May

20

The Secretary problem, i.e the optimal number of applicants to interview for a secretary job before quitting would seem to have much applicability to the time to come into the markets these days. How many extremes should one wait for in a day before coming in one way or the other, and what is the expectation for such strategies?

Jordan Low writes:

Is it just a coincidence that 1/e is close to 0.38 or the ratio used for Fibonacci time and price projections in technical analysis?

Rocky Humbert writes:

Steve Landsburg recently wrote about a variation on the Secretary Problem. He noted that "an anonymous math department chairman reports on his own strategy for cutting down on the [interview] workload. The math professor believes that one of the most important determinants of a successful career is luck. So each year, the math professor randomly rejects half the applicants without even reading the folders. That way, he eliminates the unlucky ones."

I suspect that there may be a market analogy in this admittedly sarcastic observation.

Phil McDonnell wonders:

Is it a coincidence that Fibonacci believers always seem to use the rhetorical form of: Is it a coincidence that (fill in a single observation) came close to (fill in selected Fibonacci value)?

Russ Herrold replies:

I do not think there is a co-incidence– the human mind tries to order data, and Fib sequences crop up everywhere. It is natural to do a 'trial fit' just as the eye tries to estimate a fit for a curve [and thus the reason for ready transforms as normalization, log, 1/exp and the like in running a regression– scaling often permits identifying the 'noise' and getting a 'good enough' solution]

Doing the math, of a run of repeated application of any two integers (seemingly separated by whatever distance, although I have not done a formal analysis or proof), the series seem to converge to a Fib set reasonably monotonically after say five rounds for low integers.

I ran into Fib numbers, learning the run time pass estimates for the IBM sort-merge algorithms in in the late '60s, and it appears that Knuth found them in sequential pass sorting as well. I seem to recall a childhood cartoon called 'Donald Duck in Mathmagic Land' where that quackish fellow pointed out that golden spiral, and the perfect rectangle 


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