Balancing equity and efficiency in transboundary water systems with Atkinson’s welfare function

Equity issues are as persistent as they are complex in transboundary water resource management^1,2,3. The challenge is not merely theoretical. Power asymmetries shape real outcomes: hydrohegemons leverage their geographic and political advantages to impose inequitable arrangements^4,5,6, and even well-intentioned agreements may falter under weak institutions⁷. Yet, the hydropolitical landscape is more nuanced, given that roughly two-thirds of transboundary water interactions are documented as cooperative rather than conflictive⁸. This paradox—widespread cooperation coexisting with persistent inequity—points to a critical gap. As transboundary partnerships mature, the challenge evolves from maintaining peace to actively pursuing fair distributions of benefits and risks^9,10. Addressing this challenge demands more than aspirational statements about fairness; it requires operational frameworks that clearly define what ‘equitable’ actually means and how to achieve it in practice.

Equity is inherently multifaceted¹¹, encompassing overlapping principles of justice, human rights, equality and fairness^12,13,14,15. Each of these perspectives implies different criteria for resource allocation, making equity difficult to pin down with a simple definition. This complexity has tended to push equity to the margins of water resource systems analysis, overshadowed by utilitarian approaches that assume policies maximizing overall societal benefit will eventually compensate any losers¹⁶. Recently, however, a clear methodological gap in addressing equity is becoming apparent, spurring new research on equitable water resources planning^17,18.

Existing methods illuminate different facets of the equity challenge, but they also reveal limitations. Statistical measures, such as the Gini index, quantify inequality in resource distribution^19,20,21 but reveal little about whether a given disparity is acceptable to affected parties. Morally informed weights²² and distance-based functions²³ attempt to incorporate values more explicitly, yet often obscure the underlying judgements of whose welfare matters and by how much. A mini-max approach^24,25 appears principled but can produce perverse outcomes, as it ignores benefits to many actors while focusing on marginal improvements for a single actor. Its lexicographic extension²⁶ sequentially maximizes welfare from the worst-off actor upwards, producing a complete ordering but imposing an extreme equity preference. The sufficientarian approach^22,27 requires all actors to meet minimum thresholds before efficiency is pursued, but demands exogenous threshold selection and offers no distributional guidance above the threshold.

Game theory offers a different lens, treating equity as an emergent property of strategic interaction. Used extensively in transboundary water management problems, cooperative game theory provides allocation rules such as the Shapley value²⁸ and Nash–Harsanyi bargaining solution^29,30 and stability indicators that assess whether agreements will hold^24,31,32,33 as a function of individual and group rationality^34,35,36. Non-cooperative approaches examine unilateral behaviours, revealing how actors’ foresight, risk tolerance and mutual awareness of preferences shape outcomes³⁷. Yet, game-theoretic frameworks often struggle with dynamic, hydrologically complex systems and require non-cooperative benchmark scenarios that may poorly represent real-world complexities. Multi-objective optimization can avoid these issues^38,39, but faces computational barriers as the number of actors and objectives increases.

In summary, there is still no widely adopted approach to system-scale water resources planning that pursues distributional equity alongside cooperative efficiency without relying on questionable benchmarks, obscuring value judgements behind statistical measures⁴⁰ or hard-coding subjective weightings. This Article addresses this gap by drawing on insights from welfare economics, where the relationship between inequality measurement and social welfare has long been recognized⁴¹. Specifically, we use Atkinson’s inequality measure⁴², which makes distributional preferences explicit through a single parameter that spans from utilitarian efficiency to Rawlsian justice^43,44. This approach brings value judgement to the forefront, enabling transparent deliberation about what level of inequality a society (or parties sharing a river basin) is willing to accept¹⁸.

While social welfare functions (SWFs), such as Atkinson’s, have proven valuable in climate policy^27,45 and flood management⁴⁶, their potential for broader applications in water resources remains largely unexplored. Our ‘equitable cooperation’ framework demonstrates how formulating a river basin objective as an Atkinson SWF directly yields system-wide operational and infrastructure solutions that balance distributional concerns with collective gains. The only requirement is that actor-level distributional objectives be monotonically increasing and concave. Although equity in transboundary systems spans water access, food security and livelihoods, we operationalize it here through hydropower generation, the resource on which riparian states in our case study are most heavily dependent for meeting national electricity needs⁴⁷. First, we establish an analytical foundation for using Atkinson’s measure in a shared water system, demonstrating that, because the SWF sums concave transformations jointly over actors and time steps, intercountry equity and drought resilience emerge as two facets of a single objective. Then, through application to the Zambezi Watercourse under fully cooperative reservoir management, we optimize country-level hydroelectricity generation, finding that more equitable operations primarily reduce power generation risk during critical periods rather than redistributing average production. Finally, we examine floating solar expansion strategies, demonstrating how equity considerations redirect investments towards countries facing the highest risks in electricity production. Together, these applications illustrate how Atkinson’s measure provides a transparent and mathematically consistent framework for navigating the efficiency–equity frontier in shared water resources.

