, due to the nature of the game theory in bargaining and repetition, the game will enter its third phase. In the third phase of the game, more emphasis was placed on the bargaining power of the game theory, and the goal of the players in making decisions was set accordingly. To resolve the conflict, the players' goal was to satisfy the other side based on their required area and needs. To do this, both environmental and land development objective functions were implemented simultaneously with the designed constraints of the ecologists and economists in the WINQSB software. In the next step, based on the results of the implementation of the multi-objective linear programming model (i.e., area allocated to each land use), the goal of each player, the opinion of experts and game decision scenarios were designed (Table 3).
The results of the game solution show that the Nash Equilibrium is at the step where both players choose the designed constraints related to the environment. By negotiation between the players and satisfying each other, the result indicates that the conflict has been resolved and the process of land use allocation continues. In this way, spatial optimization with the designed constraints by environmentalists can be implemented in a way that it is acceptable to all stakeholders based on the results of the game algorithm. Due to the nature of game theory, the game can enter other stages until the decisionmaking process finally reaches the Nash Equilibrium. The essence of the game theory is that each player selects a strategy, but the end result depends on the choice of all players. Therefore, each player, to some extent, can control the outcome of the game. The end result varies from person to person, with one player being the best and the other the worst. Thus, the fundamental question is, given the different outcomes for individuals, how does game theory resolve conflict (Samsura et al., 2010)? There are different approaches to game theory; however, the concept of Nash Equilibrium is often used (Aumann, 1985). Nash Equilibrium can be introduced as a strategy choice for each player such that no player is willing to change and the strategy chosen by each player is the best response to the strategy chosen by other players. The best means that changing this strategy will not increase the result (Samsura et al., 2010). The following equation shows the Nash Equilibrium in a game: In this regard, Ui is the utility of the ith player and the actions of each player in different situations. According to the above explanations, by reaching the Nash Equilibrium, the result of the game is accepted by the majority, and the objective functions were performed with environmental constraints.
For model sensitivity analysis, it is also necessary to examine the expected outputs by changing the model inputs used in this research. As explained, the purpose of this study is to improve land allocation results from MOLA analysis with objectives such (3) U i (a * ) = U i a 1 * , a 1 * , ⋯ , a n * ≥ U i a 1 * , a 1 * , ⋯ , a n * as minimizing runoff depth and maximizing profits using linear programming, and therefore, the result of linear planning changes land allocation areas. For sensitivity analysis of the model, we implemented the L-THIA model once with the initial land allocation map (MOLA) as the input land use layer and once with maps from linear programming (one time with environmental stakeholder constraints and one time with economic stakeholder constraints). Finally, we examined the sensitivity of the model and variations in runoff depth in response to changes in land allocation (Table 8).

Results
First, for each of the seven land uses selected in the study area, we obtained the land suitability surface through the MCE method. Then, we used MOLA for the initial optimization. Figure 4 shows the MCE-derived land suitability layers for each land use category.
The input data feeding objective functions during MOLA allocation process were standardized in a range between 50 and 255. Table 4 shows the environmental and economic objective functions.
In Table 4, (aa) represents agricultural lands, (ff) forest lands, (cc) cold-water aquaculture lands, (ww) hydrothermal aquaculture lands, (rr) rangelands, (pp) protected areas, and (dd) urban and rural development. Other decision variables represent the conversion of one land use to another, e.g., (aw) is agricultural land that can be converted into hydrothermal aquaculture and (ra) is rangeland land that can be converted into agriculture.
According to Table 4, the objective functions of minimizing runoff depth and maximizing land use profits indicate that in order to optimize MOLAderived land use map, it is important to consider which land use is likely to convert to another and if these land uses are converted, how much runoff will be produced and how much profit is generated. Then, each of the environmental groups and land developers designed their own constraints. Each group accepts its own constraints to continue the multi-objective linear planning process and rejects the other party's constraints. This conflict prevents the continuation of the land allocation process and stops the algorithm. Therefore, game theory was used to resolve the conflict. Table 5 shows the designed constraints of each group. In Table 5, A indicates agricultural use, F is forestry, R is rangeland, D is urban and rural development, W is hydrothermal water use, C is cold water aquaculture, and P is conservation. Other variables indicate the conversion of one land use to another. Considering environmental constraints, constraints (1) to (5) relate to area limitations. Constraint no.
