In which matching markets does the short side enjoy an advantage?

Kanoria, Yash; Min, Seung‐Ki; Qian, Pengyu

doi:10.1137/1.9781611976465.83

Cited by 7 publications

(1 citation statement)

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“…A recent and unexpected observation in [2] was the "stark effect of competition": that in the random preferences model the short side, whether it was the proposing side or not, was the one to enjoy the Θ(ln n) rank matches. Subsequent work showed that this effect disappeared with short preference lists in a natural modification of the random preferences model [11]. Our work suggests yet another explanation for why this effect may not be present: it does not require that short preference lists be imposed as an external constraint, but rather that the preference model generates few edges that might ever be in a stable match.…”

Section: Introductionmentioning

confidence: 52%

Stable Matching: Choosing Which Proposals to Make

Agarwal¹,

Cole²

2022

Preprint

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Why does stable matching work well in practice despite agents only providing short preference lists? Perhaps the most compelling explanation is due to Lee [17]. He considered a model with preferences based on correlated cardinal utilities. Specifically, the utilities were based on common public values for each agent and individual private values. He showed that for suitable utility functions, in large markets, for most agents, all stable matches yield similar valued utilities.By means of a new analysis, we strengthen this result, showing that in large markets, with high probability, for all but the bottommost agents, all stable matches yield similar valued utilities. We can then deduce that for all but the bottommost agents, relatively short preference lists suffice.Our analysis shows that the key distinction is between models in which the derivatives of the utility function are bounded, and those in which they can be unbounded. An important instance of the first model is the linear separable model, in which the utility function is a sum of public and private values. For the bounded derivative model, we show that for any given constant c ≥ 1, with n agents on each side of the market, with probability 1 − n −c , for each agent its possible utilities in all stable matches vary by at most O((c ln n/n) 1/3 ) for all but the bottommost O((c ln n/n) 1/3 ) fraction of the agents. This result is tight, even for the linear model. When the derivatives can be unbounded, we obtain the following bound on the utility range and the bottommost fraction: for any constant > 0, for large enough n, the bound is . This bound too cannot be improved in general. These bounds continue to hold when the two sides have unequal numbers of agents. Thus, in this model, the "stark effect of competition", meaning that the agents on the short side fare much better, which is observed in the standard random preferences model, is limited to a lower portion of the agents on the longer side.We extend the study to many-to-one settings, e.g. employers and workers, where each employer has the same fixed number d of positions. Qualitatively, the results are unchanged. However, the outcomes are no longer symmetric. The range of utilities for employers is unchanged for small d, while the range for workers is increased, compared to the one-to-one setting. We completely characterize these outcomes as a function of employer size.In the bounded derivative model, we also show the existence of an -Bayes-Nash equilibrium in which agents make relatively few proposals. Specifically, there is an equilibrium in which no agent proposes more than O(ln 2 n) times, and all but the bottommost O((ln n/n) 1/3 ) fraction of the agents make only O(ln n) proposals, and = O(ln n/n 1/3 ).These results all rely on a new technique for sidestepping the conditioning between the matching events that occur over the course of a run of the Deferred Acceptance algorithm.We complement these theoretical results with a variety of simulation experiments.

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Section: Introductionmentioning

confidence: 52%