A Systematic Review of Agent-Based Modelling in Agricultural Water Trading

Abstract

Agricultural water trading is typically considered an effective water management mechanism, and decisions made by agricultural agents highly influence its effectiveness. Agent-based modelling (ABM) simulating agricultural agents in the water trading context has drawn attention due to its distinguishable features driven by interactions, heterogeneity, independence, and the evolving characteristics of the decisions of agents. Given its strengths and potential to simulate a complex water trading system, the objectives of this study are to (a) provide a comprehensive review of the status of ABM applications in agricultural water trading through a systematic review and (b) identify the primary trends of the empirical nature of ABM studies, approaches to modelling agricultural agent decisions, uncertainty assessments, and validation approaches in ABM studies. The results show that there is a relationship between the empirical nature of the ABM studies, selected decision models to describe agricultural agents, analysed uncertainties, and the validation approaches employed in ABM studies. This study also provides a future research agenda, including exploring attributes with a direct influence on agent trading decisions and integrating the effects of uncertain trading decisions, long-term water availability changes, and water quality into ABM outcomes.

1. Introduction

Water trading is considered an effective water demand management mechanism, especially during drought events, due to its ability to increase the profit of water users, protect the environment, increase agricultural production [1,2] and cope with climate change and the increasing water demand of the growing population [3]. Globally, surface water markets are more common and well-developed than groundwater markets [4]. This situation may be attributed to the lack of monitoring for groundwater [4,5], the provision of groundwater at free or low prices [5], and the complex hydrological difficulties faced in groundwater management [6].

Despite its wide adoption [7] and research on trading structures, such as double-sided structures with multiple sellers and multiple buyers [1,2,8,9] and single-sided structures with a single buyer and multiple sellers [10] or vice versa [8,11], water trading systems may not function as expected in the real world [1,3,12]. This difference is due to the behaviour of actual water traders [1] and uncertainties associated with climate change effects and water allocation, among others [1,3]. Thus, an appropriate approach to addressing the suitability of water trading in a study area [13] and identifying areas for improvement in this system in terms of target groups of water users is critical [14,15].

Top-down and bottom-up approaches are typically applied to evaluate water management mechanisms [16], including water trading studies [1,17,18]. However, the top-down approach can be considered to result in overestimated model outcomes [1,12,19], possibly due to a lack of the integration of complex system features [1,12,19,20]. In contrast, as shown in Table 1, as a bottom-up approach, agent-based modelling (ABM) is considered a powerful approach in terms of its ability to integrate the following four interrelated features of complex systems [20]:

• Heterogeneity represents the diversity of agents (e.g., the diverse attributes of each agent [21]).
• Independency allows for each agent’s independent decisions and goals [21] through micro-level dynamic decisions [20].
• Dynamic feedback enables modelling two-way interactions [22] or feedback loops [21].
• Evolution allows modelling changes in system behaviour over time [21] as a result of the adaptation, learning, and interactions of the agent [20,21]. These changes can be modelled through evolutionary algorithms [12,23,24], neural networks, reinforcement learning [24] and other machine learning techniques [24,25].

ABM is widely applied in many water management areas, such as agricultural water management [7,12], urban water management [26], and flood management [27]. Since water trading is affected by the individual decisions of traders [1,9,12], ABM has recently been applied to agricultural water trading, where heterogeneous attributes of each agent and its independent micro-level decisions and dynamic interactions are integrated [9,20,28].

Table 1. Features of complex modelling systems.

Approach Type	Approach	Features of Complex Modelling Systems	Yes (✓)/No (X)	Reference
Bottom-up	ABM	H: High importance of agent diversity	✓	[21]
		I: Presence of micro-level dynamics	✓	[21]
		DF: High importance of agent interactions	✓	[22]
		E: Absence of fixed decisions	✓	[21,23]
Top-down	Hydro-economic model with single-objective optimisation	H: Aggregated agents are preferable	X	[29]
		I: Single objective for all agents set by a central mechanism	X	[21,30]
		DF: Absence of feedback between the water source and agent	X	[31]
		E: No evolution over time due to the absence of feedback between the water source and agent	X	[31]
Top-down	System dynamics	H: Aggregated features are preferable	X	[20,21]
		I: Absence of micro-level representation	X	[21]
		DF: Feedback loops	✓	[20,21]
		E: Limited ability to evolve over time	X	[20,21]

Notes: H: heterogeneity, I: independency, DF: dynamic feedback, E: evolution, ABM: agent-based modelling.

Table 1. Features of complex modelling systems.

Approach Type	Approach	Features of Complex Modelling Systems	Yes (✓)/No (X)	Reference
Bottom-up	ABM	H: High importance of agent diversity	✓	[21]
		I: Presence of micro-level dynamics	✓	[21]
		DF: High importance of agent interactions	✓	[22]
		E: Absence of fixed decisions	✓	[21,23]
Top-down	Hydro-economic model with single-objective optimisation	H: Aggregated agents are preferable	X	[29]
		I: Single objective for all agents set by a central mechanism	X	[21,30]
		DF: Absence of feedback between the water source and agent	X	[31]
		E: No evolution over time due to the absence of feedback between the water source and agent	X	[31]
Top-down	System dynamics	H: Aggregated features are preferable	X	[20,21]
		I: Absence of micro-level representation	X	[21]
		DF: Feedback loops	✓	[20,21]
		E: Limited ability to evolve over time	X	[20,21]

Notes: H: heterogeneity, I: independency, DF: dynamic feedback, E: evolution, ABM: agent-based modelling.

1.1. The Major Features of Agent-Based Modelling in the Context of Irrigation Water Trading

Figure 1 illustrates the features of ABM, comprising four primary interconnected parts: heterogeneity, independence, two-way feedback driven by interactions [20], and outcomes [1]. Heterogeneity can be reflected in ABM by assigning different attributes to each agent [1,12] or applying different decision-modelling approaches to each agent type to represent varied decision-making processes [32]. Heterogeneity also contributes to the formation of independent decisions and interactions [1,32]. Various interaction mechanisms, such as interactions between water and the agent [16,22] and trading [11,12], can be employed in ABM (Figure 1). In the context of water trading, studies simulated via ABM can be related to centralised [1,33,34] or decentralised water markets [12,35]. Centralised water markets can rely on auctions, water banks, or buybacks, whereas decentralised water markets are relatively more flexible and based on bilateral agreements or negotiations between agents (i.e., no need for a central authority) [36]. In cases where the interactions of the agents with each other are highlighted, such as auctions or negotiations between agents, the terms ‘multi-agent’ or ‘multi agent’ can also be used to refer to ABM [11,22].

