The framework considers UWMs and WMs within the watershed as individual agents. It includes (1) two agent-based models for urban water and watershed manager agents to realistically represent UWM and WM decision-making processes and (2) two hydrologic models – the Urban Water Balance Simulation Model (UWB-SM) and the Muskingum–Cunge routing model. The UWB-SM describes the dynamics of urban water balance induced by city-scale IGWM decisions, while the Muskingum–Cunge routing model simulates changes in streamflow in river reaches connecting adjacent urban areas. By integrating these components, we build an agent-based model and two multiagent systems to simulate GI-driven socio-hydrologic dynamics at three spatial scales for addressing IGWM issues (see Fig. ). Specifically, the following is done:

An agent-based model combines the UWM agent-based model with the UWB-SM to address city-scale IGWM (see Fig. a).

Detailed introductions of the model components and structure are provided in the following sections.

Figure a illustrates the decision-making process of city-scale IGWM by an UWM agent and the resulting changes in urban water cycles. To simulate the UWM's behavior in city-scale IGWM, an agent-based model is developed by coupling the agent-based model for UWM with the UWB-SM. In this model, annual decisions on GI construction made by the UWM agent are first used as inputs to the UWB-SM. Subsequently, monthly water supply portfolio selections are represented by the UWM agent, while the resulting changes in urban water cycles are simulated in the UWB-SM. The agent-based model for UWM interacts with the UWB-SM through a coupling strategy. This coupling strategy enables us to use a simulation-based optimization approach to estimate and predict the UWM's decision-making in city-scale IGWM and the resulting urban hydrologic dynamics under different socioeconomic and hydroclimatic conditions.

The data exchange between the UWB-SM and the agent-based model for UWM occurs in two phases to ensure feasible solutions (Fig. b in Appendix ). In Phase 1, an annual data exchange generates feasible GI construction decision variables for the UWM agent and initializes the UWB-SM. Initial data, including urban water demand, land and water characteristics, and hydroclimatic data, are input into both models. Annually available construction areas for infiltration-based GI, rainwater, and stormwater harvesting systems are determined by GI construction constraints and used to update the urban land features in the UWB-SM. In Phase 2, a monthly data exchange generates feasible water supply decision variables for the UWM agent. Each month, the UWB-SM runs at least twice, exchanging data with the agent-based model for UWM. The UWB-SM updates the urban hydrologic state based on the storage levels of seven storage units from the previous month and computes available runoff for stormwater harvesting. This information, along with the storage levels of four storage units (river, aquifer, rainwater, and stormwater harvesting systems) from the previous month, is used by the agent-based model for UWM to generate feasible monthly water supply decision variables. These variables are then transferred back to the UWB-SM to simulate urban hydrologic variables for the current month. The updated hydrologic variables are sent back to the agent-based model for UWM to verify the feasibility of the decision variables. If the decision variables fail the check, they are regenerated by the agent-based model for UWM. If they pass, the relevant IGWM cost is calculated, and the updated storage units' data are used to initialize the UWB-SM for the next month. This process continues until the termination criteria are satisfied, generating a feasible IGWM decision variable and associated annual cost.

The UWB-SM and the agent-based model for UWM are tightly coupled at the source code level, with subroutines of the agent-based model embedded into the UWB-SM. The primary data exchanged between the two models include water supply portfolios, GI construction plans, and hydrologic states. Given that some parameters of the agent-based model for UWM are computed by the UWB-SM, a corresponding solution approach is detailed in Appendix .

Figure b illustrates the river connections between urban areas, which significantly affect the impact of GIs on urban and watershed hydrology in terms of water resource allocation. These river connections mean that all UWM agents must share surface water resources, and the decisions made by upstream agents can affect downstream agents . A multiagent system is constituted by considering multiple urban areas and interconnected river networks. In this system, all UWM agents are linked by a river network. Each UWM agent independently makes city-scale IGWM decisions – such as water supply portfolios and GI construction – based on its current hydrologic states and upstream inflow, which is influenced by the outflows from upstream areas and the decisions of associated UWM agents. The interactions among multiple UWM agents can be depicted as a sequence of city-scale IGWM decisions along river networks, as water is transported downstream. This interaction process exhibits the Markov property , where the hydrologic state of a downstream urban area depends only on the hydrologic state of the adjacent upstream area after making its own IGWM decisions and its own decisions. This state transition process continues sequentially along the river.

