Mechanism Design without Money via Stable Matching

Mechanism design without money has a rich history in social choice literature. Due to the strong impossibility theorem by Gibbard and Satterthwaite, exploring domains in which there exist dominant strategy mechanisms is one of the central questions in the field. We propose a general framework, called the generalized packing problem (\gpp), to study the mechanism design questions without payment. The \gpp\ possesses a rich structure and comprises a number of well-studied models as special cases, including, e.g., matroid, matching, knapsack, independent set, and the generalized assignment problem. We adopt the agenda of approximate mechanism design where the objective is to design a truthful (or strategyproof) mechanism without money that can be implemented in polynomial time and yields a good approximation to the socially optimal solution. We study several special cases of \gpp, and give constant approximation mechanisms for matroid, matching, knapsack, and the generalized assignment problem. Our result for generalized assignment problem solves an open problem proposed in \cite{DG10}. Our main technical contribution is in exploitation of the approaches from stable matching, which is a fundamental solution concept in the context of matching marketplaces, in application to mechanism design. Stable matching, while conceptually simple, provides a set of powerful tools to manage and analyze self-interested behaviors of participating agents. Our mechanism uses a stable matching algorithm as a critical component and adopts other approaches like random sampling and online mechanisms. Our work also enriches the stable matching theory with a new knapsack constrained matching model.

💡 Research Summary

The paper tackles the challenging setting of mechanism design without monetary transfers, where agents must be incentivized to report their private information truthfully while the designer seeks to maximize social welfare. Recognizing the impossibility of dominant‑strategy truthful mechanisms in unrestricted domains (Gibbard‑Satterthwaite), the authors restrict the environment by introducing the Generalized Packing Problem (GPP). In GPP there is a public set of items A, each with a known value v_j, and n agents each privately holding a subset A_i of items. The mechanism must select a feasible collection of items S_i ⊆ A_i under a public feasibility constraint (e.g., matroid independence, graph matching, knapsack capacity, or the Generalized Assignment Problem – GAP). Two demand models are considered: unit (at most one item per agent) and multi‑unit (no per‑agent limit beyond the global feasibility).

The central technical contribution is the use of stable matching theory as a design tool for truthful, approximation‑guaranteed mechanisms in this multi‑parameter, money‑free setting. The authors adapt the Gale‑Shapley deferred‑acceptance algorithm so that jobs (or items) propose to machines (or bins), defining appropriate preference orders for both sides. For many‑to‑one settings (machines can accept multiple jobs) the machines’ preferences are over subsets of jobs, not just individual jobs. When the stable matching algorithm is run on the full set of jobs, the resulting assignment A* provides a structural upper bound on the optimal solution because any unmatched job‑machine pair would constitute a blocking pair, contradicting stability.

For several special cases (matroid‑unit, matching‑unit) a simple greedy algorithm already yields a deterministic, truthful mechanism with constant‑factor approximations (2‑approx for matroid‑unit, 3‑approx for matching‑unit). However, for knapsack and GAP the greedy approach fails both in approximation quality and truthfulness. To overcome this, the authors devise a random‑sampling + online‑allocation framework:

1. Randomly select half of the jobs to form a test set T.
2. Run the stable‑matching algorithm on T, obtaining a stable assignment A_T. With high probability, A_T closely mirrors the structure of the optimal solution.
3. Process the remaining jobs in a predetermined order, using A_T as a guide to decide where to place each job. The decision rule is designed so that any deviation in a job’s reported compatibility cannot improve its expected utility, ensuring universal truthfulness (truthfulness holds for every random coin flip, not just in expectation).

This two‑phase mechanism achieves a constant‑factor approximation for both the knapsack‑unit/mul and GAP‑unit/mul problems, improving upon the prior best results. Notably, for GAP the authors resolve an open question from Dughmi and Ghosh (2015), who only obtained a logarithmic‑approximation truthful‑in‑expectation mechanism. The new mechanism is universally truthful, a strictly stronger guarantee.

The analysis leverages key properties of stable matchings: every unassigned pair is blocked by at least one side’s current assignment, allowing the authors to bound the loss relative to the optimal solution. By comparing the three assignments—A* (stable on all jobs), A_T (stable on the sample), and the final online assignment A—they show that each is within a constant factor of the optimum, yielding the desired approximation bound.

Beyond the algorithmic results, the paper contributes to theory by introducing a knapsack‑constrained many‑to‑one matching model into stable‑matching literature, expanding the applicability of stable‑matching concepts to combinatorial optimization problems with capacity constraints.

In summary, the paper presents a unified framework (GPP) for a broad class of combinatorial allocation problems without money, demonstrates how stable matching can be harnessed to design truthful mechanisms, and provides constant‑approximation, polynomial‑time mechanisms for matroid, matching, knapsack, and the generalized assignment problem—thereby advancing the frontier of approximate mechanism design without monetary transfers.