[Paper Review] Stable schedule matchings by a fixed point method

Vilmos Komornik, Zsolt Komornik+1 more|arXiv (Cornell University)|May 12, 2010

Game Theory and Voting Systems13 references3 citations

TL;DR

This paper generalizes stable schedule matching by introducing a fixed point method based on Fleiner's theorem, enabling the construction of revealing choice maps that ensure stable, worker- or firm-optimal allocations under complex scheduling and preference constraints. The key contribution is a flexible, algorithmic framework that extends prior models to handle part-time work, overlapping schedules, and capacity constraints via a consistent, revealing choice map construction.

ABSTRACT

We generalize several schedule matching theorems of Baiou-Balinski (Math. Oper. Res., 27 (2002), 485) and Alkan-Gale (J. Econ. Th. 112 (2003), 289) by applying a fixed point method of Fleiner (Math. Oper. Res., 28 (2003), 103). Thanks to a more general construction of revealing choice maps we develop an algorithm to solve rather complex matching problems. The flexibility and efficiency of our approach is illustrated by various examples. We also revisit the mathematical structure of the matching theory by comparing various definitions of stable sets and various classes of choice maps. We demonstrate, by several examples, that the revealing property of the choice maps is the most suitable one to ensure the existence of stable matchings; both from the theoretical and the practical point of view.

Motivation & Objective

• To extend stable matching theory to complex scheduling problems involving part-time work, overlapping days, and capacity constraints on both workers and firms.
• To develop a unified framework that generalizes Baiou–Balinski and Alkan–Gale models by incorporating preference and scheduling constraints into a single formalism.
• To demonstrate the existence and constructibility of worker- or firm-optimal stable matchings using a fixed point approach.
• To clarify the role of the 'revealing' property in choice maps for ensuring stable matching existence, both theoretically and practically.

Proposed method

• Adapts Fleiner’s fixed point theorem to construct stable matchings via a generalized choice map construction that respects scheduling and preference constraints.
• Introduces a revealing choice map construction using disjoint subsets of acceptable matches (e.g., firms, days, quotas), ensuring consistency and stability.
• Applies a step-by-step iterative algorithm to compute the fixed point of the choice map, yielding a stable allocation of workers to firms across time slots.
• Uses a quota-based filtering mechanism where each agent selects from acceptable combinations of firms and days, constrained by individual and institutional limits.
• Employs set-theoretic conditions (e.g., |C_k(A) ∩ Y_n| ≤ q_n) to enforce capacity and preference constraints during the fixed point iteration.
• Demonstrates that consistent choice maps satisfying the revealing property yield stable matchings, with optimality (worker- or firm-optimal) derivable via algorithmic construction.

Experimental results

Research questions

• RQ1 Can stable schedule matchings be guaranteed under complex scheduling constraints, such as part-time work across multiple firms on the same day?
• RQ2 What properties of choice maps are necessary and sufficient to ensure the existence of stable matchings in generalized schedule matching models?
• RQ3 How can worker- or firm-optimal stable matchings be algorithmically constructed in settings with overlapping work schedules and capacity limits?
• RQ4 To what extent does the 'revealing' property of choice maps outperform other properties (e.g., consistency) in ensuring stable matching existence?
• RQ5 How do the theoretical results of Baiou–Balinski and Alkan–Gale extend to models with incomplete preferences and dynamic scheduling constraints?

Key findings

• The fixed point method based on Fleiner’s theorem guarantees the existence of stable matchings in generalized schedule matching problems with complex constraints.
• The revealing property of choice maps is both theoretically and practically optimal for ensuring stable matching existence, outperforming other properties like consistency alone.
• A worker-optimal or firm-optimal stable matching can be algorithmically computed via iterative construction of the fixed point of a revealing choice map.
• The construction of choice maps using disjoint subsets of acceptable matches (e.g., Y_n) ensures that capacity constraints (q_n) and total quotas (q) are respected at each step.
• Counterexamples show that without the disjointness condition in the choice map construction, stable matchings may fail to exist, even with consistent maps.
• The framework generalizes and subsumes prior results by Baiou–Balinski and Alkan–Gale, extending them to settings with incomplete preferences and dynamic scheduling.