Two-way feedback, which can be modelled through adaptation or learning algorithms [1,10,14,37,38], occurs between micro- and macro-level outcomes and other features of ABM (Figure 1). Micro-level outcomes, which result from each agent’s independent interactions [1,20,39], refer to agent-level outcomes, such as the profit change of each agent [9] and groundwater level change at each agent’s location [7,40]. Macro-level outcomes, which measure or affect the performance of the whole ABM system [1,40], refer to system-level outcomes [20,40] or aggregated micro-level outcomes [1], such as total crop production amounts [1], emerged social norms [40], and average water transaction prices [1]. These outcomes can influence the other remaining features of ABM [9,28], such as a less profit-oriented agent that is in line with changed attributes via adaptation [28] and the exploration of a new and more profitable bidding strategy via learning and changed interactions [10]. Adaptation and learning are employed to explore dynamic aspects of ABM [22]. Despite the lack of a consensus distinguishing the learning and adaptation of agents [22,37], adaptation is considered to adhere to the same decision or rule, whereas learning explores a new decision or rule over time [37].

1.2. Study Purpose

In the context of agricultural water management, a recent review [16] examined the application of ABM on a very broad scale ranging from adaptation to drought by irrigation water users and land-use change to irrigation water-saving technology diffusion and cropping strategies. Similarly, a review by Kremmydas et al. [41] focuses on ABM applications in a wide range of agricultural policy analyses related to land use, biodiversity conservation, diffusion of innovation, and livestock and biogas production. In Kremmydas et al. [41], only data-driven ABM applications were analysed, with real-world data or observations being used. Another review [22] examined only ABM-based groundwater management systems, including urban water supply and agricultural water use. These reviews [16,22,41] analysed model transparency [22,41], agents’ rationality [16,22,41], the initialization of an agent population [16,41], and the integration of ABM with hydrological modelling [16,22].

Despite the wide application of ABM, as outlined in review papers [16,22,41], a comprehensive review of ABM applications to agricultural water trading focusing on their climate change and water quality aspects, water trading setting, uncertainty assessment, and validation is lacking. This paper is the first to review ABM studies applied to irrigation water trading fully, excluding subsistence irrigation and including surface water or groundwater as an irrigation water source. First, this paper provides a comprehensive review of data-driven and non-data-driven ABM applications based on the studied regions, time-series of publications, market structure, model documentation, climate change considerations, water quantity and quality aspects, ex post/ex ante characteristics of the study, trading setting, approaches to agent decision-making, and problems associated with uncertainty and validation. Second, this paper identifies research gaps based on this examination and develops a validation framework applicable to ABM studies.

2. Methods

A systematic quantitative literature review was conducted to examine ABM-based agricultural water trading studies. This approach involves a systematic search and categorisation of the relevant literature to assess the status of the research field [42]. A series of pilot analyses were conducted to set the keyword combination for this review.

The reviewed papers were found in the Scopus database dated up to June 2024 using the combination of ALL (“agent-based” OR “agent based” OR “multi-agent” OR “multi agent”) AND TITLE-ABS-KEY (“water market” OR “water buyback” OR “water trade*” OR “cap and trade” OR “water bank” OR “irrigation management”) AND (LIMIT-TO (DOCTYPE, “ar”)) AND (LIMIT-TO (LANGUAGE, “English”)). As demonstrated in the keyword combination applied for this search, the search was limited to studies published as articles in English. Review papers, conference papers, and book chapters were excluded, because the aim was to examine the primary literature applying the ABM approach. As illustrated in Figure 2, 178 papers were initially identified.

At the initial screening stage, the 178 papers were screened based on the title, keywords, and abstract to remove out-of-scope results. For this stage, the criterion was set as follows:

• The research must have been conducted in the water management area (excluding studies focused on carbon or greenhouse gas emission trading).

For the second screening stage, the remaining 108 papers received a full-text assessment. Of these papers, 87 were excluded using the following criteria, resulting in 21 relevant papers identified and considered for review:

• The research must have applied the ABM approach to a case study (excluding studies that applied different approaches, such as hydro-economic modelling, or that focus on developing the ABM algorithm or its implementation as an example to demonstrate its performance).
• The research must have simulated water trading (excluding studies that solely focus on irrigation management or water supply projects).
• The research must have covered irrigation water trading among farmers/irrigation districts or irrigation water trading between the agricultural sector/farmers and other organisations/water user groups (excluding studies related to water quality trading and urban-focused water trading between households, such as rainwater trading). Thus, throughout this paper, the term agent must refer to the farmer, irrigation district, or agricultural sector.
• The research must have included water trading that can be formal or informal (excluding studies on option contracts, where water transactions occur in the future, e.g., [43]).

The final sample of publications was categorised based on the following factors:

• Study characteristics: this category includes information about the location of the studies, the factors driving water trading, water market types, and the temporal distribution of the studies.
• Model documentation: this category covers information about use of the Overview, Design concepts and Details + Decision (ODD+D [37]) protocol.
• Climate change dimension: this category involves information about the integration of the climate change dimension into ABM studies.
• Interactions between the water source and agent: this category covers information about water quantity and quality aspects, such as the type of hydrological models used and the water quality parameters considered.
• Interactions between the agents: this category includes information about water trading settings, including the studied pricing mechanisms and the water market performance indicators.
• Individual decision-making of agents: This category involves information about the approaches employed to model the rules, decisions, or strategies of the agents. This category also includes information about the classification of the studies as empirical or non-empirical, the agent heterogeneity, the agent attributes, the ex ante or ex post analysis of water trading and the additional behaviours included in the reviewed studies.
• Uncertainty and validation aspects of the studies: this category comprises information about the types of uncertainty focused on and the study categorisation based on the developed validation framework.

3. Results and Discussion

3.1. Geographic and Temporal Distribution

Figure 3 illustrates the geographic distribution of the ABM-based agricultural water trading studies. The largest portion of the studies (seven) were performed in Iran; however, a significant number of studies were conducted in China (five), the United States (three), and Australia (three). In contrast, only one relevant ABM study was conducted in Spain, New Zealand, and Taiwan. As demonstrated in

Table 2. List of interrelated critical features of the reviewed studies by country.

Countries	Critical Features	Reference
Iran	Arid or semi-arid climate	[2,9,35,40,44]
	Growing water demand of the agriculture	[2,7,9]
	Over-extraction of water	[40,45]
China	Arid or semi-arid climate	[33,46]
	Growing water demand of the sectors, including agriculture	[33,34]
	Ineffective irrigation technology use	[13,34]
	Low irrigation water price	[13]
	Over-extraction of water	[13,17]
United States	Drought or severe drought	[1,28]
	Semi-arid climate	[38]
	Growing water demand of the agriculture	[28,38]
	Over-extraction of water	[38]
Australia	Drying climate and dry season	[10,47,48]
	Growing water demand of the agriculture	[10,47]
	Over-extraction of water	[10,47]
New Zealand	Drought	[11]
	Growing water demand of the agriculture	[11]
	Over-extraction of water	[11]
Spain	Growing impacts of climate change	[49]
	Over-extraction of water	[49]
Taiwan	Growing water demand of the sectors, including agriculture	[3]
	Over-extraction of water	[3]

, water scarcity, driven primarily by climate impacts, growing agricultural demand, and the overuse of water sources, is the most critical problem in the reviewed studies that the developed ABM approaches attempted to address.