The multiagent system for inter-city-scale IGWM is formulated by integrating the agent-based model for UWM (Eq. ) with the UWB-SM and the Muskingum–Cunge routing model (Eq. ), leveraging its Markov property. This system is modeled as a special type of multi-stage decision system , where the sequence of decision-making for each UWM agent – city-scale IGWM – follows their spatial locations along river networks, from upstream to downstream. The hydrologic variable, specifically the upstream inflow for each UWM agent, is considered the state variable that describes interactions between agents. This can be expressed as follows: 1agent-based model for UWM i+UWB-SM,∀i(01)Muskingum–Cunge routing modeliint,∀i,t(02)qri1(t)=Qt1,andqrii(0)=Q0i,∀i,t(03), where the third row of Eq. () shows initial conditions for the multiagent system for UWMs, and Qt1 and Q0i are the initial amounts of the upstream inflow for UWM agent 1 in month t and UWM agent i in month 0, respectively. The details of the multiagent system and the corresponding solution approach are presented in Appendix  and .

In the described multiagent system, each UWM agent minimizes its own IGWM costs without directly considering the external effects on other UWMs due to the Markov property of the system. This might lead to inequalities in water resource sharing, where upstream users might consume more surface water than downstream users, potentially increasing the IGWM costs for downstream areas . To address this inequality and promote sustainable water use in watershed-scale IGWM, a WM can implement policies such as a streamflow penalty strategy.

Under such policy interventions, each UWM agent must balance the costs of GI construction, water supply, and wastewater drainage with the potential penalty fees imposed by the WM for not meeting specified low streamflow thresholds. Consequently, the agent-based model for UWM is extended to include these penalty fees in the annual IGWM cost function. Details of this model extension are provided in Appendix .

As illustrated in Fig. c, a streamflow penalty strategy may prompt upstream UWM agents to adjust their IGWM decisions to increase outflow, thereby minimizing penalty fees and benefiting downstream agents. These adjustments can shift the interactions within the multiagent system, impacting water resource distribution within the watershed, which can be quantified using the water allocation Gini coefficient set by the WM. The WM can assess the policy's effects on the watershed by evaluating this metric and iteratively adjusting the policy to find the optimal solution. This process represents an interaction between the WM and UWMs in watershed-scale IGWM under a streamflow penalty strategy. This interaction is not solely determined by the WM or the UWMs; both parties aim to optimize their objectives (equity for the WM and cost minimization for UWMs) under respective constraints (streamflow for the WM and GI construction, water supply, and demand constraints for the UWMs) and the reactions of the other party. This decision-making process follows the Stackelberg game theory , where the WM makes the initial decision, and UWMs respond to optimize their objectives with full knowledge of the WM's decision. The WM then optimizes its objective based on the rational reactions of the UWMs.

Leveraging the Stackelberg game framework , we construct a bi-level multiagent system by combining the agent-based model for WM with the multiagent system comprising multiple extended agent-based models for UWMs and the relevant hydrologic models. This system can be formulated as follows: 2agent-based model for WM;whereWi,GIisolvesextended agent-based model for UWM i+UWB-SM,∀iMuskingum–Cunge routing modeliint,∀i,tqri1(t)=Qt1,andqrii(0)=Q0i.∀i,t, where [-]* represents the parameter being from the simulation calculation of the UWB-SM. Wi and GI_i represent the decision variables of water supply portfolios and GI construction for UWM agent i. The details of the bi-level multiagent system and the pertaining solution approach are illustrated in Appendix  and .

As Fig.  shows, the Upper Mississippi River basin ranges in latitude from 47 to 37° N, and it flows roughly 2092 km, from Lake Itasca (northern Minnesota) to the Ohio River (southern Illinois), which covers seven states of the USA, such as Illinois, Iowa, Minnesota, and Wisconsin, and has a watershed area of 489 508 km². The main river and its tributaries have an average annual discharge of 3576 m³ s⁻¹, which has three high-flow (late April, late June, and October) and two low-flow (midsummer and late winter) periods as a result of varied rain and snow conditions . In the watershed, more than 70 % of the area is used for agriculture and animal husbandry. Only 5 % of the site has been converted to urban areas. However, it has a population of about 24 million, especially in the metropolitan high-density regions (> 100 000 people km⁻²), such as Minneapolis–St. Paul, Minnesota, and La Crosse, Wisconsin. It is estimated that over 5.3×106 m³ of water is withdrawn from the Upper Mississippi River each day in the 60 counties for municipal and public supplies.

