How to distribute the European regional development funds through a combination of egalitarian allocations: the constrained equal losses min

The European Regional Development Fund (ERDF) is a key instrument of the EU Cohesion Policy aimed at reducing regional disparities by financing projects that support SMEs, health systems, digital infrastructure, clean transport, and emissions reduction. The ERDF allocation is typically guided by the Berlin method across NUTS2 regions classified as less developed, transition, and more developed according to GDP per capita. Viewing this allocation as a conflicting claims problem—where aggregate claims exceed the available budget—enables the use of axiomatic rules for fair division. Prior work shows the Constrained Equal Losses (CEL) rule promotes convergence effectively by favoring larger (per capita) claims typical of less developed regions, but it can assign zero funds to some regions, which is impractical. To address this, the paper proposes CELmin, a rule that guarantees a minimal positive allocation to all claimants and then applies CEL to the remaining estate, thus balancing egalitarian awards with egalitarian losses to preserve convergence incentives while avoiding zero allocations. The paper formalizes claims problems and classic rules, defines CELmin, presents an axiomatic analysis, evaluates convergence via Lorenz dominance and a Divergence Ratio, and applies the approach to the ERDF case.

The study builds on the bankruptcy/claims-problem literature (O’Neill, 1982; Young, 1987; Thomson, 2003, 2019) and on applications of claims approaches in education (Pulido et al., 2002), fisheries (Iñarra and Prellezo, 2008; Iñarra and Skonhoft, 2008; Kampas, 2015), rural development funds (Kiryluk-Dryjska, 2014, 2018), and CO2 emissions allocation (Giménez-Gómez et al., 2016; Duro et al., 2020). Within EU funds allocation, Fragnelli and Kiryluk-Dryjska (2019) and Solís-Baltodano et al. (2021) model ERDF as a claims problem and compare rules such as Proportional (P), Constrained Equal Awards (CEA), Constrained Equal Losses (CEL), and αmin-Egalitarian; CEL performs best at promoting convergence but yields zero awards to some regions. The paper also relates to frameworks that mix allocation principles via baseline/composition operators (Hougaard et al., 2012; 2013a; 2013b) and to the Min lower bound of Dominguez and Thomson (2006). Lorenz dominance comparisons of rules (Bosmans and Lauwers, 2011) show CEA is most equitable and CEL most inequitable, motivating a compromise rule like CELmin.

- Formalization of claims problems: Agents N={1,...,n} each have a claim c_i on estate E, with a claims problem when sum_i c_i > E. Solutions (rules) map (E,c) to awards φ(E,c) satisfying non-negativity, claim-boundedness, and efficiency. - Classic rules: Proportional (P) allocates proportionally to claims; Constrained Equal Awards (CEA) equalizes awards up to claims; Constrained Equal Losses (CEL) equalizes losses (sum of claims minus E) ensuring non-negativity. The αmin-Egalitarian rule guarantees the smallest claim to all, then distributes the remainder proportionally if possible, else equally. - New rule (CELmin): Guarantees each agent the Min lower bound min{c_1, E/n} (sustainable bound) as a minimal positive allocation; if E is insufficient, it divides E equally. Otherwise, it subtracts this guaranteed amount from both E and claims and applies CEL to the residual problem. Formally: CELmin(E,c) = min(E,c) + CEL(E − MIN(E,c), c − min(E,c)), with min(E,c) = (min{c_1, E/n},..., min{c_1, E/n}) and MIN(E,c) = n·min{c_1, E/n}. The rule can be seen as CEL with a baseline equal to the smallest claim via the composition operator. - Illustrative example: For (E,c)=(2000; (500,2000,2400)), CELmin=(500,2500,1000), ensuring a minimal right (500) to all before applying CEL to the remainder, contrasting with CEL that gives (0,800,1200). - Axiomatic analysis: The paper evaluates CELmin against axioms including Equal treatment of equals, Anonymity, Order preservation, Resource monotonicity, Super-modularity, Order preservation under claims variations, Min lower bound, and shows where CELmin differs from CEL (Limited consistency, Composition up/down). Proof sketches demonstrate satisfaction or violation. - Convergence assessment: Uses Lorenz dominance to compare equity across rules and computes a Divergence Ratio (DR) to measure how allocations reduce the GDP per capita gap between the least-developed and most-developed regions. The DR is defined as d(αβ)=1−GDP_α/GDP_β using updated GDPs after allocations. - Empirical setting: ERDF 2014–2020 budget (~€182.15 billion) for NUTS2 regions, grouped into 47 claimant categories (R1/R2/R3) as in Solís-Baltodano et al. (2021). Claims are constructed as a common base per capita plus an amount increasing with the GDP per capita gap from the richest region, so less developed regions claim more. The study computes allocations under P, CEA, CEL, αmin, and CELmin and compares DRs.

