Stanford professor Al Roth recently gave a market design speech that focused on matchmaking and unusual exchange markets. Roth discusses common matchmaking markets and uses applying to Stanford as an example. While one can apply to Stanford, they can't just show up. People have to be accepted by Stanford before the exchange is complete. Roth also describes dating and marriage as a matchmaking market. One can't just date whoever they want to, because relationships are mutual.
Al Roth spends a great deal of time talking about the kidney exchange as being an unusual market. It is illegal for organs to be exchanged for money or valuables, however people often work out a different kind of exchange when it comes to kidney transplants. If one needs a kidney, but their spouse is not a match, they find someone else on the kidney transplant list who is compatible with their spouses kidney and arrange a kind of trade.
Roth discusses the way in which people don't think of certain things, such as marriage or kidney exchanges as markets, but they are.