Atkinson’s inequality measure⁴² provides a welfare-based approach that moves beyond statistical dispersion as a measure of inequality. Under the general assumption of a monotonically increasing and concave utility function (indicating risk aversion), Atkinson identified a close parallel of inequality measurement with comparing performance distributions in decision-making under uncertainty. Combining this insight with mean-preserving transfers under the Pigou–Dalton principle⁴⁸ enabled ranking distributions without restricting the precise form of the utility function. This allowed the direct expression of social welfare as the sum of identical concave transformations of individual incomes, with a single parameter ϵ representing the degree of inequality aversion (Methods).

Figure 1 schematically illustrates the concept of equitable cooperation using a simplified two-country (upstream and downstream riparian), three-reservoir system. We consider a system-wide aggregated objective J (substituted for economic welfare W) and individual performance measure y that is monotonically increasing and concave. In this setting, we identify four characteristic operating solutions, ranging from low to high degrees of coordination, cooperation and distributive equity. Without coordination (a), the upstream riparian maximizes its own performance, and the downstream riparian can do no better as it is constrained by upstream operations. Coordination (b) allows the downstream riparian to ‘see’ upstream decisions and objectives, thus improving its performance without compromising the upstream riparian.

Full cooperation (c) maximizes the basinwide objective as an unweighted sum. In our conceptual example, downstream system characteristics (for instance, storage and megawatts) influence the shape of the trade-off, favouring downstream riparian performance. The utilitarian solution assumes that system-wide surplus (that is, J_c versus J_b) is redistributed by other means, thereby indirectly compensating the loser. However, redistribution in multi-actor transboundary systems requires complex international compensatory schemes that could prove fragile and contentious. This motivates the search for other solutions that, although having lower total performance, provide a fairer balance of performance. We call these ‘equitable cooperation’ (d) solutions, which are generated by optimizing the system-wide objective J using Atkinson’s SWF. As ϵ-inequality aversion increases, the basin objective isoline intersects with solutions that have more equalized performance (in Fig. 1, the circle size of equitable solutions (d) become larger than that of full cooperation (c)). Importantly, these equitable cooperation solutions lie along the Pareto frontier between riparian states; equality is sought alongside system-wide efficiency, not for its own sake.

The power of this approach lies in what it makes visible. Rather than hiding equity considerations behind complex multicriteria weightings or statistical indices, the framework reduces the essential trade-off to a single, interpretable parameter. How much inequality are river basin actors willing to accept in exchange for greater total benefits? This transparency transforms equity from an afterthought to a design criterion, enabling stakeholders to explicitly negotiate the balance between collective gains and their fair distribution.

The Zambezi Watercourse is a complex transboundary system with large hydropower dams serving as the dominant energy supply. The Watercourse faces considerable infrastructure gaps: merely 3.6% of 5.2 million hectares of arable land is irrigated, and only 38% of ~13 GW hydropower potential has been developed⁴⁹. Substantial future energy infrastructure investments are needed to close electricity access gaps (15–75% across Watercourse countries⁵⁰) and serve a population expected to double by 2050. While investments in mini-grid and standalone systems can support lower-tier electrification, on-grid supply remains critical for higher-tier and urban areas^51,52.

Following a wave of major dam construction in the mid-twentieth century, cooperative water management efforts among the riparian states remained incomplete and lacked cohesion. Only recently, all eight riparian states within the Watercourse are actively engaged through the Zambezi Watercourse Commission (ZAMCOM) to ‘promote equitable development and utilization’⁴⁹. This evolution raises potential for coordinated operations and Watercourse infrastructure investments emphasizing equity.

