Value-based Resource Matching with Fairness Criteria: Application to Agricultural Water Trading

TL;DR

This work tackles value-based resource matching for agricultural water trading under drought, modeling a two-sided market of seniority-ranked sellers and buyers connected by geographic compatibility. When value functions are monotone, the problem MaxWelfare is solvable in polynomial time by reducing to a maximum weighted bipartite matching; otherwise, buyer-side non-monotonicity leads to NP-hardness. The paper further introduces fairness-aware variants, including MaxWelfareFair (randomized rounding with lower-bound constraints in expectation) and MaxLeximin (single-seller lexicographic fairness), with accompanying complexity and algorithmic results. Experiments on synthetic data and real Washington basins (Touchet and Yakima) illustrate how drought severity, seniority, and crop portfolios shape trade structure, welfare, and buyer satisfaction. Overall, the work provides tractable algorithms and hardness results for fiscally and socially mindful water allocation in realistic, constraint-rich market settings, with practical implications for water-rights markets and policy design.

Abstract

Optimal allocation of agricultural water in the event of droughts is an important global problem. In addressing this problem, many aspects, including the welfare of farmers, the economy, and the environment, must be considered. Under this backdrop, our work focuses on several resource-matching problems accounting for agents with multi-crop portfolios, geographic constraints, and fairness. First, we address a matching problem where the goal is to maximize a welfare function in two-sided markets where buyers' requirements and sellers' supplies are represented by value functions that assign prices (or costs) to specified volumes of water. For the setting where the value functions satisfy certain monotonicity properties, we present an efficient algorithm that maximizes a social welfare function. When there are minimum water requirement constraints, we present a randomized algorithm which ensures that the constraints are satisfied in expectation. For a single seller--multiple buyers setting with fairness constraints, we design an efficient algorithm that maximizes the minimum level of satisfaction of any buyer. We also present computational complexity results that highlight the limits on the generalizability of our results. We evaluate the algorithms developed in our work with experiments on both real-world and synthetic data sets with respect to drought severity, value functions, and seniority of agents.

Figures (5)

• Figure 1: Example. Going from left to right, the first panel shows a sample stream flow and points of diversion for different trading agents. In the second panel, each directed edge $(a,b)$ indicates that $a > b$, that is, $a$ has higher priority than $b$. When there is no directed edge between two nodes, the interpretation is that they belong to different streams. The drought cutoff corresponds to the scenario where water is available to agents $1$, $2$, and $3$. In the compatibility graph shown in the third panel, water units and their values have not yet been factored in. The rightmost panel shows the corresponding resources--needs bipartite graph along with the water units of the buyers and sellers and the corresponding compatibility graph. Here, Seller 3 has, and Buyer 4 needs, two water units, while the others have or need only one water unit each. The values of the water units are shown in parentheses.
• Figure 2: Panels (a) and (b) show structural properties of the resources--needs graph for the synthetic datasets with respect to increasing water availability. In Panels (c) and (d), Y-axis gives the total value $\sigma(\mathcal{T}^*)$ and $\mathop{\mathrm{welfare}}\nolimits(\mathcal{T}^*)$ corresponding to an optimal assignment $\mathcal{T}^*$ respectively, normalized by $\sigma_0$, the total value when 100% of water is available. All results are for $N=10$ and the number of units per agent $k=5$. The results are shown for different values of $\lambda$ and $\beta_{h}$.
• Figure 3: Panels (a) and (b) show structural properties of the resources--needs graph for real-world datasets with respect to increasing water availability. In Panels (c) and (d), Y-axis gives the total value $\sigma(\mathcal{T}^*)$ and $\mathop{\mathrm{welfare}}\nolimits(\mathcal{T}^*)$ corresponding to an optimal assignment $\mathcal{T}^*$ respectively, normalized by $\sigma_0$, the total value when 100% of water is available. All results are for varying sizes for water units.
• Figure 4: The loss in welfare as the lower bound on the number of water units for each individual buyer is increased. The welfare-fairness tradeoff is the ratio $\mathop{\mathrm{welfare}}\nolimits(\mathcal{T}_r^*)$/$\mathop{\mathrm{welfare}}\nolimits(\mathcal{T}^*)$, where $\mathcal{T}_r^*$ is an optimal solution which satisfies the fairness criteria that every buyer is matched with at least $r(\{b\})=r$ water units and $\mathcal{T}^*$ is an optimal solution when no such constraints are imposed. The results are for the synthetic graphs for different values of $\lambda$ and $\delta$ over 100 replicates.
• Figure 5: A boxplot of buyer satisfaction (water units matched/required water units) distribution with respect to water availability. Given the optimal solution from Algorithm \ref{['alg:monotone']}, for each buyer, the satisfaction is computed. In some cases, the boxes are not visible. These correspond to a median of either zero or one.

Theorems & Definitions (13)

• Remark 3.1
• Theorem 4.1
• Theorem 4.2
• Remark 4.3
• Theorem 5.1
• Corollary 5.2
• Theorem 5.3
• Remark 5.4
• Proposition C.1
• Lemma C.2