Four main types of water market structures can be employed to deal with water shortages and protect water resources. One is a buyback programme, where the water authority buys irrigation water rights to protect the water balance through, for example, annual lease arrangements [10]. The second is setting up an agricultural water market, where irrigation water is traded between farmers or irrigation districts [1,7]. The third is setting up an intersectoral water market, where the irrigation water is traded between farms and industries [9]. The fourth is a water bank, where the authority buys water rights from the sellers and sells them back to the buyers [34,36].

In this review, water banks were embedded in two intersectoral water market studies (i.e., [33,34]). However, this review does not explicitly use the term water bank to make the market structure categorisation simple. Figure 4 demonstrates that most of the ABM studies focus on the agricultural water market (14), followed by the intersectoral water market (five) and the buyback programme (four).

Compared to agricultural water market studies, the reason ABM-based intersectoral water trading has been relatively less studied could be attributed to the order of priorities for various water-consuming sectors. Based on the importance of the sectors, water allocation can be prioritised in the following order: domestic, industry, and agricultural use [17]. Water trading between medium-priority groups (i.e., medium-level water supply reliability), such as agriculture, and high-priority groups (i.e., high-level water supply reliability), such as industry, can make the water amount relatively small, which may not meet the high-priority water group requirements, making trading impractical between these two groups [50]. In contrast, none of the ABM studies focusing on the intersectoral water market mention trading water between various priority groups.

Regarding the buyback programme, only a few ABM studies have been completed, according to this review. In addition to not being specifically related to ABM, water buyback programmes can generally be considered relatively difficult to implement due to their dependence on the budget of the water authority [51]. This dependence may cause this topic to receive less attention from scholars. In terms of the geographic distribution of the water buyback studies, most were from Australia and Iran. Studies from Iran compared the effectiveness of buyback programmes with various agricultural water markets (e.g., [2]), whereas studies in Australia focus on improving buyback programmes (e.g., [47]).

Overall, ABM has become more popular since the 1990s [52]. Hence, the application of this approach to the field of agricultural water trading is relatively recent (Figure 5). However, this approach continues to be popular in this research area, especially after 2018.

3.2. Documentation of Agent-Based Modelling Studies

The descriptions of ABM are typically incomplete and not transparent [37]. Thus, scholars are often encouraged to implement the Overview, Design concepts and Details + Decision (ODD+D) protocol to meet the requirements for the transparent documentation of model design and outcomes [37,53]. The ODD+D protocol enables modellers to document ABM using a more structured and systematic method [22,41]. This approach ensures that each part of the model is explained step by step by following this protocol [22,41], easing the comparison of ABM approaches [37]. Given its importance in terms of model documentation, this study examined the use of the ODD+D protocol applied in the reviewed papers.

The ODD protocol was first introduced in 2006 to document ecological ABM [54] and was updated in 2013 to the ODD+D protocol to include the decision-making process of agents (i.e., human decision-making processes) [37]. Therefore, papers published before 2013 and during 2013 were excluded from this examination. Compared to the first protocol, the ODD+D protocol asks for more information about the decision-making rules and algorithms of agents, their adaptability and learning, the uncertainty in their behavioural rules, the interaction rules between agents, and their level of heterogeneity [37].

Out of the 18 studies reviewed after the release of ODD+D in 2013, only three (i.e., [28,40,49]) included an ODD+D protocol for model documentation. Regarding model documentation, this review reinforces the work by [22], supporting the idea that using a common language for model documentation offers a systematic structure.

3.3. Climate Change Considerations in Agent-Based Modelling Studies

Climate change can exacerbate water shortage problems [55], such as the prolonged droughts experienced in Australia [3] and expected in Middle East countries [56]. Therefore, the climate change dimension of ABM was investigated by differentiating between long-term and short-term decision-making. Short-term decision-making refers to the in-season process under short-term weather conditions that can be wet or dry [15]. In contrast, long-term decision-making describes strategic decision-making under long-lasting dry or wet conditions [15,35].

Table 3. Categories of considered climate change dimensions and interrelated agent decision-making.

Climate Change Dimension	Number of ABM Studies	Reference
Short-term decision-making	8	[1,2,3,10,17,28,40,46]
Long-term decision-making	3	[35,38,48]
Extreme weather conditions not assessed explicitly	10	[7,9,11,13,33,34,44,45,47,49]

reveals that most of the ABM studies (10) did not explicitly assess the impacts of extreme weather conditions on trading outcomes. In addition, most of the ABM studies (eight) focus on water market performance on a short-term basis, whereas a few studies (three) incorporated the influence of long-term weather changes on water market performance. Although the effectiveness of any water management intervention is more likely to be determined under long-lasting climate conditions rather than the short-term decision-making of agents [15,35,57], studies have rarely considered long-term weather changes in ABM-based agricultural water trading simulations.

3.4. Water Quantity and Quality in Agent-Based Modelling Studies

The interaction between agents and the water source has commonly been studied. Given its importance, this study examined whether two-way feedback between the water source (i.e., the attributes of water sources: water quantity and quality parameters) and agents (i.e., agent attributes and decisions) was included in ABM outcomes to identify unexplored areas. Then, the type of hydrological models representing this feedback was identified. Finally, this study investigated the signs of the externality of water trading relative to business as usual (i.e., no trading).

Hydrological components can be integrated into ABM in three ways:

• A simple water mass balance equation without modelling subsurface runoff or surface runoff [1,58];
• A simple water flow route with ecosystem agents or reservoir agents with flow targets [10,17,59];
• Semi-distributed or distributed hydrological models [16,38,58,59,60].

Of these approaches, hydrological models are more suitable for combining with ABM, exploring interactions between the water source and agents on smaller spatial scales, and integrating hydrological or hydrogeological heterogeneity among agents [16,38,58,60,61,62]. These models investigate the agents’ influence on the water flow and their response to it (i.e., two-way feedback [58,60,62,63]). These models can also explore the negative or positive hydrological externalities of water trading settings [9,38].

The focus of this section is to examine ABM combined with a hydrological model, where two-way feedback is included; hence, this examination excluded studies (i.e., [45]) that integrated ABM with a hydrological model unidirectionally. Similarly, this paper also excluded studies where the information on the hydrological model was too limited to infer whether ABM was combined with a hydrological model (i.e., [48]). Regarding the reviewed studies,

Table 4. List of hydrological models combined with agent-based modelling in a bidirectional manner.

