For a set A of n applicants and a set I of m items, we consider a problem of computing a matching of applicants to items, i.e., a function M mapping A to I; here we assume that each applicant x∈A provides a preference list on items in I. We say that an applicant x∈A prefers an item p than an item q if p is located at a higher position than q in its preference list, and we say that x prefers a matching M over a matching M' if x prefers M(x) over M'(x). For a given matching problem A, I, and preference lists, we say that M is more popular than M' if the number of applicants preferring M over M' is larger than that of applicants preferring M' over M, and M is called a popular matching if there is no other matching that is more popular than M. Here we consider the situation that A is partitioned into A1,A2,...,Ak, and that each Ai is assigned a weight wi>0 such that w_{1}>w_{2}>...>w_{k}>0.Forsuchamatchingproblem,wesaythatMismorepopularthanM′ifthetotalweightofapplicantspreferringMoverM′islargerthanthatofapplicantspreferringM′overM,andwecallMank−weightedpopularmatchingifthereisnoothermatchingthatismorepopularthanM.Inthispaper,weanalyzethe2−weightedmatchingproblem,andweshowthat(lowerbound)ifm/n^{4/3}=o(1),thenarandominstanceofthe2−weightedmatchingproblemwithw_{1} \geq 2w_{2}hasa2−weightedpopularmatchingwithprobabilityo(1);and(upperbound)ifn^{4/3}/m = o(1),thenarandominstanceofthe2−weightedmatchingproblemwithw_{1} \geq 2w_{2}$ has a 2-weighted popular matching with probability 1-o(1).
