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
re: Charlie I wonder... by Charlie)
Well, I agree, because:
(a) For any given racer, the expected average score per race is 2.5.
(b) For any given racer, we can pick a number of races such that the
standard deviation of the average score is arbitrarily low.
(c) There are a fixed and finite number or racers
(d) I think that this implies that the limit of the expected average
score of the winning racer is the same as the expected average score
for all racers. Since, for instance, I can pick a number of races
such that each racer has a very very very low probability of a 2.6
average, I think the limit of the best racer has to be less than
2.6. And the same applies for any suspected limit that is not
2.5. So the limit is 2.5. I'm sure I could make this
rigorous, but ...
re(2): Charlie I wonder...
