On the probabilities of the mutual agreement match (Q1098152)

A set of n men is to be matched to a set of n women in marriage, subject to the mutual agreement of the individuals to be married. Each agent lists the available members of the opposite sex in descending order of preference and accepts to marry only the member ranked as his (resp. her) first choice. Whenever all these agents match on their own, the resulting match is called a mutual agreement match. We calculate the probability \(p_ n\) that such a match occurs by deriving a recursion equation which the \(p_ n's\) satisfy, \(n=1,2,...\), provided certain stochastic assumptions on the preferences of the agents are satisfied, and show that \(p_ n\to 0\) as \(n\to \infty\).