Stable Matching: Choosing Which Proposals to Make
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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 (Lee, 2016). He considered a model with preferences based on correlated cardinal utilities. 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 matchings 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. 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((cln n/n)^1/3) for all but the bottommost O((cln n/n)^1/3) fraction of the agents. When the derivatives can be unbounded, we obtain the following bound on the utility range and the bottommost fraction: for any constant ϵ>0, for large enough n, the bound is ϵ. Both these bounds are tight. In the bounded derivative model, we also show the existence of an ϵ-Bayes-Nash equilibrium in which agents make relatively few proposals. Our 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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