Ex-post Stability under Two-Sided Matching: Complexity and Characterization

A probabilistic approach to the stable matching problem has been identified as an important research area with several important open problems. When considering random matchings, ex-post stability is a fundamental stability concept. A prominent open problem is characterizing ex-post stability and establishing its computational complexity. We investigate the computational complexity of testing ex-post stability. Our central result is that when either side has ties in the preferences/priorities, testing ex-post stability is NP-complete. The result even holds if both sides have dichotomous preferences. On the positive side, we give an algorithm using an integer programming approach, that can determine a decomposition with a maximum probability of being weakly stable. We also consider stronger versions of ex-post stability (in particular robust ex-post stability and ex-post strong stability) and prove that they can be tested in polynomial time.

💡 Research Summary

The paper investigates the computational problem of verifying whether a given random (fractional) matching can be implemented as a lottery over weakly stable deterministic matchings – a property known as ex‑post stability. This question is central to many real‑world allocation mechanisms (school choice, resident matching, housing allocation) where each realized outcome must satisfy stability for fairness. The authors first formalize the model: there are n agents and n items, each agent i has a preference relation ≿_i over items, each item o has a priority relation ≿_o over agents. Preferences and priorities may contain ties; a dichotomous relation partitions the other side into at most two indifference classes. A random matching p is an n×n bistochastic matrix, and by the Birkhoff–von Neumann theorem any such matrix can be expressed as a convex combination of deterministic matchings. Ex‑post stability requires that all matchings in the decomposition be weakly stable (i.e., no blocking pair where an agent prefers an item and the item prefers the agent over its current match).

When preferences and priorities are strict, ex‑post stability coincides with fractional stability, and a simple linear‑time test exists (via checking linear constraints (2.4)). Moreover, any fractionally stable matching can be decomposed efficiently into deterministic stable matchings. The paper’s main contribution, however, is to show that as soon as ties appear, the verification problem becomes computationally intractable. The authors prove three NP‑completeness results:

1. Strict agents / dichotomous items – even if agents have strict preferences while items have only two‑level priorities, deciding ex‑post stability is NP‑complete.
2. Both sides dichotomous – when both agents and items have dichotomous preferences/priorities, the problem remains NP‑complete.
3. Bounded list length (≤3) – even if every preference and priority list contains at most three items/agents, the problem is NP‑complete.

The reductions are from classic NP‑complete problems such as Exact Cover by 3‑Sets or 3‑SAT. The construction maps variables and clauses to agents and items, using ties to encode the logical structure so that a feasible ex‑post‑stable decomposition exists if and only if the original formula is satisfiable. These results demonstrate that the difficulty persists under highly restricted and practically relevant settings (short lists, binary classifications).

Beyond the basic notion, the authors explore two stronger variants:

• Ex‑post strong stability – the decomposition must consist of strongly stable deterministic matchings (no weakly blocking pair where at least one side strictly prefers the deviation). Strong stability is stricter but, crucially, checking whether a strongly stable matching exists can be done in linear time (e.g., Irving’s algorithm). Consequently, verifying ex‑post strong stability is polynomial.

• Robust ex‑post stability – every possible decomposition of the random matching must consist solely of weakly stable matchings. This property can be tested by simply verifying the fractional stability inequalities (2.4), which is again polynomial.

Thus, while the basic ex‑post stability problem is hard, its robust and strong counterparts are tractable.

Recognizing that NP‑completeness does not preclude practical solutions, the paper proposes an integer programming (IP) formulation to compute a “best‑possible” decomposition when exact ex‑post stability fails. Variables λ_j represent the weight of each deterministic matching M_j in the convex combination. Constraints enforce λ_j ≥ 0, Σ λ_j = 1, and that each selected M_j satisfies the weak stability inequalities. The objective minimizes a measure of instability, such as the total violation of the stability constraints or the probability of selecting an unstable matching. This IP can be solved with modern MILP solvers for moderate‑size instances (hundreds of agents/items), offering a pragmatic tool for designers who need to approximate ex‑post stability.

The paper situates its contributions within a rich literature: classic stable marriage theory, the impact of ties on computational hardness, random matching models (Bogomolnaia & Moulin), and recent work on ex‑ante stability, claimwise stability, and Pareto optimality for random assignments. It resolves an open question posed by Kesten and Ünver (2020) regarding the characterization and complexity of ex‑post stability.

In summary, the authors establish that verifying ex‑post stability is NP‑complete under realistic tie structures, while robust and strong versions are polynomially solvable. They complement the hardness results with an integer‑programming approach that can find a decomposition minimizing instability, thereby bridging theory and practice. The findings have direct implications for the design of fair randomized allocation mechanisms, suggesting that practitioners either restrict to the tractable strong/robust notions or employ optimization heuristics when full ex‑post stability cannot be guaranteed.