Consider the standard Diamonds coconut economy. One could imagine that people who search longer, have a different likelihood of matching someone to trade with.
Has there been something done akin this (in any part of matching theory)?
Second Example Say, we want to match liver transplants between donors $d$ and recipients $r$. The longer anyone waits, the higher the likelihood of being matched with someone from the other side.
Hence, we would have a distribution of $r$ and $d$ w.r.t the time that they have already been waiting. And instead of an aggregate matching function $M(r, d)$ we would have something that takes into account these distributions.
What has been done in this area so far?