Figure 2 illustrates the Watercourse’s diverse operational dynamics, with active storage periods ranging from under 1 month to 20 months, reflecting a reliance on both run-of-river hydropower operations and substantial seasonal and interannual regulation. The main Zambezi (970 m³ s⁻¹ annual flow) is supplemented by the Kafue (300 m³ s⁻¹), Luangwa (780 m³ s⁻¹) and Shire (640 m³ s⁻¹) rivers. Kariba (shared by Zimbabwe and Zambia) and Cahora Bassa (Mozambique) provide the highest regulative storage capacity. Current irrigation use is estimated at 6.2 Bm³ annually, while reservoir surface evaporation losses remain the dominant consumptive use at 7.9 Bm³.

The proposed dams trigger three hydropower expansion scenarios distinguished by geographic and institutional dimensions of Watercourse development: Upper (Zimbabwe and Zambia), Lower (Mozambique) and Basinwide development. Presently, the system offers ~5.9 GW of capacity, with a 35%/65% Lower/Upper Basin distribution, and Zimbabwe holding the smallest share at 18% (Fig. 2c). The Lower Basin scenario increases Mozambique’s capacity by 1.5 GW, probably reinforcing its dependency on upstream management. The Upper Basin scenario increases Zambia and Zimbabwe’s capacity by 2.8 GW (split evenly), potentially exacerbating a trade-off with downstream uses. The Basinwide expansion (4.3 GW) provides a nearly even split of new capacity across the three riparian states.

Despite these opportunities for new hydropower, falling costs of solar and wind⁵³ and climate change impacts^54,55 undermine its competitiveness⁴⁷. A promising alternative could be up to 9 GW of floating photovoltaics (FPV) at existing reservoirs⁵⁶. We incorporate potential FPV expansion here, considering how inequality aversion affects the provisioning of FPV capacity across the Watercourse countries.

In the first application, we examine how incorporating distributional equity via the Atkinson SWF impacts hydropower generation for Zambia, Zimbabwe and Mozambique. We consider the existing dam network as well as the major hydropower expansion scenarios (Lower Basin, Upper Basin and Basinwide additions). Each country relies on hydropower for the bulk of its electricity, so generation is monotonically increasing in value with diminishing marginal returns and severe consequences when supply falls short, satisfying the properties required by the Atkinson SWF. The SWF thus applies concave power transformations directly to these generation quantities, with ϵ controlling the distributional preference across countries, not an assumed individual utility function over production. Although individual turbine efficiency curves can be non-concave at low flows, the relevant quantity is aggregate monthly production across each country’s dam portfolio, which naturally smooths plant-level nonlinearities.

Figure 3 shows the resulting efficiency–equity solution frontier of problem (7) where maximization of total hydropower generation is split into a two-objective problem by ϵ-inequality aversion (Methods). Connecting back to the conceptual representation in Fig. 1, the upper right-most solution (for each scenario) in Fig. 3 is equivalent to a full cooperation solution (c) and the lower left-most solution is equivalent to an equitable cooperation solution (d); because the dual-objective formulation optimizes J_ϵ=0 against J_ϵ=2, the resulting Pareto front already spans an efficiency–equity continuum, and any intermediate ϵ selects a preferred operating point directly on the frontier (Fig. 3a highlights solutions for ϵ ∈ {0, 0.5, 1, 1.5, 2}). Figure 3 shows that distributional equity in country-level hydropower generation requires minimal sacrifices in total system hydropower generation: 1.5% (0.36 TWh yr⁻¹) under Existing, 2.7% (0.78 TWh yr⁻¹) under Lower Basin, 1.0% (0.38 TWh yr⁻¹) under Upper Basin, and 2.6% (1.06 TWh yr⁻¹) under Basinwide scenarios. These modest efficiency losses result in reductions in country-level hydropower inequality, ranging from 3 to 8 percentage points on the Atkinson index. Moderate inequality aversion (ϵ = 0.5–1.0) already captures much of these gains, moving well along the frontier from the efficiency endpoint, while beyond ϵ = 1.5 additional aversion yields diminishing returns and the ϵ = 2 solution approaches the limit of what operational changes alone can achieve (Supplementary Fig. 1).