There are three primary reasons for selecting the Upper Mississippi River basin as the case study for conducting numerical experiments. First, although the widespread use of GIs for rainwater harvesting is not prevalent in the USA, the study has demonstrated the potential for rainwater harvesting in this watershed, attributed to its humid climate and abundant annual precipitation. However, this study also indicates that seasonal variations in water demand necessitate a deeper exploration of rainwater harvesting potential. Applying the socio-hydrologic framework to the basin can provide valuable insights into this potential. Second, despite being a water-rich basin, water sharing among the riparian urban areas remains a significant concern due to the high demands for environmental and agricultural purposes, strict water level regulations for navigation, and high-density urbanization. Climate change is expected to drastically alter the probability distribution of streamflow, increasing the frequency and magnitude of both high and low streamflow extremes. Consequently, even high-flow river basins also might face nonstationary drought risks . A simulation study showed that climate change could result in streamflow decreases in all seasons except winter within the basin . Third, there are notable similarities between the actual case and the simulated case by the proposed model. For instance, in fact, the Upper Mississippi River Basin Association as WM aims to implement a water level management policy , which mirrors the multi-player management structure assumed in the proposed model. Additionally, surface water and groundwater are crucial sources of water widely used by urban areas, and green infrastructures are gradually encouraged and expanded to manage urban stormwater by local communities and authorities .

In this study, the Minneapolis–La Crosse section – approximately 236 km long – of the Upper Mississippi River basin, a high-density urban area, is considered the study area (see Fig. ). Note that the study only focuses on the urban water use and allocation, which perhaps is a small percentage of the basin water resource in the study region. Figure  indicates that there are nine main riparian urban areas along the section, i.e., i=1, 2, …, 9. Some are metropolises with a large population for these urban areas, such as Minneapolis, St. Paul, and La Crosse. The others are city groups that consist of multiple small cities, such as Red Wing and Bay City. In short, the basic features of urban areas in the study site are shown in Table .

Given the background of the study area, the all urban area – metropolises or city groups – is assumed as a UWM agent that makes city-scale IGWM decisions individually, which can be formulated by Eq. (), and the Upper Mississippi River Basin Association is regarded as the WM agent that regulates these urban areas, and the associated interactions among UWMs and between WM and UWMs are formulated by Eqs. () and (), respectively. In the case study, we address the above issues of IGWM at three spatial scales through observing and comparing the results of the decision making of WM and UWM agents and interactions in IGWM simulated by the proposed model under different socio-hydrologic settings. Therefore, three numerical experiments to IGWM at city, inter-city, and watershed scales are designed in the following:

In the proposed framework to the case study, some parameters of the two agent-based models and two hydrological models need to be determined by collecting, processing, and estimating actual data from diverse sources.

For the parameters of UWB-SM, as detailed in Table  of Appendix , the urban water demand data (36×9) were derived from urban populations and layouts using the method of . The hydroclimatic data inputs (27×9) were estimated using raw data on streamflow, precipitation, and temperature from the USGS Current Water Data for the Nation (https://waterdata.usgs.gov/nwis/rt, last access: 1 May 2021) and the Global Historical Climatology Network daily (GHCNd) databases (https://www.ncei.noaa.gov/). Specifically, upstream inflow data were identified from GIS maps of urban areas and estimated using the map correlation method with monthly streamflow observations from USGS stations because of the location difference between the urban area's inlet and the associated USGS stations. Evaporation and evapotranspiration data were calculated using the method of with temperature data from the GHCNd database. The urban area data (1×9) were sourced from the U.S. Census Bureau (http://www.census.gov/, last access: 12 June 2021), and urban land features (4×9) were derived from remote sensing images , as shown in Table  of Appendix . Urban-depth-related data were obtained using various methods: mean aquifer depths at low topographic points (1×9) were assumed to be 10 m plus riverbed depths from the National Elevation Dataset (NED; https://datagateway.nrcs.usda.gov/, last access: 25 June 2021), while aquifer depths at high points (1×9) were calculated using linear fitting to urban hypsometric curves . These assumptions are believed to have minimal impact due to the region’s abundant water supply. Mean well depths for groundwater withdrawal (1×9) were averaged from nearby wells in the USGS Groundwater Data for the Nation database (https://waterdata.usgs.gov/nwis/gw, last access: 4 July 2021). Maximum ratios for constructed areas of rainwater, stormwater harvesting systems, and infiltration-based green infrastructure (3×9) were set at 50 % to test maximum urban rainfall utilization potential, though actual ratios may be lower. Mean depths for shallow soil layers (1×9) and wastewater pipe networks (1×9) were set at 3 and 2 m, respectively, based on similar settings . Mean effective porosity (1×9) was estimated at 10 % from the report by . Storage capacity data (4×9) were determined based on urban layouts using the method of .