Hydrological Model Type	Traded Water Source	Model Calibrated in Another Study	Two-Way Feedback Parameter	Sign of the Highlighted Externality	Conjunctive Water Resource Use	Reference
MODFLOW (distributed groundwater model)	Groundwater	Yes	Groundwater level	Negative (nonuniform water level in the irrigated area)	Yes	[7]
	Groundwater	Yes	Groundwater level/pumping cost	Negative and positive (nonuniform water level in the irrigated area and improved streamflow)	Yes	[38]
	Surface water and groundwater	No information on calibration	Groundwater level/pumping cost	Negative (decreased streamflow)	Yes	[46]
FlowLogo (distributed groundwater model)	Groundwater	No	Groundwater level/pumping cost	Negative (increased water drawdown)	No	[9]
	Groundwater	No	Groundwater level/pumping cost	Positive (improved aquifer condition)	No	[2]
	Groundwater	Yes	Groundwater level/pumping cost	Positive (improved aquifer condition)	No	[40]
	Groundwater	Yes	Groundwater level/pumping cost	Positive (improved aquifer condition)	No	[44]

reveals that the integration of ABM with a hydrological model was more popular in groundwater market systems. Most of the studies (Table 4) focus on the groundwater market, except for the study by [46], where surface water and groundwater were traded simultaneously. In contrast, this study found that surface water market systems (e.g., [1,11,17,28]) preferred not to combine ABM with any distributed or semi-distributed hydrological models. Regardless of the type of water market systems that can be surface water-based, groundwater-based, or both, the reason why water market systems do not integrate their ABM with any distributed or semi-distributed hydrological models can be attributed to the high data requirements of these models [62]. A substantial amount of data are needed for representing the spatial variability of the study area, calibrating and validating the hydrological models [62]. As demonstrated in Table 4, most of the ABM studies used hydrological models that were calibrated in another study, easing their data requirements.

Especially for cases where conjunctive groundwater and surface water use occurs (e.g., [46]), using integrated surface and groundwater models is more suitable than using only groundwater models [63,64] or surface water models [64]. Integrated surface water and groundwater models, such as the Ground-water and Surface-water FLOW model (GSFLOW [65]) and integrated SWAT-MODFLOW [64], can also be combined with ABM for exploring interconnected externalities at the agent level (e.g., decreased surface water availability for downstream agents and lower pumping costs for agents close to surface water) [63,64]. However, in terms of conjunctive water use, where both surface water and groundwater can be used for irrigation [38,63,64], Table 4 demonstrates that ABM studies, including conjunctive water use, did not prefer to combine ABM with integrated surface water and groundwater models. Not specific to just conjunctive water resource use, integrated surface water and groundwater models are considered more appropriate for fully examining surface water–groundwater interactions and managing water effectively [63,64,66]. However, regardless of conjunctive water use, the studies listed in Table 4 primarily relied on groundwater models.

In Table 4, the groundwater level is the most popular heterogenous hydrogeological attribute for agents and the water source. This popularity is because the groundwater level affects the agent attributes, such as the pumping cost (e.g., [38,46]), and affects agent decisions, such as changing the water-overuse decision based on the previous groundwater level (e.g., [7]). Similarly, the studies focused on water quantity-based positive and negative externalities and ignored the importance of water quality parameters, such as salinity, on water trading outcomes (Table 4). Only one of the reviewed studies (i.e., [45]) explored the influence of water quality on water trading outcomes. Water quality parameters were added as agent attributes by Sharghi and Kerachian [45] (i.e., a farmer group with high or low salinity in the water and the bid price based on the water quality). In contrast, in [45], salinity values were not directly associated with changes in groundwater levels, which could have provided water-quality-related two-way feedback for ABM.

The water quality can change based on water-level variations due to interactions between groundwater pumping, variable seasonal irrigation, and precipitation patterns [67]. Water quality considerations have been included in ABM in fields other than irrigation water trading. For example, in groundwater-based energy systems, bidirectional feedback between (1) a cold/warm well operator’s ABM decision model and (2) the water quality and quantity in aquifers (e.g., groundwater level, temperature, and potentially salinity values) was determined in a study by [61] by combining MODFLOW with the MT3DMS model (a modular three-dimensional multispecies transport model [68]). Given that irrigation water salinity thresholds and their corresponding effects on yields have been published for each crop type [69], especially for areas sensitive to salinity, including salinity values in ABM is strongly recommended. The primary problem that can hinder this process is the lack of data on water quality for calibrating related models, such as MT3DMS and MODFLOW [70], and for developing the agent decision rules or attributes (e.g., the heterogenous bid price of each agent) that can influence water quality parameters.

3.5. Water Trading Settings of Agent-Based Modelling Studies

Irrigation water trading is the focus of this study, and the trading setting can influence ABM trading outcomes [2,11,12]; hence, this study examined the primary water trading settings, including the water transaction prices applied to the reviewed studies and water market performance indicators.

ABM can be applied to two primary water trading settings:

• A centralised water market, where an authority regulates water transactions, matching or ranking offers (e.g., auction-based systems, water banks, or buybacks [12,36,49]);
• A decentralised water market, such as bilateral negotiations, where buyers and sellers directly interact with each other to trade water (e.g., the trading network of the agent [12,36,49]).

Apart from this categorisation, water markets can also be divided into two other broad categories: (1) formal or legal water markets and (2) informal or illegal water markets, based on the relationships and negotiations between buyers and sellers [36]. Due to the direct trading interaction between buyers and sellers in decentralised and informal water markets, in this paper, informal water markets have been considered as a subcategory of decentralised water markets. This study examined whether the reviewed ABM studies simulated a centralised or decentralised water market and addressed additional interaction mechanisms employed to complement decentralised water markets. Further, this paper explores the relative influence of both trading settings on water market results where possible.

Among the reviewed studies, most of the ABM studies (19) focus on various types of central water markets, such as multi-bid auction (e.g., [45]), a multiunit or single-unit auction (e.g., [11]), a single-sided auction (e.g., [10,11]), and a double-sided auction (e.g., [1]). In contrast, much fewer studies (two studies, i.e., [35,49]) explored the effects of a decentralised water market on market performance.

Regarding the interaction mechanism employed for the decentralised water market, Matinju et al. [35] developed an empirical social network model to formulate the interaction size of the agent as the number of people that the agent interacts with for trading. Similarly, Sapino et al. [49] explored the effects of randomly assigned interactions between trading agents on ABM results (i.e., random trading partners). Both studies noted that more interactions between farmers improve water market performance, for example, increasing the number of farmers whose water needs are met [35] and increasing the market surplus (i.e., increased total benefits of sellers and buyers, dollars/year [49]).

Sapino et al. [49] further explored the influence of centralised vs. decentralised water markets on their performance and the estimated transaction costs (i.e., reflecting the difference in monetary benefits between these two market types as the transaction costs). Similar to the findings of studies conducted in another field (i.e., water quality trading [12]), Sapino et al. [49] found that market monetary efficiency is lower for decentralised water markets relative to centralised ones due to the higher uncertainty regarding trading partners (i.e., random matches of traders), adding additional transaction costs for agents in the decentralised water market. This finding may also explain the greater popularity of ABM studies based on the centralised water market in this review.

Similarly, the efficiency of water market performance has also been studied in the design of a centralised water market setting. In [11], a single-unit auction (i.e., the winning buyer is obliged to buy the entire volume of water offered by the seller, even if it exceeds the buyer’s water requirements) and a multiunit auction (i.e., the total amount of water is split into multiple units by the seller) were compared. Chiewchan et al. [11] found that from an economic perspective, a single-unit auction can be more profitable than a multiunit auction. In contrast, a multiunit auction can be more beneficial from an environmental perspective [11].

