Saturday, October 18, 2014

Mathematical Meetup

As some long time explorers in this blogs will know, Wanderers was blessed with several sons of Cal Tech, which makes sense given ISEPP's history and ties to Linus Pauling, x2 Nobel Prize winner.  I recently had lunch with one of these alums (the reflected ambient light is from a Thai restaurant in the PSU area downtown).

How do judges decide who wins in science fairs?  That's a deceptively simple question, which I will couch in the isomorphic namespace of a "beauty contest".  First assume the impossible:  all the judges completely agree because they're all clones of each other.  As omniscient onlookers, we have ahead-of-time knowledge of how the ranking should go.  And the judges, in retrospect, would all agree with said ranking.  Like I said, impossible.

Now here's the wrinkle:  you have sixty "beauty queens" (assuming nothing about gender) or "prima donnas" and each of ten judges only gets to interview twelve of them.  No judge interviews them all (too many contestants, not enough judges -- a realistic constraint based on actual science fair data).  Every candidate is interviewed twice (10 * 12 == 60 * 2).  So what affect on final results does the initial random assignment of each judge's twelve make?

Running such analysis thousands of times suggests noise is greatest in the middle, as certain pairings will not have been made.  No one will have compared X to Y, by happenstance, and this missing puzzle piece degrades the result.  A runner-up always has the excuse (legitimized by this study):  "just bad luck, I should have ranked higher and the judges would agree with me had they seen us all." That's what the math says too.

I should hasten to add though, that these "zombie-clone judges" who all agree do not represent the typical science fair judge, or even beauty contest judge.  For one thing, with proper mathematical tools it's possible to compensate for these "luck of the draw" issues.  Judges caucus precisely for this reason:  to avoid robotic behavior and thereby falling victim to the exigencies of pure mathematics.

Good job explaining David!  I hope I captured the essence of what you're finding.