The efficiency–equity trade-off is not uniformly distributed across countries. As shown in Fig. 3b, Zimbabwe’s total generation is held constant in the Existing and Lower Basin scenarios and reduces by a maximum of 1.2% with added capacity. Meanwhile, Zambia and Mozambique shoulder the majority of the trade-off, facing reductions in total hydropower generation of 0.5–4.7%. As shown in Fig. 3c, equity manifests primarily through increases in firm hydropower generation (‘firm’ power is the reliable electricity available under low-flow conditions, here calculated as the average of the three lowest annual minimum values). Under the Existing scenario, all three countries gain firm generation: 62 GWh per month (+10%) for Mozambique, 64 GWh per month (+66%) for Zimbabwe and 250 GWh per month (+145%) for Zambia. When new dams expand capacity, however, the system has more scope to rebalance generation across countries, so the equitable solution shifts Mozambique’s firm generation downwards by 40–61 GWh per month (–4% to 9%) while delivering larger gains to Zimbabwe (66–122 GWh per month (+48–59%)) and Zambia (28–282 GWh per month (+5–82%)). Thus, equitable cooperation alters reservoir operating strategies to prioritize lifting minimum generation levels during critical periods. To contextualize these gains, even a modest increase of 100 GWh per month in firm generation could reliably serve an additional 550,000 people during a month of drought at the tier-5 level of electricity supply (2,195 kWh per capita per year as defined in ref. ⁵⁷). These firm generation gains are economically important given that power outages cost an estimated 5–7% of gross domestic product in Southern African economies, and the 2015–2016 El Niño drought at Kariba alone prompted Zambia to reduce its gross domestic product growth forecast from over 7% to 5.8% in anticipation of hydropower rationing⁵⁴.

The hydropower infrastructure expansion scenarios demonstrate how physical capacity distribution shapes equity outcomes. In the Existing and Lower Basin scenarios, which do not expand Zimbabwe’s limited 18% share of total capacity, inequality reductions (8 and 3 Atkinson points, respectively) are primarily based on operational changes that boost firm hydropower generation for Zimbabwe and Zambia. Upper Basin and Basinwide scenarios achieve greater equity improvements by addressing the capacity imbalance, adding generation infrastructure to Zimbabwe and Zambia alongside operational optimization. This dual approach (infrastructure plus operations) results in substantially lower inequality indices (0.14 for Upper Basin and 0.13 for Basinwide) in the equitable solutions compared with the Existing (0.32) and Lower Basin (0.40) scenarios.

For the second application, we examine how inequality aversion alters FPV capacity provisioning among countries and include a hydropower-competing objective of maintaining natural Delta flows for ecosystem services⁵⁸. The multi-objective problem incorporates Atkinson inequality aversion in country-level power generation (hydropower plus FPV), Delta flow deficit and capital cost (US$1 per watt FPV) (Methods). Because environmental flow objectives may exhibit threshold-dependent ecological responses that violate the monotonic concavity assumption, they are not candidates for aggregation through the Atkinson SWF. Here, the Delta flow deficit enters as a separate minimization objective (equation (8)), while any hydropower trade-off it induces is mediated across countries by the SWF.

Figure 4 shows the resulting system-wide efficiency (ϵ = 0) and equitable (ϵ = 2) solution sets of problem (8). Up to 8.4 GW FPV capacity installed at Kariba and Cahora Bassa could increase the Watercourse’s total production by ~12 TWh yr⁻¹. The efficient solutions deploy FPV at Kariba first (Fig. 4b, top). This is more cost-effective because, as Cahora Bassa operates closer to its hydropower capacity, even modest levels of FPV on Cahora Bassa can cause frequent curtailment of hydropower generation due to transmission line congestion. Because transmission line constraints at Kariba are nearly equivalent for both Zimbabwe and Zambia connections, the distribution of the first ~3 GW FPV capacity to Zambia and Zimbabwe is driven mostly by random selection in the evolutionary optimization.

The equitable solutions deliberately allocate new FPV capacity in order from the lowest- to the highest-producing country (Fig. 4b, bottom) and remain especially sensitive to firm production levels. Zimbabwe receives the first 2 GW capacity added, followed by a roughly equal split of the next 1 GW to Zambia and Mozambique. Driven by Zimbabwe’s gains, inequality reduction is the largest (up to 25 Atkinson index points) for the first 3 GW of FPV capacity added. Throughout the remaining 3–8.4 GW FPV investment, inequality remains 5–20 Atkinson index points lower in the equitable solutions.

As shown in Fig. 5, supporting the Delta flow costs 4.5 TWh yr⁻¹, which is about one-fifth of the hydropower-maximizing efficiency solution. Because modifying Cahora Bassa’s operations is the most efficient way to support the Delta flow objective^38,59, Zimbabwe and Zambia are barely affected by the change in operations, while Mozambique’s mean hydropower generation falls by 375 GWh per month (−41%) with a major uptick in the frequency of months producing less than 100 GWh (Fig. 5c–e).