(1) relates to the amount of required area (cell) from the entire study location. Constraint no. (2) indicates the amount of areas that should be converted to agriculture that have the required area (same as the area obtained from MOLA) and constraint no.
(3) refers to areas that should be converted to forests that have the required area (area obtained from MOLA). Restriction no. (4) shows areas that need to  be converted to hydrothermal aquaculture that has the required area (MOLA area) and restriction no. (5) is areas that should be converted to conservation that has the required area (MOLA area) constraint. Other limitations relate to environmental and technical constraints. In terms of economic constraints, constraints nos.
(1), (2), (4), (5) are related to area constraints and other constraints are related to economic, social, and technical constraints. Table 5 shows that there should be constraints to achieve the desired objective functions. The unit of all variables in Table 5 is area (cell). The pixel size in this research is 30 m × 30 m. In this research, the prisoner puzzle game was used for the land use allocation game. Based on game elements, action A indicates the choice of its designed constraint and action B indicates the choice of the constraint designed by the other player. The third element of the game is the payoff of each player at each stage of the game. After running the game in Gambit 13.1 software, a stage of the game called Nash Equilibrium was achieved (Fig. 3). The concept of Nash Equilibrium indicates each player makes rational decisions, meaning that each player seeks to maximize his profit. In other words, Nash Equilibrium is a point at which if a player changes his game, his profit does not change (assuming the rest of the game is fixed). In the second game, the goal for each player was primarily to achieve environmental and economic goals and then to select the constraints designed by the players. To do this, first environmental objective function was implemented based on environmental and economic constraints separately designed in WINQSB software. Then, the same procedure was repeated for the economic function. Table 6 shows the results of the objective functions in the WINQSB software. Table 6 shows the unitless values of the objective functions.
In the next step, players make decisions according to the designed scenarios. The results of the game indicate that each player has chosen his own designed restrictions (Fig. 3). In the third game, the players' goal was to satisfy the other side based on their required areas and needs. To do this, both environmental and economic objective functions were implemented simultaneously with the designed constraints of the environment and in another stage with the designed economic constraints in WINQSB software. The results of performing the objective functions are presented in Table 7. The pixel size of each of the maps used in this research is 30 × 30 m. After running the game in Gambit 13.1 software, the third stage of the game was obtained under the Nash Equilibrium state. At this stage, each player selects the constraints designed by the environment, and the payoff is the environmental player ten and the economic player five. Due to the solution of the game, the environmental player is declared as the winner of the game, and with this result, the conflict is created, and then, it is possible to resolve and continue the process of allocating land use. Accordingly, land use optimization continues throughout the multi-objective linear programming process with constraints designed by environment (Fig. 4).
Linear programming is not an originally spatial method and the results are in the form of changing the area of each land use and converting one land use to another during the final optimization according to the defined set of goals. Locations with the least suitability for the initial land use and the highest suitability for the converted land use should be changed. In order to find these areas, the map layers were cross-tabulated. In the next step, there were different solutions for locating these areas. In this regard, we extracted the land use whose area needs to be changed and then we ranked the MOLA-derived map and extracted the desired area from this layer. These areas were then reclassified based on land use category, and finally overlaid with MOLA map. This process was implemented for the results of multi-objective linear programming and the final land use optimization map was configured using goal programing and GIS. Table 8 shows the comparison between new MOLA areas and the initial optimization areas and the probability of runoff depth of each land use based on old and new MOLA maps with L-THIA method.
In Table 8, A means the initial MOLA land use map that has been used as the input of the L-THIA model. In addition, B and C are the land allocation maps resulted from linear programming model with environmental  Protection Protection (710,000 cells), protection to forestry (100,000 cells), protection to rangeland (100,000 cells) Protection (810,000 cells), protection to forestry (100,000 cells) stakeholders' constraints and constraints of economic players, respectively. The L-THIA model was implemented again with an end-use allocation map to determine the effect of the objective function (minimizing the probability of runoff depth) on reducing runoff depth. The results are presented in Table 8 and Fig. 5. Figure 6 shows the game elements and the form of games played during the execution of the algorithm.
