## The Annals of Probability

- Ann. Probab.
- Volume 21, Number 3 (1993), 1441-1477.

### The "Stable Roommates" Problem with Random Preferences

#### Abstract

In a set of even cardinality $n$, each member ranks all the others in order of preference. A stable matching is a partition of the set into $n/2$ pairs, with the property that no two unpaired members both prefer each other to their partners under matching. It is known that for some problem instances no stable matching exists. What if an instance of the ranking system is chosen uniformly at random? We show that the mean and the variance of the total number of stable matchings for the random problem instance are asymptotic to $e^{1/2}$ and $(\pi n/4e)^{1/2}$, respectively. Consequently, $\operatorname{Prob}$ a stable matching exists) $\gtrsim (4e^3/\pi n)^{1/2}$. We also prove that, given the last event, in every stable matching the sum of the ranks of all members (as rank ordered by their partners) is asymptotic to $n^{3/2}$, and the largest rank of a partner is of order $n^{1/2}\log n$, with superpolynomially high conditional probability. In other words, stable partners are very likely to be relatively close to the tops of each other's preference lists.

#### Article information

**Source**

Ann. Probab., Volume 21, Number 3 (1993), 1441-1477.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989126

**Digital Object Identifier**

doi:10.1214/aop/1176989126

**Mathematical Reviews number (MathSciNet)**

MR1235424

**Zentralblatt MATH identifier**

0778.60005

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 60F99: None of the above, but in this section 05A05: Permutations, words, matrices 05C70: Factorization, matching, partitioning, covering and packing 05C80: Random graphs [See also 60B20] 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)

**Keywords**

stable matching random preferences asymptotic mean variance ranks order conditional probability

#### Citation

Pittel, Boris. The "Stable Roommates" Problem with Random Preferences. Ann. Probab. 21 (1993), no. 3, 1441--1477. doi:10.1214/aop/1176989126. https://projecteuclid.org/euclid.aop/1176989126