Another critical component of the water trading setting is the pricing mechanisms employed to determine the market clearing or water transaction price. Among the reviewed papers where water transaction price changes were examined, this price was applied to explore the stability of the water market over time (e.g., the effects of the agents’ learning behaviour on the stability of water transaction prices [2,9]) and to examine the relation between water transaction price changes and diverse water availability or supply–demand conditions (e.g., [1,7,10,13,17,38,40,44,46]). In addition, the water transaction price has been used to explore the critical price points that can lead to water-efficient technology use (e.g., [34]).

Three common pricing mechanisms have been employed in the ABM-based irrigation water trading literature:

• The equilibrium price, where the tendency of sellers and buyers change based on exogenous water price changes (i.e., the identification of the point at which the water supply and demand of the agents are equal [2,7,17]);
• The discriminatory price, in which each transaction has a different transaction price [1], such as the water transaction price being averaged between the bid and ask price of each transaction [1,2];
• The uniform price, which can be the most expensive winning ask price for a single-sided auction [10] or the lowest winning bid price for a double-sided auction [2].

In cases where more heterogeneity was introduced into the model [1] or the bidding or trading behaviour of the agents was modelled separately (i.e., [1,2,9,10,11,13,28,35,40,44,47,48]), the discriminatory price was the most common pricing mechanism in the reviewed ABM studies (i.e., [1,2,9,13,28,40,44,47]). Further, heterogenous attributes of agents not supported by empirical data can misrepresent agent behaviour and ABM outcomes [71,72]. In this review, of the studies (12) that separately simulated the bidding behaviour of farmers, few supported the behavioural parameters of trading (e.g., rent-seeking coefficients or learning rates of sellers or buyers) with empirical data collected from the study area (e.g., determination of the site-specific distribution of these values or calibration of the agent bidding behaviour [9,35,48]). Despite the difficulty in collecting empirical data [72], it is highly recommended to do so for agent attributes to increase the validity of ABM trading outcomes.

The primary indicators for evaluating water market performance in the reviewed studies (

Table 5. Main performance indicators.

Water-Trading-Related Performance Indicators Relative to “No Trading” or Other Scenarios	Number of ABM Studies	Reference
Change in area allocated to crop type (e.g., hectares), average crop production (e.g., tons per hectare), or total crop production (e.g., tons)	6	[1,7,13,28,35,40]
Change in total consumed water amount (e.g., cubic meters) or total excess water remained/consumed (e.g., %, cubic meters)	10	[3,7,11,17,33,34,38,40,45,46]
Change in groundwater level (meters), streamflow (cubic meters), or water amount transferred to environment (cubic meters)	9	[2,7,9,13,17,38,40,44,46]
Change in total profit (e.g., dollars or dollars per hectare) or profit of each agent group	18	[1,2,3,7,9,10,11,13,17,33,34,38,40,44,45,46,48,49]
Change in traded amount of water (e.g., cubic meters or cubic meters per crop type)	13	[1,2,9,10,13,17,28,40,44,45,46,47,48]
Change in water transaction price or bid price (e.g., dollars per cubic meter)	11	[1,2,7,9,10,13,17,34,40,44,46]
Change in the number of traders/transactions (dimensionless)	5	[2,13,35,40,49]

) were changes in the irrigated area, groundwater level or streamflow, total consumed water, crop production, profit, amount of traded water, number of traders, and water transaction prices. Additionally, the principal scenarios for running water trading simulations comprised a baseline with no trading (e.g., [9]), different penalties for overuse (e.g., dollars per cubic meter [7]), various degrees of monitoring water overuse (e.g., [40]), diverse water availability conditions (e.g., [40]), and different water trading settings (e.g., [11]) and agent population sizes (e.g., [10]).

3.6. Approaches to Model Agent Behaviour

The ABM approach aims to mimic the decision-making process of the individual agent [14,37]. The validity of ABM simulations depends on the type of selected decision-making model, including the selected attributes (i.e., the ability of the selected model to model human behaviour [71,73]). Thus, studies were classified as empirical or non-empirical ABM approaches based on the chosen methods for the derivation of the behavioural rules. Then, ABM applications were assessed in terms of the approaches to modelling agent behaviour and the attributes influencing the agent behaviour, including the heterogeneity and interactions. Finally, the additional behaviour, simulated to complement the water trading simulation, was evaluated.

According to [25,74], various interrelated methods of describing or deriving agent behaviour rules exist. Based on the classification proposed by [25,74], the following categories are most relevant for examining the approaches to modelling agent decision models:

• Microeconomic models: maximising the farmer’s utility or profit [74] by selecting the maximum utility decision among the decision options [75] or applying optimisation techniques, such as mathematical programming, to determine the optimum decision that maximises utility [76];
• Cognitive models: a mental model of the agent, applying, for example, the theory of planned behaviour [9,15], participatory mapping [57], or fuzzy or Bayesian cognitive mapping [39,77];
• Rules of thumb can be categorised as follows:
  • Rules from the literature [74];
  • Empirically informed rules, where straightforward or self-evident rules are derived from qualitative or quantitative data, such as if/then or yes/no rules [74];
• Participatory ABM: capturing real-time decision-making of the agents by, for example, implementing role-playing games [25,48,74];
• Empirical or heuristic rules: derivation of the decision rule that is not strongly grounded in a theory through complex data compilation and statistical analysis, such as logistic regression, neural networks, decision tree methods, and evolutionary programming (i.e., deriving rules that are not straightforward or self-evident) [74];
• Pure data-driven approach: a selection of the most representative technique governing the decision rule by analysing the empirical data through various machine learning techniques, such as logistic regression and Bayesian belief network (e.g., selecting a technique with a lower root-mean-squared error, among other techniques) [25].

Additionally, the following condition was applied to categorise the ABM studies as empirical or non-empirical. In cases where behavioural rules are structured based on empirical data collected via questionnaires, interviews, lab or field experiments, or stakeholder workshops [20,63] or where the rules are derived from applied statistical models, such as probabilistic graphic models and boosted regression analysis [73], the ABM study was considered empirical [20,63,73,78]. Otherwise, the ABM study was considered non-empirical, in which case the behavioural rules are based entirely on assumptions [78] and lack observed facts in the study area [33,73]. To clarify, the ABM studies were considered non-empirical studies where empirical data were used only to quantify the agent attributes in decisions (e.g., [1,10]) rather than to calibrate the simulated decision against any observed behaviour. In this respect, the calibration of decisions can be conducted quantitatively [79] or qualitatively with the direct involvement of the agents, such as in participatory modelling [20].

Regarding the categorisation of the studies as empirical or non-empirical, as listed in

Table 6. Empirical ABM studies, including the main approaches used for modelling farmer behaviour, additional modelled behaviour, and interaction-related attributes of the farmer. “-“ = Not applicable.