For the parameters of the agent-based model for UWM, as shown in Table  in Appendix , the construction cost data for three types of green infrastructure (GIs) (3×9) were estimated based on the EPA cost databases (https://www.epa.gov/, last access: 23 July 2021) following the method of . The associated cost scaling coefficients (3×9) were derived using a non-linear fitting method . Cost-related parameters for surface water and groundwater supply (4×9) were set according to the recommendations by , while those for stormwater and rainwater harvesting (2×9) followed the recommendations of . Parameters for sewage drainage costs (2×9) were set based on . Due to a lack of specific data for each urban area, these cost parameters were uniformly applied. Water-availability-related parameters (24×9) were determined by setting the minimum storage levels for surface water withdrawals based on the minimum monthly historical streamflow from the USGS datasets using the above same method. Minimum storage levels for groundwater withdrawals were set based on the mean depths of wells. Water-supply-capacity-related parameters (2×9) were set equal to the corresponding maximum monthly indoor water demand. The ratio of soil moisture for plant demand (1×9) was uniformly set at 0.31 based on recommendations. Additionally, for the parameters of agent-based model for WM, as shown in Table  in Appendix , minimum and maximum historical streamflow data (24×9) were obtained from the USGS datasets using the above same method.

In the framework, the other parameters of the UWB-SM and the Muskingum–Cunge routing model were evaluated through model calibration and validation using estimated historical streamflow data via the above same method.

For the calibration parameters of UWB-SM, as detailed in Table  in Appendix , these were obtained by calibrating and validating the UWB-SM against estimated historical monthly outflow data derived from USGS datasets. The calibration process utilized estimated monthly inflow from USGS databases and monthly rainfall data from NOAA databases as inputs for the UWB-SM. During calibration, rainwater and stormwater supplies were excluded, and urban water demands (both indoor and outdoor) were assumed to be met solely through surface water and groundwater supplies, with a set ratio of 1:1. Monthly surface water and groundwater supply amounts were determined based on this setting. The simulated monthly outflow data from the UWB-SM were then compared with the estimated outflow data for calibration and validation. Detailed calibration and validation results are presented in Table .

For the calibration parameters of the Muskingum–Cunge routing model, as shown in Table  in Appendix , these were obtained by calibrating and validating the model using estimated historical monthly outflow data from the upstream urban area and inflow data for the adjacent downstream urban area. Detailed calibration and validation results for the Muskingum–Cunge routing model are also provided in Table .

The classified results of UWMs' decisions of IGWM to different environments are illustrated in Fig. a. As Fig. a shows, there are four IGWM patterns for UWMs in response to the different combinations of upstream inflow and rainfall inputs. The region of light-blue dots is defined as pattern 1, which represents the similar reactions of UWMs in the decision-making of IGWM to the high upstream inflow and rainfall inputs. The region of green dots indicates pattern 2, indicating the consistent behavior of IGWM for UWMs to the low upstream inflow and high precipitation settings. Similarly, patterns 3 and 4 are set in the region of dark-blue and yellow dots, which denotes the analogous decision-making of IGWM for UWMs in the study site under high (low) inflow and low (low) rainfall conditions respectively. The result indicates the homogeneous behavior of city-scale IGWM for all UWMs in the relatively extreme hydroclimatic conditions in the study region.

The characteristics of the four IGWM patterns mentioned above are shown in Fig. b–f. The ratios of water supply portfolios and GI construction in the four patterns are illustrated in Fig. b–d. There are large distinctions in the IGWM between the four patterns; in pattern 1, centralized and decentralized water account for 52.9 % and 47.1 % of total water supply, respectively, which means that stormwater and rainwater are widely utilized. More than 80.0 % of centralized water supply is from surface water. In the aspects of GI construction, over three-fifths of available urban areas is used to develop stormwater and rainwater harvesting systems, which accounts for 96.2 % of total GI construction areas. These results show the responses of IGWM to the high upstream inflow and rainfall inputs – UWMs prefer to use stormwater and rainwater directly to meet urban non-potable water demand by stormwater and rainwater harvesting systems for the sake of cost and to supply surface water to meet potable water demand. In pattern 2, similar to pattern 1, stormwater and rainwater are also heavily used directly to satisfy non-potable demand. At the same time, groundwater is the main potable water resource due to the low upstream inflow input. Accordingly, over 75 % of available areas is utilized for GI development, 35.9 % of which, unlike pattern 1, is for infiltration-based GIs for groundwater recharge. In pattern 3, stormwater and rainwater are hardly used (only 2.9 % of total water supply), and the construction of the associated GIs (only 3.9 % of available urban areas) is also limited due to the scarcity of precipitation. The surface water resource is dominant in urban water supply, accounting for 87.2 % of total water withdrawals because of the high upstream flow inputs. In comparison, stormwater and rainwater are partially used (22.2 % of total water supply) directly or indirectly by constructing GIs at a moderate level (41.2 % of available urban areas) in pattern 4. This might be because UWMs have to collect and use all kinds of available water resources as much as possible when system water inputs – rainfall and upstream inflow – are low. Hence, the groundwater is supplied more extensively for maintaining water supply steadily. It is worth mentioning that infiltration-based GIs, which account for 16 % of total GI construction areas, are, to a great extent, built to enhance aquifer recharge for reducing the costs of groundwater abstraction. In addition, Fig. d also illustrates the mean IGWM costs per unit of water in the four patterns. Undoubtedly, the costs of the pattern under high system water input conditions are lower than those under low water inputs.