FPV expansion represents one way to recover this trade-off between hydropower and Delta flow maintenance. At 2.9 GW FPV expansion, the efficient solution deploys the full 2.9 GW (100%) to Kariba, raising Zimbabwe and Zambia’s output by ~200 GWh per month each (Fig. 5c–e). Although this more than recovers the total hydropower trade-off with the Delta flow and reduces inequality by 23 Atkinson index points, Mozambique receives none of the benefit despite bearing the full hydropower cost. This is what the ‘equitable cooperation’ solution can address through its innate targeting of the most at-risk producers. In the equitable solution, 1.14 GW (39%) of FPV is deployed to Cahora Bassa, recovering nearly half of Mozambique’s hydropower foregone to maintain the Delta environmental flow and reducing the frequency of months with very low production. The remaining 1.76 GW (61%) of FPV is deployed to Kariba and 100% dedicated to Zimbabwe. By reducing Zimbabwe and Mozambique’s power production risk relative to Zambia’s, the equitable solution reduces inequality by an additional 25 Atkinson index points for a total production trade-off of 1.1 TWh yr⁻¹ (4.2%). This demonstrates how the Atkinson welfare function approach inherently adapts to shifting inequities when optimizing a system across conflicting objectives that have disparate impacts across system actors.

This study demonstrates how distributive equity can be operationalized alongside cooperative efficiency in system-scale optimization of water resources. The Atkinson SWF-based framework builds in equity as a design criterion with explicit normative assumptions and an adjustable parameter for degrees of inequality aversion. We apply the framework to the Zambezi Watercourse, finding that inequality aversion towards country-level hydropower generation generates modest system-wide losses of 0.36–1.06 TWh yr⁻¹ (1.0–2.7%) while achieving inequality reductions of 3–8 Atkinson index points. Because physical constraints of water availability and turbine capacities limit the potential for rebalancing total hydropower generation, inequality aversion manifests primarily through increases in firm generation (48–66% for Zimbabwe, 5–145% for Zambia and −9% to +10% for Mozambique). This suggests that operational strategies targeting the most vulnerable during critical periods are a practical means to achieve equity^60,61. This finding is not surprising, given the concavity assumption that drives inequality aversion parallels risk aversion⁴², implying that equitable resource allocation and robust system design are closely related objectives, and that what appears as reduced efficiency is actually improved system resilience when accounting for uncertainty.

A key strength of the framework is the inherent adaptability during optimization. Unlike static allocation rules or predetermined weights, the Atkinson approach automatically adjusts distributional priorities in response to changing system conditions. In the FPV multi-objective application, Mozambique bears the full trade-off of maintaining a more natural Delta environmental flow, reducing its hydropower generation by 41% and increasing its periods of critically low production. FPV investment priorities shift accordingly (from Zambia to Mozambique) without requiring prior knowledge of the actor-level effects and reduce inequality by 25 Atkinson index points for a total production trade-off of 1.1 TWh yr⁻¹ (4.2%). This responsiveness addresses a need for adaptive mechanisms that accommodate shifting power dynamics and environmental conditions in transboundary water management⁵.

Setting ϵ-inequality aversion equal to 2 in our analysis represents a strong preference for equity (Fig. 6), yet stakeholders may reasonably disagree about appropriate values. However, because the dual-objective formulation optimizes J_ϵ=0 against J_ϵ=2, the resulting Pareto front already spans an efficiency–equity frontier, and any intermediate ϵ selects a preferred operating point without additional optimization. We therefore propose framing ϵ not as a technical calibration but as a deliberation parameter that surfaces distributional value judgements for explicit negotiation. Stakeholders need not understand the mathematical formulation; structured choice experiments can present alternative generation distributions (for example, one option that maximizes total basin power versus another that sacrifices a small percentage to equalize firm generation during droughts) and recover implicit ϵ preferences from stated choices^18,45. In the Zambezi, ZAMCOM could deploy such experiments alongside the efficiency–equity frontier (Fig. 3) in participatory workshops, allowing negotiators to identify acceptable inequality ranges. Careful attention to whose values are represented in such processes remains essential^17,27, particularly as the framework extends beyond hydropower to broader development objectives where the cross-sectoral implications of equity must be negotiated.

