Cyclic-Coalition Stable Allocations

• Cyclic-coalition stable allocations are defined as a robust stability criterion that prevents cyclical reassignments where every agent improves, ensuring strict Pareto optimality.
• The research introduces two MILP formulations—the basic and reduced models—that enforce capacity, preference, and sequential fill-up constraints for stable and efficient allocations.
• Computational experiments confirm that the reduced formulation offers up to 17% lower total cost and tighter LP relaxations, making it highly competitive in complex matching markets.

Cyclic-coalition stable allocations generalize the concept of stability in cooperative matching and allocation problems by requiring robustness not only against single-agent or pairwise deviations, but also against coalitional improvements that are structured as cycles. This concept is especially significant in models where agents or customers are matched to resources or facilities under preferences and capacity constraints, and where ensuring Pareto efficiency and groupwise non-dominance is critical. In recent research, cyclic-coalition stability has been formalized and incorporated into mixed-integer programming frameworks, yielding both stronger theoretical guarantees and practical, computationally competitive models (Domínguez et al., 29 Jul 2025).

1. Definition and Theoretical Foundations

Cyclic-coalition stable allocations are defined to prevent improvements by any subset of agents that could permute their assignments in a cycle, with each agent strictly preferring the alternative allocation. Formally, given an allocation $A$, stability against cycles requires:

• No blocking customer exists: no customer can be reassigned to an open, undersubscribed facility they prefer more than their current one.
• No blocking pair exists: no two customers can mutually swap facilities such that both improve their allocations.
• No blocking set exists: for any group of $k$ customers, each assigned to a different facility, it is impossible to reallocate them via a cyclic permutation so that every agent strictly improves.

This enforces that every groupwise, cyclical deviation is ruled out. The property is strictly stronger than individual or pairwise stability and is also equivalent to Pareto optimality with respect to global (customer) preference maximization in the Capacitated Facility Location Problem with Customer Preferences (CFLCP) and closely related problems (Domínguez et al., 29 Jul 2025).

2. Mixed-Integer Programming Formulations

To operationalize cyclic-coalition stability, new mixed-integer linear programming (MILP) formulations were introduced. Two main formulations are provided:

(a) Basic Formulation ((CFLCP)_{CC})

• Introduces binary variables $y_j$ (facility open/close), and $x^\sigma_{ij}$ encoding customer $i$ being assigned to facility $j$ in round $r$ according to a permutation $\sigma$.
• Key constraints include capacity ($$\sum_{i, r'} x^r_{ij} \leq (c_j-1) y_j + \sum_{r' \leq r} u^r_j$$) and order-respecting assignments to prevent any customer being assigned to a less-preferred facility while a more-preferred open, available facility exists.
• Additional variables $u_j^r$ capture the fill-up sequence of facilities as customers are assigned.

(b) Reduced Formulation ((CFLCP)_{CC}^R)

• Reduces the number of variables and constraints by collapsing round-related indices and leveraging the typically smaller number of facilities $|J| \ll |I|$.
• Maintains auxiliary variables $u_j^r$ for capacity tracking.
• The main constraints enforce:
  • $\sum_{i} x_{ij} \leq (c_j-1) y_j + \sum_r u^r_j$ [facility capacity]
  • $$y_j - \sum_r u^r_j \leq \sum_{j': j' \preceq_i j} x_{ij'}$$ [customer preference consistency]
  • Sequential fill-up and unique round assignments per customer.

This reduction exploits total unimodularity to yield tighter linear relaxations, computational tractability, and less branching in practical solvers.

Formulation	Variables Per Customer	MILP Size	Key Features
Basic (CFLCP)_{CC}	$O(|J||I|^2)$	Large	Explicit permutation/round encoding
Reduced (CFLCP)_{CC}^R	$O(|J||I|)$	Moderate	Collapsed rounds, fewer auxiliary variables

3. Pareto Efficiency and Solution Quality

Cyclic-coalition stable allocations achieved via these models are Pareto optimal in the sense that no customer can move to a strictly better facility (according to their preferences) without causing a worsening for at least one other customer. In the context of CFLCP with global preference maximization, this non-domination property aligns with the strongest fairness and efficiency requirements typical in house allocation, school choice, and market design (Domínguez et al., 29 Jul 2025). The models avoid the artifacts of numeric-preference encodings and instead work directly with strict, ordinal preference rankings.

4. Computational Results

Extensive computational experiments compared these MILP-based cyclic-coalition stable allocation models against prior approaches (notably, the bilevel model [CGICC, Calvete et al. 2020]). Key empirical findings include:

• The reduced formulation outperforms the more extensive basic model in both branch-and-bound node count and wall-clock running time.
• Cyclic-coalition stability, as enforced in these models, produces allocations with up to 17% lower total cost than previous methods relying only on customer or pairwise stable allocations.
• The reduced model consistently delivers tighter LP relaxations and better lower bounds, which translates into faster and more robust convergence when solving large instances.

5. Extensions to House Allocation and Matching

By adapting the reduced cyclic-coalition stability mechanism, the same framework provides a formulation for maximum-cardinality Pareto optimal matchings in the Capacitated House Allocation (CHA) problem. In this context, all facilities are considered always open, and the objective is to maximize the number of customers served subject to Pareto optimality—again corresponding to cyclic-coalition stability.

This cross-applicability highlights the generality of the modeling approach and its close connection to allocation problems in matching theory where stability against cyclic (groupwise) deviations is synonymous with global non-domination.

6. Distinctions from Prior Stability Concepts

Cyclic-coalition stable allocations strictly refine individual and pairwise stability. While customer stability only guards against unilaterally dissatisfied agents, and pairwise stability blocks only bilateral improvement swaps, cyclic-coalition stability eliminates all possible improving cycles or groupwise reallocations involving strictly more than two customers.

A key implication is that allocations deemed stable under weaker concepts may still be dominated by a Pareto optimal (cyclic-coalition stable) allocation. This highlights a vulnerability to group manipulations or sequencing of deviations in systems governed only by customer or pairwise stability.

7. Bias Avoidance and Preference Representation

The proposed models directly encode agents' ordinal preferences in the constraints, circumventing the need for arbitrary numeric preference scores and avoiding biases inherent in such cardinal representations. This fidelity ensures the solution's stability properties are properly aligned with stakeholders' reported priorities rather than artifacts of their numerical encoding (Domínguez et al., 29 Jul 2025).

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In conclusion, cyclic-coalition stable allocations, as formalized in the context of capacitated facility location and allocation problems, impose a stringent form of groupwise Pareto efficiency and non-blocking under cycles, realized efficiently via advanced MILP formulations. These models set a new standard for fair, robust allocation in complex matching markets and extend naturally, via direct modeling of cyclic-blocking, to applications in capacitated matching, school choice, and housing assignments.