When 20 single women show up but only 19 single men, one woman is going to go home empty-handed. The odd woman will make the obvious rational decision to muscle in on an existing pairing, offering a deal better than a 50-50 split, perhaps accepting only 40 dollars. Her rival, being a rational sort, will counter-bid, maybe offering just 30 dollars. ... The bids will fall until the woman who faces leaving alone is offering to walk through the checkout with some luck guy and accept 1 cent as the price. He will get $99.99. She will get 1 cent, but it is still better than nothing.
The trouble doesn’t end there. No matter what deals are agreed, there will always one girl left over, offering to pair up for just 1 cent. One cent is what all of them will get: anyone on the verge of offering a better offer will be undercut. The 19 men will all get $99.99. 19 women will get 1 cent each, and the last woman will get nothing.
A shortage of just one man gives all the other men massive scarcity power.
In reality, the bargaining process is probably not as calculating, and it is not as easy for suitors to bid against one another as marriage prospects. But it is just as brutal.