Figure e and d illustrate the four indexes used to measure urban water cycle defined by under these patterns, which show that the impacts of the four IGWM patterns on the urban water cycle. As Fig. e shows, the ratios of system water output to input and rainfall harvesting vary from pattern to pattern. Except pattern 4, the ratio of system water output to input of the rest of the patterns is smaller than 1, which means that the relevant IGWM patterns can increase water storage in an urban system, especially the pattern 1 and 2 with relatively large areas of GIs – 17.5 % and 12.6 % of water inputs are stored, respectively, which, to some extent, decreases the outflow of the urban catchment. The result is consistent with the results. However, in contrast to the other patterns, the ratio of system water output to input of the pattern 4 indicates the decreases in urban water storage. In this circumstance, to build a small range of infiltration-based GIs (i.e., the ratio of rainfall harvesting = 17.6 %) to increase groundwater recharge is limited. In addition, as shown in Fig. f, the ratios of water cycle increase in order from pattern 1 to 4, and the associated ratios of change in stored water have an opposite trend. These results also are consistent with the above results in Fig. e.

Our simulation study demonstrates the reasonable responses of UWM agents to different hydroclimatic settings in city-scale IGWM. Specifically, the hydroclimatic environment determines the availability of the four types of water sources, thereby influencing UWMs' selection of IGWM patterns from an economic perspective. Furthermore, the IGWM patterns chosen by UWMs can alter the urban water cycle, which, in turn, affects the hydrologic environment of the watershed.

The results of the inter-city-scale IGWM in the two cases are shown in Table . The ratios of water supply portfolios of the all urban area in the two scenarios are shown in the 2nd–5th and 9th–10th rows of Table . In the case without GIs – a control group – surface water use fractions gradually decrease from the upstream to the downstream areas. For example, the ratios of surface water in urban areas 1, 5, and 9 are 0.85, 0.79, and 0.73, respectively. It might be because of a trend that the available amounts of surface water gradually decrease along with the river (see the blue line in Fig. a). These results indicate the adverse impacts of upstream water users on the downstream water users in surface water withdrawals. That is, water extraction from a stream in the upstream areas reduces the available amounts of surface water in the downstream regions, which would increase the costs of surface water withdrawal for the downstream urban areas, thereby forcing them to substitute other water resources, such as groundwater. In contrast, in the case with GIs – an experiment group – the fractions of surface water use appear to be independent of the study area locations due to stormwater and rainwater use via GIs. However, it might aggravate the adverse impacts, i.e., upstream–downstream imbalance of available surface water. For example, the ratios of surface water use markedly decrease in the downstream urban areas, especially in urban areas 6, 8, and 9, which have adopted IGWM pattern 4 – the high ratio of groundwater use. This may be because the upstream UWMs prefer to use stormwater and rainwater resources through developing GIs, such as IGWM pattern 1 and 2, to minimize the costs of water use . However, as mentioned before, IGWM pattern 1 and 2 can reduce the outflow of the urban subcatchment (see Fig. e), which might worsen the decrease in available surface water in the downstream region (see the red line in Fig. a). In addition, the 8th and 12th rows of Table  also show the Gini coefficients (Eq. ) in the two scenarios – 0.0129 and 0.0169, indicating that the GI construction to use stormwater and rainwater intensifies the imbalance in water use in the study area.