The elements of the first game were as follows: After running the game, step A of the game obtained the game's balance. At this stage, each player selects the constraints designed by the environmentalists, and the payoffs are the environmental player ten and the economic player five. According to the solution of the game, the environmental player is declared as the winner of the game, and with this result, it is possible to resolve the conflict and continue the process of land use optimization. Accordingly, land use assessment continues with constraints designed by the environmentalists. Each stage of the game had its own Nash Equilibrium, but it should be noted that once the game reaches the Nash Equilibrium, the player does not change his strategy in any way. In the first and second games, during stage B of the game, each of the stakeholders selects his own designed constraints (Fig. 6). The output of this stage of the game was the Nash Equilibrium but the game conflict was still unresolved and it was not possible to continue the process of land use optimization. Figure 7 shows the final land use map of the study area using the land allocation model based on linear programming and game theory.

Discussion
In the third phase of the game, more emphasis was placed on the bargaining power of the game theory, and the goal of the players in making decisions was grounded on this basis. Table 7 shows the results of the execution of the objective functions with their corresponding constraints. Each player must first prioritize the needs of the other party based on the required area, which is accordingly translated as a constraint. With environmental constraints, 70,000 cells should be allocated from agricultural to hydrothermal and cold With this result, attention has been paid to the needs of the economic players considering production of labor, increasing profits and reducing the initial cost of establishing each land use. Under such designed limitations, the economic player of 15,000 cells from agriculture should be allocated to cold water and hydrothermal aquaculture. Environmental constraints have also improved economic landscape of the area by changing agricultural land uses, while not reducing forest and natural ecosystems, which play a key role in reducing runoff and erosion. In addition, in the second stage, with this change in agricultural use, the concerns of environmentalists for protecting forests are also taken into account. With the limitation of the environmental player, about 60,000 cells should be allocated to forest protection, and consequently, proper attention has been paid to such environmental aspects. But according to designed limitations, the economic player of forestry will remain intact. Namely, more attention has been paid to the economic aspects of forestry than to the environmental concerns of the other side. Cold-water aquaculture will remain unchanged in both constraints. 10,000 cells of hydrothermal water use should be allocated to urban and rural development based on the designed constraints of the environment (considering socio-economic aspects through changing economic uses). A total of 15,000 cells of this land use should be converted to urban and rural development according to the designed constraint of economic domain. The use of urban and rural development will remain unchanged in both constraints. Due to its role in reducing runoff and erosion, as well as its economic aspects such as forage production, livestock grazing and beekeeping, the use of rangeland habitat will remain unchanged in the designed environmental constraints. But with the economic constraints, 25,000 cells of this land use must be allocated to urban and rural development. Therefore, less attention has been paid to the needs of the other side and mostly the socio-economic aspects are considered. Finally, with the designed environmental constraints, 100,000 cells should be allocated from conservation land use to forestry. With this allocation, attention has been paid to both socio-economic and Fig. 6 Elements and form of the games environmental aspects of forestry. Forestry use will reduce runoff and erosion, and it provides shelter and habitat for wildlife. On the other hand, 200,000 protection cells should be allocated for rangeland and forestry use. Therefore, the area of environmental protection activities will be reduced. In stage B of the third game, each player will choose their own designed restrictions. According to the rules of the game, the environmental player will get 8 points and the economic player will get 3 points assuming that the environmental player will not change his game, if the economic player wants to change his game and choose the designed environmental constraints (moving from stage B to stage A of the game). In this case, his score will increase from 3 to 5 points. Therefore, according to the definition of Nash Equilibrium, stage B of the game cannot maintain the equilibrium state.
In stage A of the game, each player will choose the designed environmental constraints. Based on game scenarios, the environmental player gets 10 points and the economic player gets 5 points. Assuming that the environmental player is not willing to change his game, if the economic player wants to change his game (moving from stage A to stage B) and his score will be reduced from 5 to 3; so refuses to change his game. Also, assuming the economic player is fixed, if the environmental player wants to change his game and chooses the economic constraint (moving from stage A to stage C of the game), then his score will be reduced from 10 to 3 and the environmental player will not be willing to change his game. As mentioned, because none of the players are willing to change their game, stage A of the game will be considered as the equilibrium state and steps C and D, as defined by Nash, cannot reach the Nash Equilibrium.
The preferences of the environmental player on each of the outputs in the third phase of the game are as follows: Step A > Step B > Step C > Step D.
