43 research outputs found
Unbalanced Random Matching Markets with Partial Preferences
Properties of stable matchings in the popular random-matching-market model
have been studied for over 50 years. In a random matching market, each agent
has complete preferences drawn uniformly and independently at random. Wilson
(1972), Knuth (1976) and Pittel (1989) proved that in balanced random matching
markets, the proposers are matched to their th choice on average. In
this paper, we consider markets where agents have partial (truncated)
preferences, that is, the proposers only rank their top partners. Despite
the long history of the problem, the following fundamental question remained
unanswered: \emph{what is the smallest value of that results in a perfect
stable matching with high probability?} In this paper, we answer this question
exactly -- we prove that a degree of is necessary and sufficient.
That is, we show that if then no stable matching is
perfect and if , then every stable matching is
perfect with high probability. This settles a recent conjecture by Kanoria, Min
and Qian (2021).
We generalize this threshold for unbalanced markets: we consider a matching
market with agents on the shorter side and agents on the
longer side. We show that for markets with , the sharp threshold
characterizing the existence of perfect stable matching occurs when is .
Finally, we extend the line of work studying the effect of imbalance on the
expected rank of the proposers (termed the ``stark effect of competition''). We
establish the regime in unbalanced markets that forces this stark effect to
take shape in markets with partial preferences
A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings
Stable matching is a classical combinatorial problem that has been the
subject of intense theoretical and empirical study since its introduction in
1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new
upper bound on , the maximum number of stable matchings that a stable
matching instance with men and women can have. It has been a
long-standing open problem to understand the asymptotic behavior of as
, first posed by Donald Knuth in the 1970s. Until now the best
lower bound was approximately , and the best upper bound was . In this paper, we show that for all , for some
universal constant . This matches the lower bound up to the base of the
exponent. Our proof is based on a reduction to counting the number of downsets
of a family of posets that we call "mixing". The latter might be of independent
interest
An Approximate "Law of One Price" in Random Assignment Games
Assignment games represent a tractable yet versatile model of two-sided
markets with transfers. We study the likely properties of the core of randomly
generated assignment games. If the joint productivities of every firm and
worker are i.i.d bounded random variables, then with high probability all
workers are paid roughly equal wages, and all firms make similar profits. This
implies that core allocations vary significantly in balanced markets, but that
there is core convergence in even slightly unbalanced markets. For the
benchmark case of uniform distribution, we provide a tight bound for the
workers' share of the surplus under the firm-optimal core allocation. We
present simulation results suggesting that the phenomena analyzed appear even
in medium-sized markets. Finally, we briefly discuss the effects of unbounded
distributions and the ways in which they may affect wage dispersion
Approximately Stable, School Optimal, and Student-Truthful Many-to-One Matchings (via Differential Privacy)
We present a mechanism for computing asymptotically stable school optimal
matchings, while guaranteeing that it is an asymptotic dominant strategy for
every student to report their true preferences to the mechanism. Our main tool
in this endeavor is differential privacy: we give an algorithm that coordinates
a stable matching using differentially private signals, which lead to our
truthfulness guarantee. This is the first setting in which it is known how to
achieve nontrivial truthfulness guarantees for students when computing school
optimal matchings, assuming worst- case preferences (for schools and students)
in large markets
Stable Secretaries
We define and study a new variant of the secretary problem. Whereas in the
classic setting multiple secretaries compete for a single position, we study
the case where the secretaries arrive one at a time and are assigned, in an
on-line fashion, to one of multiple positions. Secretaries are ranked according
to talent, as in the original formulation, and in addition positions are ranked
according to attractiveness. To evaluate an online matching mechanism, we use
the notion of blocking pairs from stable matching theory: our goal is to
maximize the number of positions (or secretaries) that do not take part in a
blocking pair. This is compared with a stable matching in which no blocking
pair exists. We consider the case where secretaries arrive randomly, as well as
that of an adversarial arrival order, and provide corresponding upper and lower
bounds.Comment: Accepted for presentation at the 18th ACM conference on Economics and
Computation (EC 2017