The 7th and 11th rows of Table  illustrate the IGWM cost per unit of water for each urban area in the two scenarios. There is a trend that the unit water costs constantly increase from the upstream to the downstream region, indicating the adverse impacts of the upstream–downstream imbalance of the surface water on the cost of water use for the downstream urban areas. In comparison, the costs of water use in the multiagent systems with GIs are smaller than those without no GIs for the all urban area. In contrast, the differences in these costs between the two scenarios continuously decrease from urban area 1 to 9 (see the red line in Fig. b). The reasons behind these results might be that there are two main factors – upstream inflow and GIs – affecting the costs of water use in the multiagent system for UWMs, especially for the downstream UWM agents. In general, the upstream inflow reduction can decrease the amounts of surface water, thereby increasing the cost of water use for the downstream UWMs. Conversely, rainwater and stormwater use via GIs can increase the available amounts of water resources and decrease water supply costs. In the multiagent system for UWMs, especially for the downstream UWM agents, the cumulative effect of the upstream IGWM decision behavior on streamflow gradually amplifies along the river. In contrast, the impact of the GIs on rainfall resource use generally remains stable due to the limitations of climatic, physical, and socioeconomic conditions. Therefore, the combination of the two effects might lead to a gradual decrease in the impact of GIs on the cost of water use along with the river.

Our findings reveal a persistent upstream–downstream imbalance in water resource access, exemplified by stark contrasts in surface water reliance between urban areas – from 85 % in upstream regions (e.g., urban area 1) to 73 % in downstream zones (e.g., urban area 9) in without GI scenarios. This disparity reflects systemic inequities: downstream areas face heightened costs due to diminished surface water availability, causing reliance on costlier groundwater sources. Crucially, GI adoption intensifies these tensions – while reducing city-scale costs (e.g., via stormwater harvesting in pattern 1), it inadvertently reduces downstream inflows by 12 %–18 % (Fig. a), amplifying inter-urban conflicts akin to a “tragedy of the commons” . These results underscore the dual-edged impact of GIs: though locally cost-effective, watershed-scale implementation risks entrenching inequities by privileging upstream urban areas. To reconcile this tension, policies must integrate negotiated water-sharing agreements that harmonize localized GI benefits with watershed-wide equity, ensuring sustainable and equitable resource allocation across urban boundaries.

The mean unit costs and the Gini coefficients in the multiagent system for UWMs under different precipitation, upstream inflow, and urban water demand inputs are illustrated in Fig. a and b, which show the influences of different social and hydroclimatic settings on the socio-hydrologic dynamics of the watershed in the context of inter-city-scale IGWM. In Fig. a, the blue and grey lines represent the changes in the mean costs per unit of water use in the all urban area under different precipitation and upstream inflow inputs. It shows a non-linear inverse relationship between the cost of water use and the system water inputs – the mean cost of water use in a watershed-scale IGWM decreases with the increases in both watershed upstream inflow and rainfall. Notice that the cost of water use in the multiagent system for UWMs is more sensitive to the precipitation than upstream inflow. Notably, the mean cost of water use decreases from USD 3.01 per cubic meter to USD 2.96 per cubic meter when the rainfall increases from 150 % to 175 %. It appears that rainfall inputs, when they go beyond a certain threshold, might have a profound effect on the IGWM cost of the multiagent system for UWMs. This might be due to the fact that more and more UWMs can switch IGWM patterns from pattern 3 or pattern 4 to pattern 1 or pattern 2, with the available rainfall exceeding a given threshold, leading to a significant decline in water use costs. These results are consistent with the results in Fig. d. In comparison, as the yellow line in Fig. a shows, the cost of water use in the all urban area is highly sensitive to the urban water demands. The increases in urban water demands are equivalent to the decreases in available water resources within an urban area, which greatly adds to the costs of water use.

Figure b shows the changes in the equity level of water use in the all urban area under different hydroclimatic and socioeconomic settings. The blue and grey lines in Fig. b represent the trends of Gini coefficients under mixed precipitation and upstream inflow conditions – it tends to proportionally decrease as the upstream inflow increases, while there is an inverted U-shaped curve for the Gini coefficient when the precipitation increases; it reached a peak (0.0171) when the ratio of the rainfall to the baseline is equal to 125 %. On the side of the upstream inflow, these results show that the reduction in watershed upstream inflow harms the equity levels of water use in watershed-scale IGWM. The possible reason is that the Markov property of the multiagent system for UWMs has a cumulative effect on the reduction in surface water availability along with the river. It can, to some extent, amplify the surface water conflicts between the upstream–downstream urban areas as the watershed upstream inflow decreases. On the side of the precipitation, the impact of rainfall on the equity of water resources distributions among urban areas is more complicated. The reasons for the increase in the Gini coefficient as the precipitation increase slightly might be that the water use costs for the upstream UWMs decrease remarkably by substituting stormwater and rainwater for surface and groundwater water as the rainfall increases. In contrast, in the downstream regions, the reduction in the relevant costs is limited because the cumulative effect of upstream inflow mentioned above limits the intention of the corresponding UWMs to switch IGWM patterns to use more rainfall resources economically. However, as the precipitation increases significantly, the abundant rainfall encourages the downstream UWMs to use stormwater and rainwater via GIs cost-effectively, reducing water use costs sharply, thereby making the watershed-scale IGWM more equitable. In comparison, the Gini coefficient in the all urban area is more sensitive to upstream inflow than rainfall. This is because the cumulative effect of upstream inflow might offset the impact of rainfall due to the Markov property of the multiagent system for UWMs.

