Yankee Swap: a Fast and Simple Fair Allocation Mechanism for Matroid Rank Valuations

We study fair allocation of indivisible goods when agent valuations are matroid rank functions (MRFs). Our main contribution is a simple algorithm based on the colloquial Yankee Swap procedure that computes provably fair and efficient Lorenz dominating allocations. While there exist polynomial time algorithms to compute fair and efficient allocations for MRF valuations, we improve on them in two ways: (a) Our approach is easy to understand and does not use complex matroid optimization algorithms as subroutines. (b) Our approach is scalable; it is provably faster than all known algorithms to compute Lorenz dominating allocations. These two properties are key to the adoption of algorithms in any real fair allocation setting; our contribution brings us one step closer to this goal.

💡 Research Summary

The paper addresses the problem of fairly allocating indivisible goods when agents’ valuations are matroid rank functions (MRFs), a class of binary submodular valuations that capture many real‑world constraints such as course‑schedule conflicts or capacity limits. Existing polynomial‑time algorithms for MRFs (e.g., Babaioff et al. 2021) can compute allocations that are simultaneously leximin, EFX, max‑utilitarian social welfare (MAX‑USW), and max‑Nash social welfare (MNW), but they rely on sophisticated matroid‑optimization subroutines and have high polynomial exponents, making them impractical for large instances.

The authors propose a new, conceptually simple algorithm inspired by the colloquial “Yankee Swap” (also known as “Nasty Christmas” or “White Elephant”). The algorithm proceeds in rounds according to a fixed priority ordering π over agents. Initially every agent holds an empty bundle. In each round the current agent first tries to pick an unassigned good that yields marginal utility 1. If none exists, the agent initiates a transfer path: a sequence of steals where each intermediate agent gives up a good of marginal utility 1 and receives another good, preserving its own utility, while the initiator’s utility increases by one. The key technical contribution is a combinatorial proof that such a transfer path always exists for the agent with the lowest utility (Lemma 3.1) and that once no more transfer paths can be created, the remaining agents’ bundle sizes are almost equal (Lemma 3.4). These properties guarantee that the final allocation is Lorenz‑dominating (Theorem 3.5), which in the MRF setting is known to coincide with leximin, EFX, MAX‑USW, MNW, and ½‑MMS (Theorem 2.2).

To compute transfer paths efficiently, the authors construct a good‑exchange graph whose vertices are goods and directed edges represent feasible exchanges (an edge from g to h exists if the owner of g values h with marginal utility 1). Finding a shortest path from a good owned by the lowest‑utility agent to any unassigned good yields a valid transfer path (Algorithm 2, Theorem 3.9). The overall runtime is (O((n+m)(n+\tau)m^{2})), where τ is the time to evaluate a valuation on a bundle. This improves dramatically over the previous best bound (O(n^{6}m^{7/2}(m+\tau)\log nm)). When the number of goods m is Θ(n), the algorithm also outperforms matroid‑intersection based methods for computing MAX‑USW.

Beyond the deterministic guarantee, randomizing over the priority ordering π provides ex‑ante envy‑freeness and proportionality, because all allocations that share the same sorted utility vector become equally likely, equalizing expected utilities. The authors are currently implementing Yankee Swap for course‑section assignment at the University of Massachusetts Amherst, collaborating with the registrar’s office and the CS department, and report promising preliminary performance gains.

In summary, Yankee Swap offers a fast, easy‑to‑implement, and theoretically optimal mechanism for fair allocation under matroid rank valuations. It eliminates the need for heavy matroid optimization, achieves the strongest known fairness and efficiency guarantees (leximin, EFX, MNW, MAX‑USW, ½‑MMS), and runs in significantly lower polynomial time, making it a strong candidate for real‑world deployment in settings such as university course allocation, resource scheduling, and other combinatorial assignment problems.