Tim Harford's "Marriage supermarket" from "The logic of life"

I have read Tim's "Undercover Economist" (In its Hebrew translation) and enjoyed it very much so I was happy to find (again the Hebrew translation of) "The Logic of Life" at the book store.

NOTE: In some of what I am about to write I am translating back to English from the Hebrew translation of the book, I hope it will not be far from the original.

I am still reading it and I just read the part about the "marriage supermarket" in chapter 3.

In this part Tim describes the following scenario:
20 single males and 20 single females are in a room, the "marriage supermarket", each pair that decides to "get married" gets paid 100 Pounds. (They don't actually get married they only pair up and are not required to do anything else).
Tim goes on and explains that when there is an equal number singles from both genders We should expect the money to be divided 50%-50%. (This does sound reasonable).

But then Tim Harford goes on and asks what will happen if on one night on of the males does not show for some reason. Tim's analysis suggest that after 19 pairs will be formed the last woman that could currently expect to get 0 Pounds (because she does not have someone to "get married to" should/will suggest to one of the males that he abandons his current "wife" and take her instead, for this she will promise him that she would be satisfied with receiving only 40 pounds. This would leave another woman at a disadvantage and cause he to make a similar suggestion, a process that will continue until 19 of the females will get 1 penny and the 20th will go home even without that single penny when the man will each get 99.99 Pounds each.

Well, I think that this will not be what will actually happen.
If you assume that all the females know what they are doing the 20th should talk to the other 19 and tell them something along the following lines:
"Listen Girls, We can either start a bidding war and end up with a penny each or we can split the 19*50 we can get from the males and share it among us leaving us each with 47.5 pounds each".

I believe this will/should be the stable solution and it s a much better one for the females involved.

More that that - If this was not the case - then it would be a good idea for the group of man to decide to leave one of them at home to begin with (hopeing that the women did not think about it as well).

The only reason to accept Tim's solution IMHO would be if this kind of money sharing among the women would just not be allowed in the "GAME".