Our formulation treats each country as a single, equally weighted unit in the welfare function, embedding a sovereignty-based normative choice in which each riparian state receives equal standing regardless of population, consistent with the ‘one state, one vote’ principle in international water law⁶². In the Zambezi, ZAMCOM’s mandate is explicitly interstate, while transboundary agreements and infrastructure investments operate at the national scale⁴⁹. Two extensions could incorporate population at this scale, each encoding a distinct normative commitment⁴¹. A population-weighted SWF^42,45 would scale each country’s welfare contribution by its population, counting each person equally in the welfare sum. Alternatively, using per-capita generation as the welfare input would retain equal country weights but measure equity in generation per person, prioritizing countries where output is spread thinly across large populations. Both extensions reframe equity around individuals rather than states, but embed population at different points in the welfare function. In basins where the most populous country is also the lowest per-capita producer, as in the Zambezi where Mozambique has the largest population and the lowest per-capita generation, both extensions reinforce each other. Where population and per-capita rankings diverge, the two extensions would pull in opposite directions, making the choice between them a substantive policy discussion.

The country-level analysis is defensible when interstate and intrastate equity are recognized as distinct subjects of distributive justice³. Nevertheless, country-level gains can mask within-country disparities. Extending the framework subnationally requires coupling river basin models with power system dispatch⁶³ and geospatial electrification models that capture off-grid options^51,52 at compatible resolutions. Reference ²¹ takes a step in this direction by coupling river basin and power system simulators to minimize regional electricity access inequality in Ghana, but the Gini index carries no adjustable inequality aversion parameter, precluding deliberation over how much priority the worst-off should receive, and equal-population regionalization embeds a per-capita equity norm in geographic preprocessing rather than the welfare function, where it could be inspected and varied. An Atkinson SWF would address both limitations while scaling readily to finer-grained actors and larger basins such as the Mekong or the Nile; in data-scarce transboundary settings, the binding constraint on subnational equity analysis is the integrated modelling infrastructure, not the welfare aggregation.

In this framework, distributive justice concerns the equality of welfare contribution to ends that matter for each actor^3,22, which raises the question of whether the framework should accommodate group-specific inequality aversion across subregions, vulnerability classes or gender-differentiated impacts. We find this idea methodologically uneasy, as the nested formulation violates anonymity across actors, complicates Pigou–Dalton consistency at group boundaries and introduces as many unaccountable value choices as there are groups, so we prefer to keep a single ϵ and let disaggregation do the work where scope and data permit. A complementary direction would combine the SWF with a needs-referenced threshold, in the prioritarian spirit of ref. ⁴⁵. This, while more methodologically sound, faces a mathematical constraint, because Atkinson’s transformation requires strictly positive inputs, and an evidential one, because defending distinct thresholds per group is difficult outside narrow cases of legal or vulnerability classification, although a common reference is more tractable. Ultimately, the framework’s contribution to distributive justice depends on what is disaggregated and made operationally visible, not on layering additional value parameters onto the aggregation.

Finally, the framework extends to any multi-actor optimization problem with monotonic, concave objectives. Objectives with threshold effects or increasing returns cannot enter the SWF directly and must be handled as separate optimization criteria, as we do for environmental flows. In practice, however, many resource utilization objectives naturally exhibit diminishing marginal returns^64,65, making concavity a reasonable default for welfare-based aggregation of resource allocation outcomes. This includes renewable energy deployment, conservation planning or climate adaptation funding, particularly in common-pool resource contexts where side payments prove unreliable and distributional outcomes affect system stability. As pressures on shared water resources intensify from changing precipitation patterns and increased extremes, frameworks that explicitly balance efficiency and equity can help maintain cooperation under stress. This Atkinson SWF-based equitable cooperation approach is a way to operationalize ‘equitable and reasonable utilization’³ towards real decisions with real consequences for human welfare and environmental sustainability.

Atkinson’s inequality measure mathematical formulation

Reference ⁴⁸ pioneered the welfare-based approach to inequality measurement, distinguishing economic inequality from mere statistical dispersion by focusing on the underlying social welfare implications of income distributions. Dalton’s work formalized the Pigou–Dalton transfer principle, but the approach required specifying the precise functional relationship between income y and utility U, as different and empirically unverifiable assumptions about the utility function U(y) yielded different inequality measures. Reference ⁴² resolved this limitation. First, by assuming U(y) is monotonically increasing and concave, indicating risk aversion, Atkinson identified a close parallel with comparing performance distributions in decision-making under uncertainty. This insight, combined with a sequence of mean-preserving transfers in accordance with the Pigou–Dalton principle, is all that is needed to rank two distributions in terms of inequality without restricting the form of U(y) (ref. ⁶⁶). Second, Atkinson recast Dalton’s welfare-based approach entirely within income space through the concept of ‘equally distributed equivalent income’ (y_EDE), the income level that, if equally distributed, would yield the same social welfare as the actual distribution:

Because the concavity of U(y) ensures that equally distributed equivalent income is less than the mean income, the proportional shortfall (Atkinson’s inequality index) quantifies the welfare loss due to inequality. On this basis, Atkinson’s key innovation was to express social welfare directly as the sum of identical concave transformations of individual incomes:

where the parameter ϵ captures the degree of inequality aversion by determining the relative weight given to transfers at different points in the distribution f(y). Figure 6 illustrates this preference for redistribution as a function of the disparity between two actors and the acceptance of a ‘leaky transfer’ between them⁶⁷, varying according to different degrees of ϵ.

Zambezi Watercourse system model

We use the Zambezi Watercourse system model (ZW model) to capture the dynamics of reservoir operation, hydropower generation, irrigation use and environmental flow over a historically observed 20-year (1986–2006) sequence of inflows (catchment hydrology is directly measured rather than simulated). The ZW model has been applied in previous studies to explore the synergies and trade-offs across water, environmental, energy and food objectives for the Watercourse^59,68,69, and its physical components (mass balance, storage–area–elevation relationships, evaporation and power production functions) have been validated across these studies. The model includes the five major existing dams, one run-of-river hydropower plant at Victoria Falls, eight irrigation areas and (up to) three of the major planned dams. System state transitions are captured through mass balance equations of the form

where s_t is reservoir storage at the beginning of month t, ({q}_{t+1}^{mathrm{up}}) is inflow to the reservoir from one or more upstream tributaries, ({r}_{t+1}^{mathrm{up}}) is the volume of water released from the upstream reservoir(s), ({omega }_{t+1}^{mathrm{up}}) is the water abstracted (and fully consumed) by upstream irrigation diversion(s), and e_tS_t is the water evaporated in the time interval [t, t + 1) where e_t is the mean monthly evaporation rate and S_t is the reservoir surface area (determined by a nonlinear relation given s_t). The reservoir release is defined as r_t+1 = f(s_t, u_t, q_t+1, e_t) where f(⋅) is a nonlinear, stochastic relation between the release decision determined by the optimal operating policy ({rho }_{theta }^{* }) and the actual release⁷⁰. In the ZW model applications of this study, we exclude Malawi’s hydropower dams along the Shire River because their operations do not affect the other countries in the Watercourse. Furthermore, we assume irrigation abstractions are fully satisfied up to the available flow at each stream diversion point (that is, no hedging policies are applied).

Direct validation of simulated hydropower against historical generation records is not feasible for two reasons. First, the simulation period overlaps with the Mozambican civil war (ended 1992), which severely disrupted Cahora Bassa operations, and with drought events and other contingencies that forced the Kariba operator to deviate substantially from prescribed rule curves⁶⁹. Second, the model is not designed to replicate historical policies; it optimizes fully coordinated, closed-loop operating policies that represent an upper bound on system performance⁶⁸, deliberately removing the institutional and geophysical factors that cause actual operations to deviate from optimal rules. Nevertheless, simulated annual hydropower production is consistent with publicly reported generation figures for the post-conflict portion of the simulation period (Supplementary Fig. 2) and the 1986–2005 hydrology (Supplementary Fig. 3).

Multi-objective optimization

We generate sets of Pareto-efficient fully coordinated multireservoir operating solutions by coupling the ZW model with the self-adaptive Borg evolutionary optimization engine⁷¹. Specifically, we use evolutionary multi-objective direct policy search⁷² to solve the following multi-objective problem:

where finding ({p}_{theta }^{* }) corresponds to finding the best parameters θ^* for the policy p_θ as measured by the objectives J_pθ and subject to (s.t.) the state transition of the system model. The closed-loop policy in the form of Gaussian radial basis functions determines a vector of reservoir release decisions u_t as a function of the state vector x_t, which we define as the month of the year (time t), the vector of storage volumes in each reservoir (s_t) and the previous month’s total Watercourse inflow (({sum }_{i=0}^{I}{q}_{t}^{i})).

Equation (4) can be extended to coupled planning and management optimization problems by incorporating planning variables into the policy description⁶⁹:

such that finding the optimal policy π^* means jointly finding the optimal planning action(s) α^* and optimal operating policy ({p}_{theta }^{* }) in a single optimization process. In our FPV provisioning experiment, α^* includes the peak capacity sizing of FPV deployed at the reservoirs (in this case, two planning decision variables).