Our findings demonstrate that inter-city-scale IGWM is shaped by dynamic interactions between hydrologic conditions and social drivers. Hydrologically, rainfall variability exerts a stronger influence on water use costs due to the cost-effectiveness of stormwater and rainwater resources, while shifts in upstream inflow disproportionately affect equity in water distribution – a consequence of the Markov property, where UWMs optimize decisions based on immediate inputs without accounting for downstream cascading effects (e.g., reduced surface water availability in downstream areas, as shown in Fig. a). Socially, rising urban water demands amplify both cost and equity challenges, as seen in fragmented governance structures that prioritize localized GI benefits (e.g., upstream adoption of pattern 1) over watershed-scale resilience, forcing downstream agents into costly groundwater-dependent strategies (pattern 4). These systemic flaws underscore the urgency of institutional reforms. For instance, interactive data platforms, akin to those in water-rights trading markets , could bridge communication gaps between urban water managers by providing real-time hydrologic and economic metrics. Such tools would enable adaptive adjustments to GI investments and withdrawal limits, aligning localized actions with watershed-wide equity goals – a critical need under climate change, where shifting streamflow distributions threaten to exacerbate existing imbalances.

The results of the policy simulation to the bi-level multiagent system are shown in Table . As Table  shows, there are two Stackelberg equilibrium situations in the bi-level multiagent system, which means the two possible designs of the streamflow penalty strategies for the WM agent can achieve the minimum equity objective in the study region. An illustration of the water supply portfolios and the relevant water use patterns in the all urban area under the two equilibrium situations is in the 2rd–6th and 10th–14th rows of Table . A spatial homogeneity in the UWMs' responses to the water policy can be observed based on the associated ratios of water supply portfolios. For example, whether it is equilibrium 1 or 2, most UWM agents prefer to adopt a similar IGWM pattern, such as pattern 3 in equilibrium 1 and pattern 2 and 4 in equilibrium 2, in contrast to the case without the water policy (see the 6th row of Table ). This might be because the streamflow penalty strategy has a similar effect on altering the IGWM cost of the all urban area in the study watershed, forcing some UWMs to change their water supply portfolio selections to a specific IGWM pattern to increase outflows of urban systems for avoiding high penalty fees. These findings can also be supported by the curves of the fractions of available surface water in each urban area (see in Fig. a). As Fig. a illustrates, the available amounts of surface water withdrawal for each urban area in the two equilibriums (i.e., the blue and green lines) are larger than in the scenario without the water strategy (i.e., the red line).

The costs and equity levels in the study area are shown in the 7th–9th and 15th–17th rows of Table . In the aspects of cost, to compare with the no policy case (see the 7th row of Table ), the mean costs per unit of water use of the study area in the two equilibriums (USD 3.06 per cubic meter and USD 3.22 per cubic meter) are higher than the cost in the no policy case (USD 3.04 per cubic meter). This is because some urban areas pay penalty fees, such as urban areas 1 and 2, and others switch IGWM patterns, such as urban area 3, which increases the water use cost. These results demonstrate the reactions of the UWMs to the water strategy set by the WM in IGWM at a watershed scale; the intervention of the water policy to the multiagent system for UWMs can force UWM agents to make a trade-off between the penalty fee and the IGWM cost in a city-scale IGWM. In the aspects of equity, undoubtedly, the Gini coefficients in the two equilibriums (0.0011) are the same, which are lower than in the no policy case (0.0169; see the 8th rows of Table ). It means that the two policy designs can mitigate the inequity of water sharing in the study region to some extent.