The preferences of the economic player on each of the outputs in the game with the third repetition are as follows: Step A > Step C > Step B > Step D.
Given the players' preferences for each output, it is clear that the strategy of designed environmental constraints is strongly dominant as an action; because it takes into account both environmental and socioeconomic aspects (through environmental economics) and the results are practical. In contrast, the strategy of choosing the constraint designed by the economic player is selected as a strongly defeated action since the other side is highly dissatisfied due to its low score in the game scenarios.
We applied a game with environmental and economic constraints for land use optimization. For this, multi-objective linear programming with environmental constraints was adopted and the result was fed into the MOLA process. Comparing the optimal output areas of the multi-objective linear programming model and game theory with the initial land use areas showed the need for changing the current area and configuration of land use categories in the region. The results of the integrated model suggested a reduction of 6,300 hectares from agricultural land use, to achieve the environmental objectives such as possible reduction of runoff depth and soil erosion. Around 3600 hectares were added to forest land use to address the needs of the environmental and socioeconomic sectors. The area of warm and cold water aquaculture as well as urban and rural land uses were increased during the optimization process to meet socio-economic goals. Eventually, 3,600 hectares were reduced from protected areas and allocated to forestry. Forestry plays an important role in safeguarding the environmental objectives (reducing the potential runoff depth and the erosion likelihood), and on the other hand, it helps achieving the socio-economic goals. According to the studied socio-economic parameters, the desirability of forestry is far greater than the protected areas, which could provide an interesting topic for further studies in the region. Future studies can evaluate the potential of ecologically-friendly activities such as forestry and recreation in improving socio-economic status of the region, while protecting their valuable ecological functions and ecosystem services. Future models can also quantify the role of tourism and its interaction with other land use categories in the region to establish a spatial decision support system for informed and multi-objective land use planning. By referring to Table 8, the sensitivity in variations of the runoff depth can be understood by changing the inputs of the L-THIA model using different inputs such as different land use maps. It is clear that with different land allocation maps, runoff outputs were expected to be commensurate with them. Also, by comparing the runoff depth of each land use, it can be found that the final land allocation map in this study will provide a more acceptable runoff depth.

Conclusions
Spatial optimization of land use is a complex decision-making problem with multiple antagonistic objectives from different parties and proper coordination between among stakeholders is a major key to solve land use conflicts and successful spatial optimization. The limited quantities of land resources and their suitability for different utilities are the main reasons for such conflicts (Chen, 2007;Maleki et al., 2020). It is very useful to provide a framework that can apply the opinions of different stakeholders in land-related decisions and at the same time resolve conflicts between them so that the results related to land allocation are acceptable to all stakeholders.
This research is one of the first attempts in combining game theory with MOLA and linear programming models and according to the results, the model can lead to conflict resolution between different stakeholders. The integrated model suggested in this study, is less complex and the results are easier to interpret compared to similar studies based on application of game theory, genetic algorithm and fuzzy sets theory (mentioned in the introduction). Therefore, as a topic of further research, the model can be implemented in other research areas with different sets of land use categories and influential stakeholders to evaluate the potential of such integrative studies in conflict resolution and sustainable development plans.
The present study adds to the existing body of research on land allocation, building upon previous studies by Hajehforooshnia et al. (2011), Shabani (2010, and Türk and Zwick (2019). The study offers a comprehensive and practical examination of different approaches and can serve as a valuable resource for organizations and entities interested in this area of research. Additionally, it introduces a novel solution Vol:. (1234567890) for conflict resolution during the land allocation process which can be further explored in future studies. This study also aligns with recent research, such as Liu et al. (2015), Maleki et al. (2020), Shafi et al. (2022), and Sadooghi et al. (2022) that examines application of game theory in land decisions. It contributes to the academic understanding of different spatial analysis approaches for the optimal allocation for economic and environmental analysis.
Our proposed model had limitations. One of them was that the players were considered to be two groups of developers (economists) and environmentalists, which were virtual. Therefore, some issues including policies and social and economic issues were implicitly considered. If these items were considered directly, the results would be more accurate. In order to advance future research, we suggest that more issues, including various land policies, various socioeconomic issues, and other current environmental issues and problems be added to the modeling process.
Data availability The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of interest
The authors declare no competing interests.