A regatta is a series of sailboat races. In the fleet where I race, the regatta winner is determined using a method called "Low Point Scoring". In any given race, the 1st place boat gets 1 point, the second place boat gets 2 points, the nth place boat receives n points. Individual races never have ties for any positions. The overall regatta is won by the boat with the lowest total number of points for all races. (If there is a tie for lowest total points, then the regatta is won by whichever of the tied boats had the better performance in the last race).

Consider a relatively small fleet of only 4 boats, each of which is equally likely to win any given race.

a) If there are only two races, the boat that wins the regatta will have a score of 2, 3, 4 or possibly even 5. What is the expected value of the winning score?

b) If there are three races, what is the expected value of the winning score? (I found even this simple case hard to calculate exactly, and I am hoping that somebody will come up with a better method than mine. And yes, I know that it is easy to simulate.)

c) If there is a large number of races, how might I approximate the expected winning score? (Among other things, I think I'd welcome a simulation here)

(In reply to Charlie I wonder... by Dan)

The numbers of tests required effectively prohibits going much beyond 100.  I've speculated the values approach, but never reach, 2.5 times the number of races, as that number is just what the average is for all the boats.  As more races are done, the lead racer gets closer and closer to the average, on average.

If someone could come up with an explicit formula, we'd of course see what it would approach.

Posted by Charlie on 2005-12-04 11:19:41