The 6th and 14th rows in Table  show that different effects of two streamflow penalty strategies on the decision making of UWM agents for city-scale IGWM. That is, most UWM agents prefer to select the IGWM pattern 3 – to withdraw more surface water – in equilibrium 1. In contrast, the UWM agents in equilibrium 2 are inclined to choose pattern 4, i.e., to extract more groundwater, which is easy to increase the outflows of urban subcatchment (see Fig. e) to avoid the penalty fees. These results are consistent with the curves of the fraction of available surface water (see the blue and green lines in Fig. a). The available amounts of surface water in the study area in equilibrium 2 are higher than that in equilibrium 1. Economically, the IGWM costs and the penalty fees per unit of water use in equilibrium 2 are greater than those in equilibrium 1. These results can be explained in part by the curves of the relative ratio of low-flow thresholds in the two equilibriums (see Fig. b); the low-flow thresholds in equilibrium 2 (i.e., the green line) are higher than those in equilibrium 1 (i.e., the blue line), which means that the policy in equilibrium 2 is more stringent than that in equilibrium 1. As a result, the UWMs in equilibrium 2 have to further change IGWM patterns – over the withdrawal of groundwater and reduction in rainfall capture – to increase the outflow of the urban subcatchments to avoid the harsh penalty policy. It may cause over-reactions of the UWM agents in IGWM to the watershed water policy, thereby leading to the unnecessary costs of water use as well as the unreasonable water supply portfolios in the watershed. Therefore, the policy design in equilibrium 1 is regarded as a good watershed policy, but equilibrium 2 is not in the study area.

Our findings demonstrate that the efficacy of penalty-driven policies hinges on reconciling the priorities of watershed-scale managers (WM agents) and city-scale managers (UWM agents). The Stackelberg framework captures inherent power asymmetries between these actors – where WM agents enforce equity through penalties, while UWM agents prioritize local cost optimization – a dynamic that risks policy failure. For instance, stringent penalty rates in equilibrium 2 (Fig. ) trigger irrational overreactions, such as excessive groundwater extraction (Table ), destabilizing aquifer sustainability and inflating regional water costs. This mismatch underscores the limitations of siloed governance: WM agents lack granular insight into local trade-offs, while UWM agents neglect downstream equity.

Figure a and b illustrate the mean unit water costs and the Gini coefficients of the study area under different upstream inflow and precipitation conditions, respectively. As Fig. a and b show, both the costs of water use and the Gini coefficients of the study area decrease as the upstream inflow and the precipitation increase. This might be because the increases in the upstream inflow and the precipitation not only enable the UWM agents to have more choices to make IGWM decisions to reduce the costs of water use but also mitigate the water conflicts among urban areas in the study site, encouraging the WM agent to loosen the water penalty policy. In addition, the cost of water use is more sensitive to the rainfall, while the Gini coefficient, in contrast, is easily affected by the alteration in upstream inflow, which ties in with the results in Fig. a and b.

The water use costs and the Gini coefficients of the equilibrium situations in the bi-level multiagent system under different penalty rates settings are illustrated in Fig. c. As Fig. c shows, the costs of water use of the study area increase as the penalty rate increases, as expected. To avoid over-high penalty fees, the UWM agents have to alter the IGWM pattern further to increase the outflows of urban systems, thereby increasing water use costs. However, there is a U-shaped curve for the Gini coefficients when the penalty rate increases; the Gini coefficient reaches a nadir (0.001) when the penalty rate is rp= USD 0.006 per cubic meter. The possible reasons for these results are that the impact of the water strategy is too weak to affect UWMs' decision behavior of IGWM as the penalty rate is too low, whereas an overly high penalty rate might force UWMs to make unreasonable IGWM decisions, both of which fail to mitigate water conflicts between urban areas. Therefore, a suitable penalty rate is crucial in achieving equity in watershed-scale IGWM under a specific socio-hydrologic environment. For example, in the study, the reasonable penalty rate is set as USD 0.006 per cubic meter in comparison with other alternative penalty rates.

Our results demonstrate that streamflow penalty strategies, while effective in enhancing equity impose a dual-edged impact, increase water costs for upstream areas but might fail to fully address systemic disparities for downstream communities. For instance, the Gini coefficient’s heightened sensitivity to upstream inflow (Fig. b) reveals that marginalized downstream regions disproportionately bear the costs of water use, even under penalty strategies. These findings also highlight the critical role of penalty rates within streamflow penalty strategies. A well-calibrated rate can simultaneously reduce water use costs and enhance equity in water allocation, whereas a poorly designed rate exacerbates imbalances – heightening risks of policy failure. To mitigate such risks, adaptive management principles, such as iterative penalty adjustments informed by stakeholder feedback (Gonzales et al., 2019), could align incentives across governance levels. By embedding real-time hydrological data and equity metrics (e.g., Gini coefficients) into penalty frameworks, these strategies would address power asymmetries inherent in the Stackelberg hierarchy, ensuring policies not only regulate upstream UWM decisions to promote equitable allocation but also minimize costs across urban areas, even under climate-change-induced hydrological uncertainties.

To summarize the results and findings, we can draw the following concluding remarks:

The cost-effectiveness of developing GIs to use rainwater in urban areas is highly dependent on the hydroclimatic conditions of the watershed.