Type of ABM	Main Approaches	Methods Applied	Additional Behaviour Included	The Main Attributes Related to the Interaction of the Farmers with Each Other or Other Types of Agents	Reference
EMPIRICAL ABM studies (6 studies)	Microeconomic model (i.e., optimisation), empirically informed rules of thumb, and empirical or heuristic rules	Linear programming, fuzzy inference system, and non-dominated sorting genetic algorithm	Cropping and overuse strategy	Lobbying power of each agent, neighbours’ crop pattern and overuse and water transaction price	[7]
	Cognitive model and literature-based rules of thumb	Structural equation modelling	-	Bid price and agent’s location as pumping cost	[9]
	Empirical or heuristic rules and literature-based rules of thumb	Regression analysis	Cropping and irrigation strategy	Interaction size of the agent as the number of people the agent interact for trading and bid price	[35]
	Microeconomic model (i.e., optimisation)	Positive multi-attribute utility programming	Cropping strategy	Interaction size of the agent as the number of people the agent interacts with for trading and physical interactions, with agents trading when they reach the trading point at the same time and bid price	[49]
	Participatory agent-based modelling	-	Off-farm income strategy	Social norm regarding water use and bid price	[48]
	Microeconomic model (i.e., optimisation)	Non-linear programming	-	Agent’s location as downstream and upstream and water transaction price	[17]

and

Table 7. Non-empirical ABM, including the main approaches used for modelling farmer behaviour, additional modelled behaviour, and interaction-related attributes of the farmer. “-“ = Not applicable.

Type of ABM	Main Approaches	Methods Applied	Additional Behaviour Included	The Main Attributes Related to the Interaction of the Farmers with Each Other or Other Types of Agents	Reference
NON-EMPIRICAL ABM studies (15 studies)	Literature-based rules of thumb	-	Irrigation strategy	Bid price and agent’s location as pumping cost	[2]
	Literature-based rules of thumb	-	Irrigation and overuse strategy	Neighbours’ crop pattern and social norm regarding water use and agent’s location as pumping cost and bid price	[40,44]
	Microeconomic model (i.e., choosing the crop with maximum utility from other crops)	-	Cropping strategy	Bid price	[11]
	Literature-based rules of thumb	-	Irrigation strategy	Bid price	[1,28]
	Microeconomic model (i.e., optimisation) and literature-based rules of thumb	Genetic algorithm	Cropping strategy and water saving technology adoption	Bid price	[13]
	Microeconomic model (i.e., optimisation) and literature-based rules of thumb	-	-	Agent’s location as the downstream and upstream and bid price	[3]
	Literature-based rules of thumb	-	-	Bid price	[10]
	Microeconomic model (i.e., optimisation)	Active-set optimisation algorithm	Cropping strategy	Agent’s location as pumping cost and water transaction price	[38]
	Microeconomic model (i.e., optimisation) and literature-based rules of thumb	Non-linear programming	Cropping strategy	Bid price	[47]
	Microeconomic model (i.e., optimisation)	Non-linear programming	-	Agent’s location as pumping cost and water transaction price	[46]
	-	Non-linear programming	-	Bid price	[45]
	Microeconomic model (i.e., optimisation)	Genetic algorithm	Water saving strategy	Water transaction price	[33]
	Microeconomic model (i.e., optimisation), literature-based rules of thumb	Genetic algorithm	Water saving strategy	Water transaction price and bid price	[34]

, a few (six) were empirical ABM studies, whereas most (15) were non-empirical. The following reasons could explain why non-empirical ABM studies were found to be more popular among scholars:

• The lack of data required to calibrate the agent behaviour, such as irrigation and trading strategies [1,13,25,28];
• The more common use of coarse data types, such as regional data [10,20];
• The convenience of assumed behaviour in terms of time and budget [25,34,74];
• The ex ante analysis of a water policy, such as water trading [2,25].

Regarding the modelling of farmer behaviour, one study (i.e., [45]) did not fall into the mentioned categories, because [45] it did not employ a specific decision for the farmer, such as a farmer with an optimising or satisficing goal [80]. Instead, in [45], the agent behaviour was described solely as crop growth (i.e., technical relation between the crop and water without costs and prices), which is not considered to be linked to microeconomics or the rules of thumb [81]. In the other reviewed studies, relatively less data-intensive approaches in terms of formulating and validating agent behaviour, such as optimisation based-microeconomic models and literature-based rules of thumb [13,25,78], were the most widely applied approaches in the non-empirical ABM studies (Figure 6), where empirical data were only employed to quantify agent attributes. In contrast, most empirical ABM studies benefited from various decision approaches (Table 6 and Figure 6).

Among the empirical ABM studies, the cropping strategy (i.e., [7,35,49]) and trading behaviour were the most common calibrated behaviour types. Trading is the primary interaction mechanism of the reviewed papers; thus, this study further explored the most common types of calibrated trading behaviours and found that calibrations of rent-seeking behaviour associated with the bid price [35,48], calibrations of the decision on how much water to trade [9], and calibrations of the social network for identifying trading partners [35] received more focus from scholars. Various approaches were applied for the modelling and calibration of trading behaviour: participatory ABM [48], cognitive modelling [9], and empirical or heuristic rules [35]. However, most of the reviewed studies did not prefer to apply empirical or heuristic rules or a pure data-driven approach for modelling trading behaviour specifically as an endogenous variable, except for a study by [35]. This situation can be attributed to the ex ante characteristic of the reviewed studies [25] or the difficulty of data collection [74], especially in the form of micro-level data [20,35]. This finding on a study’s characteristics [25] is also consistent with an empirical finding in this study, which indicates that all of the reviewed studies had an ex ante characteristic, except for that by [35], which was supported by ex post water trading.

The heterogeneity of agents or the diversity of agents can influence system outcomes, such as the observable trend between profit and the number of heterogeneous agents [14]. Hence, integrating the heterogeneity of the agents into ABM outcomes is critical [82]. The inclusion of heterogeneity also enables scholars to identify target groups of agents for relevant improvements [14,15] and address the suitability of water trading in a study area, such as the limited water transactions caused by farmers with low heterogeneity [13].

Heterogeneity can be reflected in ABM by assigning different attributes to each agent [1,12] or applying different decision-modelling approaches to each agent type to represent varied decision-making processes [32]. Regarding the methods of applying heterogeneity to the model outcomes, a small portion of the reviewed studies (nine) demonstrated a lack of heterogeneity, because they aggregated individual agents at various scales, such as the sectoral scale (i.e., [3,17,33,34]), the cluster of pumping wells (i.e., [38]), and the spatial scale (i.e., [7,45,46,49]). Although the other reviewed studies (12) modelled the agents at the farmer level, a loss of heterogeneity still occurred due to assuming the same learning rate for all agents (e.g., [10,47]) or assuming the same types of crops grown for all agents (e.g., [1]). This finding may relate to the type of heterogeneity that the studies aimed to focus on while assuming certain aspects of homogeneity for practical reasons [1], such as the homogenous crop choice of the agents in the work by [1] and the homogenous learning strategy as in the work by [10]. While all reviewed studies benefited from a certain level of heterogeneity related to agent attributes, none of the reviewed studies attempted to apply different approaches to modelling each agent group for heterogeneous decision-making processes.

