Matching Games with Additive Externalities

Two-sided matchings are an important theoretical tool used to model markets and social interactions. In many real life problems the utility of an agent is influenced not only by their own choices, but also by the choices that other agents make. Such an influence is called an externality. Whereas fully expressive representations of externalities in matchings require exponential space, in this paper we propose a compact model of externalities, in which the influence of a match on each agent is computed additively. In this framework, we analyze many-to-many and one-to-one matchings under neutral, optimistic, and pessimistic behaviour, and provide both computational hardness results and polynomial-time algorithms for computing stable outcomes.

💡 Research Summary

The paper tackles a fundamental limitation of classical two‑sided matching theory: it assumes that an agent’s utility depends solely on the partner she is matched with, ignoring the fact that other agents’ matches can affect her payoff. To capture such “externalities” without incurring exponential blow‑up, the authors introduce a compact additive model. In this framework each agent i’s utility is the sum of (1) a direct value v_i(j) from being matched with partner j and (2) a set of externality weights w_i^{jk} that quantify how the formation of any other pair (j,k) influences i. Because all externalities are additive, the whole input can be described by O(n²) numbers, a dramatic reduction from the naïve exponential representation.

Stability, the central solution concept in matching markets, becomes ambiguous when externalities are present. The paper therefore distinguishes three behavioral attitudes that agents may adopt when evaluating potential deviations:

• Neutral – agents consider only the externalities already present in the current matching; future or hypothetical externalities are ignored.
• Optimistic – agents assume the most favorable externalities will materialize for them, i.e., they evaluate a deviation under the best‑case scenario.
• Pessimistic – agents assume the worst‑case externalities, making the stability test the most stringent.

The authors study both many‑to‑many and one‑to‑one matching environments under each attitude.

Many‑to‑many matchings.
For neutral and pessimistic agents the problem of deciding whether a stable (core) outcome exists is shown to be NP‑complete. The hardness proofs reduce from classic NP‑hard problems (e.g., 3‑SAT), demonstrating that even checking existence is intractable. Moreover, when a stable outcome does exist, finding one remains NP‑hard. In contrast, under optimistic behavior the problem collapses to a maximum‑weight independent‑set problem on a suitably constructed graph. Because all externalities are non‑negative, the authors can transform the instance into a weighted bipartite matching problem and solve it in polynomial time using Edmonds’ blossom algorithm or similar techniques.

One‑to‑one matchings (stable marriage setting).
With neutral agents the authors adapt the Gale‑Shapley deferred‑acceptance algorithm: they augment each agent’s preference list with the current additive externalities and show that the resulting process still converges to a stable matching in polynomial time. For optimistic agents the stability condition is equivalent to maximizing the total additive utility of the whole market. This is precisely the classic assignment problem, solvable in O(n³) time by the Hungarian algorithm. The pessimistic case, however, requires evaluating the worst‑case externalities for any deviation, which the authors prove to be NP‑complete via a reduction from the Partition problem. Consequently, only exponential‑time or approximation algorithms are feasible in the general case.

The paper’s algorithmic contributions are twofold. First, the additive externality model linearizes otherwise complex interdependencies, enabling the reuse of well‑studied graph‑matching algorithms. Second, the systematic complexity classification across the three behavioral attitudes clarifies when efficient computation is possible and when it is provably hard. The authors argue that many real‑world markets (e.g., labor platforms where a colleague’s employment affects one’s productivity, or online dating services where friends’ matches influence user satisfaction) can be modeled within this framework, and that the choice of behavioral assumption has concrete computational implications for market designers.

In the related‑work discussion the authors contrast their approach with prior literature that either ignores externalities, assumes a full exponential utility table, or restricts externalities to very specific forms (e.g., only depending on the number of matched partners). By proposing a compact additive representation and providing a thorough hardness/algorithmic analysis, the paper advances the theory of matching markets with externalities and opens several avenues for future research, such as non‑additive externalities, dynamic or time‑varying effects, and multi‑dimensional externalities (cost, time, trust) that may require richer models.