- CELmin properties: Satisfies Equal treatment of equals, Anonymity, Order preservation, Resource monotonicity, Super-modularity, Order preservation under claims variations, and Min lower bound. It does not satisfy Limited consistency, Composition down, or Composition up. CEL satisfies those three but fails Min lower bound. - Equity comparisons: Lorenz dominance ordering confirms CEA is most equitable and CEL most inequitable. CELmin Lorenz-dominates CEL (due to its egalitarian component), and CEA > αmin > CELmin > CEL. There is no general Lorenz dominance relation between CELmin and Proportional. - Convergence (Divergence Ratio): Initial DR (before allocation) between Bulgaria R3 (least developed) and Luxembourg R1 (most developed) is 0.8054. After allocation: CEA 0.8003; Proportional 0.7987; αmin 0.7989; CEL 0.7957; CELmin ≈ 0.7957 (almost identical to CEL). Thus, CELmin matches CEL’s convergence performance. - Practical allocation impacts: CELmin preserves the strong support to less developed regions characteristic of CEL while guaranteeing every region a minimal positive allocation. Some regions that receive zero under CEL (e.g., Czechia R1, Ireland R1, Poland R1, Slovakia R1, Luxembourg R1) receive a small positive amount under CELmin (e.g., 0.69 per capita) while allocations to R3 remain very close to CEL. The approach respects order preservation, unlike some aspects of the current distribution highlighted in the paper.

The paper addresses the challenge of allocating ERDF funds to promote convergence while maintaining political and practical acceptability. CEL strongly favors larger claimants (less developed regions) and enhances convergence but can allocate zero to more developed regions, hindering real-world adoption. CELmin resolves this by guaranteeing each region a minimal right based on the Min lower bound and then applying CEL to the remainder. This compromise retains CEL’s convergence performance (nearly identical DR) and axiomatic strengths such as order preservation and resource monotonicity, while ensuring that all regions receive a positive allocation. Lorenz dominance analyses situate CELmin between egalitarian and loss-equalizing extremes, offering a policy-relevant balance that supports needs-based prioritization without entirely excluding smaller claimants. Applied to the ERDF, CELmin provides allocations comparable to CEL for R3 regions and small guaranteed awards to R1 regions that would otherwise receive nothing, thereby improving the rule’s implementability.

The ERDF’s objective of supporting less developed regions implies that an unequal allocation favoring greater needs is appropriate. Classical rules either are too egalitarian (CEA), provide moderate prioritization (Proportional, αmin), or strongly favor larger claims but can yield zero awards (CEL). The proposed CELmin rule guarantees a fixed minimal right to all regions and then applies CEL to the residual, preserving strong support for less developed regions while avoiding zero allocations. Empirically, CELmin achieves virtually the same convergence improvement as CEL (Divergence Ratio ≈ 0.7957), outperforming Proportional and CEA, and offers a feasible and principled method for ERDF allocation. The approach is also relevant for other needs-based rationing contexts, such as environmental negotiations (e.g., CO2 emissions rights distribution).

- Axiomatic limitations: CELmin does not satisfy Limited consistency, Composition down, Composition up, Reasonable lower bounds on awards, Invariance under claims truncation, Self-duality, or the Midpoint property. CEL differs by satisfying the composition properties and Limited consistency but fails the Min lower bound. - Dependence on claims specification: Results rely on the construction of claims (as in Solís-Baltodano et al., 2021). Different claim-generation methods could affect allocations and comparative performance. - Minimal awards may be very small for some regions, which, while non-zero, may have limited practical impact.