Given that agent attributes influence their behaviour, hydro-economic–social influence-related attributes were dominant attributes in agent decisions in the reviewed studies. These attributes include farm size (e.g., [47]), crop production (e.g., [11,13]), the marginal value of water (e.g., [10,49]), sectoral water prices (e.g., [34]), water permits (e.g., [28]), pumping costs (e.g., [38]), the sensitivity of the farmer to water deficiency (e.g., [40]), the rent-seeking behaviour of the farmer (e.g., [1]), the boldness of the agent regarding overuse (e.g., [44]), and the agent location with respect to the water flow (e.g., [17]).

Interaction is another distinguishable characteristic of the ABM approach [20]; hence, this paper further explores the attributes representing interactions, especially between the farmers and other types of agents. Regardless of the types of interactions that can be direct or indirect [20], Table 6 and Table 7 present similar findings, suggesting that interactions were also dominated by hydro-economic–social influence-related attributes, such as neighbours’ crop patterns, social norms regarding water use, the bid price of each agent, water transaction prices, and pumping costs. Considering the attributes specific to the agent’s decision on their participation in water trading, water quantity- (e.g., [1,2]) and quality-based attributes [45] were considered influential. In contrast, none of the reviewed studies explored other types of attributes that can directly affect the agent’s decision on their participation in water trading and examined the impacts of interactions of these attributes on ABM outcomes. As demonstrated in the studies conducted by [9,15,39,77], cognitive models could be applied to identify participation-related attributes beyond the water-related ones.

Lastly, the cropping strategy (e.g., crop-type selection and the water amount allocated to each), the irrigation strategy (i.e., the sensitivity of the farmers to a soil water deficit), and overuse and water-saving strategies were the most common additional behaviours that were examined in the ABM-based agricultural water trading studies (Table 6 and Table 7). In contrast, none of the reviewed studies explored the interaction of water trading decisions with the carryover strategies of farmers, although carryover is considered to play a critical role in the farmers’ water use decisions [83].

3.7. Uncertainty in Agent-Based Modelling Studies

Uncertainty should be explicitly addressed in ABM documentation (e.g., the ODD+D protocol) for each model phase, including the initialisation of ABM [37,84]. This approach helps scholars categorise uncertainty types [84] and understand uncertainty sources [82,84]. Various sources of uncertainty in ABM range from data input uncertainties [22], such as the choice of parameter values [85], to conceptual model uncertainties [22,85,86]. Among these uncertainty sources, the quantification of uncertainty in agent behaviour and its incorporation into the modelling approach has garnered little attention from scholars [86,87], despite its importance in terms of the reliability of modelling outcomes [86,87]. Hence, this review assesses the studies in terms of the uncertainty types and assessments focused on by following the uncertainty classification in [84]. In addition, this study explores whether the reviewed studies integrated the uncertainty of agent decision-making into ABM to identify future research needs related to this subject.

According to the study conducted by [84], uncertainty can be classified as knowledge-based or stochastic. Knowledge-based uncertainty is related to uncertainty from incomplete knowledge about the agents and model assumptions, which can be reduced with additional observed data [88]. Quantitative or qualitative calibration approaches can be applied to reduce knowledge-based uncertainty [84]. In contrast, stochastic uncertainty refers to the effects of randomness on outcomes and cannot be reduced [88]. Figure 7 illustrates that the majority of the reviewed studies (11) focus on stochastic uncertainty, whereas fewer studies (six) focus on reducing uncertainty in ABM.

The majority of the stochastic uncertainty-based studies consisted of non-empirical ABM studies (i.e., [1,2,10,11,13,28,38,40,44,47]), while only one empirical ABM study [49] integrated stochastic uncertainty into ABM outcomes. Stochastic uncertainty-based studies incorporated randomness into ABM by modelling random interactions (e.g., [49]) or by assigning random values from an assumed distribution to the attributes of the farmer’s decisions (Table 6 and Table 7) to produce heterogeneous attributes for each agent in each model run (e.g., [1]), to explore the attributes that influence ABM outcomes using sensitivity analyses (e.g., [2]), and to reflect the stochasticity in ABM outcomes (e.g., [44]). In contrast, empirical ABM studies (six) (Table 6) focus on knowledge-based uncertainty by applying quantitative and qualitative calibration approaches to reduce uncertainty in ABM. On the other hand, the remaining five papers (i.e., [3,33,34,45,46]) that were classified as non-empirical ABM approaches were not classified as either a knowledge-based or a stochastic uncertainty-based study, because some of these studies (e.g., [3]) ignored uncertainty in ABM and some of them (e.g., [33,34]) did not provide information about uncertainty assessments or randomness.

Uncertainty assessments can be conducted by quantitative uncertainty methods, such as sensitivity analyses for identifying influential model parameters, uncertainty analyses with Monte Carlo sampling for identifying the model output distribution, and optimisation for estimating model parameters, or by qualitative uncertainty methods [84]. Qualitative uncertainty methods require agents’ involvement in the conceptual model design, such as the cognitive mapping approach [84]. Uncertainty assessments in most of the reviewed ABM studies that were classified as a knowledge-based or stochastic uncertainty-based study were limited to quantitative uncertainty methods. In contrast, quantitative uncertainty methods are recommended to be combined with qualitative uncertainty methods to increase the validity of the model [84].

Regarding the uncertainty in agent decision-making that can be affected by several factors, such as climate change effects, water policy changes, water market prices, and the time of water allocation announcements [83], a few studies (i.e., [1,28,38]) highlighted the importance of uncertainty in terms of the real functioning of water trading [38]. However, none attempted to quantify uncertainty in any agent behaviour and incorporate this estimate into the modelling outcomes. Given its importance, it is recommended to integrate a Bayesian network into ABM, with farmers’ participation in water trading being made endogenous. In the broader context of agricultural policy evaluation, including land-use management, Bayesian networks are considered a powerful approach in terms of incorporating uncertainty in decisions of farmers into ABM outcomes [39,71].

3.8. Validation of Agent-Based Modelling Studies

The validation of ABM is crucial in terms of the reliability of ABM studies [25,89]. However, scholars face barriers in model validation, such as the difficulty of identifying the real attributes behind agent behaviour, the difficulty of collecting relevant empirical data [72], the absence of data for the validation of ABM outcomes (e.g., the ex ante evaluation of water policies [90]), the presence of diverse validation approaches [91] and terminology [22,92], and the lack of guidance on selecting suitable approaches [22]. These challenges make the validation process difficult for ABM studies. As a starting point for establishing the reliability of ABM studies, an agreed understanding of what the validation of ABM implies is necessary [22,93]. Hence, this review sets up a validation framework for ABM studies and evaluates the validation attempts of the reviewed studies based on this developed framework to fill this gap.

A complete validation of ABM comprises two steps:

• The validation of agent decisions, including their interactions (i.e., micro-level validation of ABM [89,90,94]);
• The validation of aggregated ABM outcomes (i.e., macro-level validation of ABM [89,90,94]).