Hydropower operations experiment

To trace the trade-off between system-wide efficiency and country-level equity in hydropower generation, we split the maximization of total hydropower generation J^W of N Watercourse countries into a two-objective problem according to the magnitude of ϵ-inequality aversion:

where H is the evaluation period (here, 20 years) and w_r,t is the hydropower generated by dam r at time step t and assigned to country i. Equation (7) shows that inequality aversion is applied across time, thus representing the risk each country faces relative to one another, a key facet of addressing equity concerns in basinwide water management. We use ϵ = 2, a relatively high level of inequality aversion, which, for reference, corresponds to preferring a solution where at least 25% of the performance lost from country A can be transferred to country B, and where country A was at least twice as well off as country B before the transfer (Fig. 6). In practice, the actual level of inequality aversion could be elicited from decision-makers and stakeholders in a participatory planning process. Because the Atkinson SWF aggregates all N countries into a single objective for any given ϵ, the optimization remains a two-objective problem regardless of the number of riparian actors, avoiding the high dimensionality of an actor-by-actor objective formulation in complex basins.

FPV provisioning experiment

In the second experiment, we construct a multi-objective problem incorporating Atkinson inequality aversion in country-level power output:

where ({J}^{{W}_{epsilon }}) includes both hydropower generation and FPV production, J^E is the average February–March Delta flow deficit, and J^C is the total overnight capital cost assuming US$1 per watt of installed FPV. Because the optimization is carried out jointly for reservoir control policies and the sizing of FPV deployed at each reservoir, the solutions represent an optimal provisioning of FPV capacity across the countries for a given level of total FPV investment. Two separate optimizations of equation (8) are conducted for the existing reservoir system: setting ϵ = 0 generates a set of system-wide efficiency solutions, and ϵ = 2 a set of equitable cooperation solutions.

Floating solar constraints and electricity system representation

We use the soft-link framework developed in ref. ⁵⁶ to model floating solar in the Zambezi Watercourse. Feasible FPV peak capacities at reservoirs are defined according to reservoir coverage limits (30 km² or 30% of reservoir surface area) or where existing transmission line capacities would severely constrain power dispatch to the grid. The latter is derived from sensitivity testing using a PowNet⁶³ electricity system model of the South African Power Pool. This process also yields curtailment factors representing grid constraints when the combined solar and hydropower availability exceeds the existing transmission line capacity. The FPV production and hydropower curtailment factors are fed back into the ZW model simulation–optimization framework to jointly identify Pareto-efficient reservoir operation policies and FPV system sizing under realistic electricity system and water-management constraints.

Simulation and postprocessing code

Postprocessing and figure generation were performed in Python (v3.11)⁷³. The following pseudocode describes the full simulation–optimization pipeline and figure postprocessing scripts.

1. (1)

   Zambezi Watercourse system model. For each candidate policy, radial basis functions map inflow, reservoir storage levels and month-of-year to release decisions, which the model integrates as reservoir mass balances and hydropower generation through the cascade at monthly time steps, aggregating to country-level production and firm hydropower by riparian state. The Borg multi-objective evolutionary algorithm (v2.0)⁷¹ searches the space of radial basis functions policy parameters over 2,000,000 function evaluations and for n = 20 independent random seeds, with each seed producing a Pareto-approximate solution archive.

2. (2)

   Experiment 1, hydropower operations (exp1.py; Fig. 3). For each of the four hydropower expansion scenarios, the simulation summary output is Pareto-sorted on total production and the Atkinson inequality index. The efficiency endpoint and equity endpoint are identified, and system-wide efficiency loss and inequality reduction between endpoints are computed and reported. The Pareto front is plotted with solutions highlighted at ε ∈ {0, 0.5, 1, 1.5, 2}, and country-level total and firm hydropower bars are plotted for both endpoints.

3. (3)

   Experiment 2, FPV expansion (exp2.py; Figs. 4 and 5). For the efficiency (ε = 0) and equitable (ε = 2) FPV optimization runs, total FPV capacity is computed as the sum of site capacities at Kariba North, Kariba South and Cahora Bassa. Solutions are filtered by a Delta flow deficit threshold and FPV capacity bounds (0.2–8.4 GW), then Pareto-sorted on total production and FPV capacity. Four representative solutions are selected along the front, and their monthly production distributions are plotted as kernel density estimates by country.