Talking about free agent pickups in the market efficency post, reminded me of a well known problem in probability, often illustrated by way of romance.

A woman (or man) dates a number of potential spouses in sequence, where N is the maximum number she will encounter in her “dating life.”

The question for our heroine (Myrtle) as phrased by John Allen Paulos in Innumeracy (1988) is this:

When should I accept Mr. X and forego the suitors who would come after him some of who may possibly be “better” than he?

The answer, based on conditional probability is wonderfully precise. After rejecting the first 37% of the potential maximum, Myrtle should pick the first one that surpasses all that came before.

For example, if Myrtle predicts she’ll encounter 10 eligible suitors in her dating life, she should reject the first three before looking for Mr. Right. If the first three rate 5 1 2, and number four rates 6, she takes him. If not, she tries number five, then number six, and so on, until she either says yes or runs out of suitors.

For fantasy baseball, as with romance, this is a totally unhelpful model. Most free agent adds are driven by multiple factors including position eligibility, injuries, and statistical needs.

But let’s try it out anyway.

We can model the question several ways. For example: Say you decide to try one new free agent each week. At the end of the week you either keep that player for the rest of the season or drop him for someone else. Assume that every player you drop is grabbed by another owner in your league and no longer available.

When should you stop churning?

In a 25 week season the answer is 25 x .37 = 9.25, or week nine. Leading into weeks one through eight, you always drop your free agent (note that for week one, you start by dropping your most unfortunate draft pick) and add a new one. Leading into week nine you evaluate your current free agent against all previous and keep him if he rates “the best” (whatever that means).

Another scenario centers on the problem of the sixth starter. Perhaps you play in a weekly head-to-head league with a seven game-start limit and playoffs. Your goal is to assemble the best starting pitching squad possible by the last week of the season, the championship. You want a sixth starter for that match to ensure that you hit the game-start limit.

Assume that you give each free agent pitcher three starts before you drop him and try another. That sets N at about 12, which means that free agent starter number five is the first of the lot you consider keeping (12 x .37 = 4.51).

Paulos gets the last word (for his scenario and mine):

Then comes the hard part: living with Mr. Right.