Random Popular Matchings with Incomplete Preference Lists
Suthee Ruangwises, Toshiya Itoh

TL;DR
This paper studies the probability of the existence of popular matchings in random incomplete preference lists, revealing phase transitions depending on the ratio of items to people and list length.
Contribution
It extends previous work on complete lists to incomplete lists with fixed length, identifying new phase transition thresholds for the existence of popular matchings.
Findings
Phase transition occurs at a specific ratio 2 for complete lists.
For incomplete lists of fixed length , a similar phase transition is identified.
The threshold depends on the list length and is characterized by a specific equation.
Abstract
Given a set of people and a set of items, with each person having a list that ranks his/her preferred items in order of preference, we want to match every person with a unique item. A matching is called popular if for any other matching , the number of people who prefer to is not less than the number of those who prefer to . For given and , consider the probability of existence of a popular matching when each person's preference list is independently and uniformly generated at random. Previously, Mahdian showed that when people's preference lists are strict (containing no ties) and complete (containing all items in ), if , where is the root of equation , then a popular matching exists with probability ; and if , then a popular matching…
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