This distinction also helps ease the validation process and categorise suitable validation techniques using a specific context, reducing the confusion about the validation process and validation techniques in ABM studies. This type of validation classification also aligns with the ODD+D protocol [37], requiring scholars to detail the agent decision-making process selected for ABM.

When developing this validation framework, for simplicity and consistency, this study ignores the distinct use of calibration and validation terms, because calibration can be considered a method of validating ABM with a different data set not used for validation [92]. In addition, this study narrowed the scope of expert/stakeholder validation, which compares ABM outcomes with the results of similar studies [95] by excluding these comparisons from expert/stakeholder validation. In this respect, the main ABM validation approaches in the literature related to this review are listed as follows:

• Expert/stakeholder validation includes the calibration of behavioural rules or the validation of ABM outcomes using participatory approaches (e.g., interviews or workshops) [20,22,95], such as the validation of conceptual models, decisions, and aggregated outcomes with the involvement of agents (e.g., [39]).
• Structural validation suggests that the behavioural rules of agents and their interactions are assumed to follow widely used rules or mechanisms employed in the literature rather than empirical data to derive the actual rules (e.g., [1]).
• Empirical output validation includes a quantitative comparison of ABM outcomes with the historical or observed data (e.g., using Pearson’s correlation coefficient [15,27]).
• Empirical input validation includes the quantitative calibration of the decision of the agent [95], such as the calibrated decision model coefficients (e.g., [96]).

Table 8. Validation framework for agent-based modelling studies.

	Structural Validation	Expert/Stakeholder Validation	Empirical Input Validation	Empirical Output Validation
Micro-validation (agent behavioural rules, including interactions of the agents)	✓	✓	✓	-
Macro-validation (aggregated or emerged outcomes)	-	✓	-	✓

Notes: The signs ✓ and - mean that the mentioned validation approach is/is not appropriate, respectively.

presents the validation framework for the ABM studies, which evaluates the reviewed studies to explore which part of ABM the scholars focused on, including the applied validation approaches.

Most of the reviewed papers (16) did not explicitly use the term validation for ABM, and only a few studies (i.e., [1,9,13,40,48]) explicitly mentioned this term. Structural validation was the dominant validation approach in the non-empirical ABM studies. In contrast, expert or stakeholder validation and empirical input validation were widely used for the micro-validation of empirical ABM studies. The reasons for why structural validation was popular in the non-empirical ABM studies may be related to time constraints [39] or the difficulty of collecting relevant empirical data to validate agent decisions [35,72], especially in the form of micro-level data [20]. In contrast, based on the validation framework in Table 8, no reviewed paper applied macro-validation for ABM outcomes. This lack may be attributed to the ex ante analysis of water trading [90] or the analysis of water trading under varied climate scenarios (e.g., [35]).

4. Conclusions and Recommendations for Future Work

A systematic literature review of ABM studies that focus on agricultural water trading was conducted to address the principal trends of the empirical nature of the studies, the approaches to modelling decisions, uncertainty assessments, and validation approaches. In addition, this study reviewed previously unexplored areas in terms of agent attributes and decisions. The study findings are listed below:

• The majority of the studies were conducted in Iran (33%), followed by China (24%), the United States (14%), and Australia (14%).
• Most of the reviewed ABM studies (67%) focus on the agricultural water market, and only a few (17%) used the ODD protocol.
• Most of the reviewed ABM studies (90%) focus on centralised water markets.
• In cases where two-way feedback between water source and the agent were included, most ABM approaches focus on groundwater market or groundwater modelling, and most such publications use existing calibrated hydrological models, easing their data requirements.
• An association exists between the applied decision model approaches and the empirical nature of ABM studies. Relatively less data-intensive approaches in terms of formulating and validating agent behaviour, such as an optimisation based-microeconomic model and literature-based rules of thumb [13,25,78], were the most widely employed among the non-empirical ABM studies.
• In line with the ex ante characteristic of the reviewed studies (i.e., ex ante analysis of water trading), most of the ABM studies (95%) did not apply empirical or heuristic rules or a pure data-driven approach for modelling trading behaviour as an endogenous variable.
• A strong relationship exists between the types of analysed uncertainty and the empirical nature of ABM studies. The empirical ABM studies tended to reduce knowledge-based uncertainty, whereas the non-empirical ABM studies tended to include stochastic uncertainty in the model outcomes. The uncertainty assessments in most of the ABM studies were limited to quantitative uncertainty assessments. In contrast, quantitative uncertainty methods are recommended to be combined with qualitative uncertainty methods to increase the validity of the model [84].
• Similarly, in terms of micro-validation, the non-empirical ABM studies followed structural validation, whereas the empirical ABM studies employed various validation approaches other than structural validation. The macro-validation of ABM outcomes was not conducted for assorted reasons, such as the ex ante characteristics of the ABM studies or the hypothetical scenarios related to water availability conditions.

Based on this systematic literature review, some research gaps for future exploration in the ABM-based agricultural water trading context were identified and are listed as follows:

• When simulating an intersectoral water market, the inclusion of the conversion of medium-priority agricultural water to high-priority water for industry use, where possible and relevant, is crucial.
• Most of the studies limited water availability conditions to short-term water availability changes. However, the effects of long-term water availability changes on agent decisions should also be integrated into ABM outcomes for future research, especially for areas experiencing long-lasting weather changes. Long-term weather changes can be represented as changes in climate data (e.g., precipitation and temperature) in the hydrological models that can be combined with ABM to include the agent’s response to water cycle components, such as evapotranspiration [38,64].
• Little to no research has been conducted on integrating water quality aspects into agent decisions. The water quality affected by changes in the groundwater level [67] should be integrated into ABM trading studies, especially in areas at a risk of high salinity.
• No research has been conducted to fully consider groundwater–surface water interactions and their impacts on ABM trading outcomes. Integrated surface and groundwater models (e.g., GSFLOW [63,66] and the integrated SWAT-MODFLOW [64]) can be combined with ABM [63,64], especially in cases where surface water and groundwater are traded simultaneously or conjunctive water use occurs.
• Heterogeneity was limited to agent attributes in the reviewed studies, and there was no attempt to apply diverse approaches to modelling each agent type for a heterogeneous decision-making process.
• No research has been conducted on exploring attributes with a direct influence on the agent’s decision on participation in water trading. Cognitive models, such as cognitive mapping [39,77] or the theory of planned behaviour [9,15], could be applied to identify these attributes.
• No research explored the interaction of water trading decisions with the carryover strategies of farmers, despite the possible interaction with trading decisions of the agent [83].
• No attempt was made to quantify uncertainty in any agent behaviour or incorporate its estimate into modelling outcomes. Bayesian network approaches could be integrated into ABM for more reliable trading outcomes in future work [71].

Additionally, the highlighted gaps listed above were mapped by using the Theory of Change framework in Figure S1 (Supplementary Materials). The Theory of Change framework enables one to identify the long-term goal of a study in the context of irrigation water trading, followed by the identification of the preconditions for this goal [97].