The math of love

Yesterday, we looked a sampling problem for boy-warriors. There is a sort of similar problem in statistics where you are trying to get the best product and need to know when to stop trying and settle down with good-enough rather than spending the rest of your life looking for The Ideal . . . house-cleaner, life-partner.  As an unBLT bloke, it is unlikely that the first girl you meet is the best possible match for the bearer of your children: those chances are 1:3,500,000,000 and falling as more potential mates get born in Africa and East Asia. You should accordingly reject that childhood sweet-heart and go on a few dates with other women.  How many other women?  There is an excellent Numberphile video which couches this problem in terms of accessing a clean toilet at a pop-festival.  I know that nobody follows my links, so I'll try to give you a flavor of Dr Symonds' arguments. Let's suppose that you've arrived on Pitcairn Island and you can date each and any of the three Tahitian girls you've kidnapped.  If you break off the relationship you cannot go back and you are trying to pick the best possible life-partner.  YMMV, of course, the girl who'd suit Fletcher Christian may not suit you. The prospect of a long-term relationship with Princess Di or "This year's Most Beautiful Woman" [People magazine says so] Sandra Bullock makes my heart sink. In what follows, a lower number is better: #1 is best but #2 is still better than #3.

If you pay your court to each girl in turn [no two-timing, there is no privacy on these desert islands], there are six possible permutations which you can experience:

First	Second	Third	Pick First	Reject 1
1	2	3	W	L
1	3	2	W	OK
2	1	3	OK	W
2	3	1	OK	W
3	1	2	L	W
3	2	1	L	OK

and let's say you have two strategies: 1) accept the first girl on the first blind date or 2) scope out the first girl but reject her and accept the next girl who is 'better' than the first.  I've set out the results in the table above.  First past the post gets the best solution [#1] 1/3 of the time and avoids disaster 1/3 [#2] of the time.  OTOH, the red number solution of reject first and take the next better wins 1/2 of the time and only draws the booby prize 1/6 of the time.  That's clearly a better solution and it scales up from a population of three to a population of 3 billion. But this simple strategy is subject to the law of diminishing returns as the number of available girls gets larger. Turns out that optimally you should sample 37% of the girls/toilets/cleaners [to see what's out there for choice] and then plump for the next option which is better than what you've already seen.  The mathematical argument is set out for those who don't suffer math-anxiety.  Another solution would be to get one of the cleaners to go to work on the crappy t'ilets.