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 a single race regatta it is known that the expected winner’s score is 1.<o:p></o:p>

<o:p> </o:p>In a multi race regatta ( assuming no DSQ or DNF) with four boats all being equally able sailors then:<o:p></o:p>

<o:p> </o:p>There are 10 point given out each race so things being equal a sailor may expect to get 2.5 points per race.<o:p></o:p>

<o:p> </o:p>

One way is 10/4 = 2.5<o:p></o:p>

<o:p> </o:p>

Another was is to do it by probability of getting 1^st 2^nd 3^rd or 4^th (again assuming equal ability)<o:p></o:p>

<o:p> </o:p>

The calculation for any place is: <o:p></o:p>

<o:p> </o:p>

     (points awarded) / (number of Boats)<o:p></o:p>

<o:p> </o:p>

So for 4 boats we would have<o:p></o:p>

<o:p> </o:p>

     (1/4) + (2/4) + (3/4) + (4/4) = 2.5<o:p></o:p>

<o:p> </o:p>

So for every race after the first the average score would be 2.5, meaning for a 2 race regatta the expected winning score should be (1 + 2.5) or 3.5<o:p></o:p>

<o:p> </o:p>

So the possible low score (one sailor takes both races) is 2 and the possible high score (one sailor loses both races) is 8 the expected average is not 5 but 3.5. (8-2=6, half of which is three added to two is five)<o:p></o:p>

<o:p> </o:p>

Another way to look at it is: in 2 races 20 points have been handed out, if there are 4 sailors then they should average 5 points each as shown above.  Yet the expected winner’s score is 3.5, well below the average.<o:p></o:p>

<o:p> </o:p>

If there are three races then the expected addition to any one sailors score is again 2.5 so (1 + 2.5 + 2.5 = 6) should be the expected score if all are equal.  An anomaly? Because out of the possible high low scores (3 being low and 12 being high) or a way to make regatta’s fair? (i.e. three races gives an even chance is you lose the first race)?  Or out of a possible 30 points given out in three races each races should expect 30/4 points (7.5) yet the expected winning score is 6.<o:p></o:p>

<o:p> </o:p>

If you extend this to 7 you get 16 as the expected score.<o:p></o:p>

(1 + ( 6 * 2.5)).  So if some one finished dead last in all 4 races their score would be 28 ( 7 * 4 ).  Look at it this way; in 7 races 70 points were handed out, if all were equal then each sailor should expect to finish the regatta with ( 70 / 4 ) points or a score of 17.5 yet the expected winning score is 16.<o:p></o:p>

<o:p> </o:p>

Extend this to 100 races:  the expected winning score should be (1 + ( 99 * 2.5 )) or 248.5.  Again if you finished dead last every time your score would be (100 * 4) or 400.  Yet the expected score is well above half that number.  On the other hand in 100 races 1000 points were handed out, each sailor should expect 1000/4 = 250 points, yet the expected winning score is 248.5.<o:p></o:p>

<o:p> </o:p>

Extend this to 1000 races and the winning score should be (1 + 999 * 2.5) or 2498.5.  Again if a sailor finished last every time their score would be 4000.  Yet the expected winning score is well over half that number.  Yet on the other hand in 1000 races 10000 points have been handed out, so each sailor should expect to receive 2500 points, which is above the expected winning score.<o:p></o:p>

<o:p> </o:p>

So if you run a million races the average score should be 250000 but the expected score will still be lower ( by a very small fraction)<o:p></o:p>

<o:p> </o:p>

<o:p> </o:p>

So in General for a 4 boat regatta the expected winning score for N races (except N = 1) is<o:p></o:p>

          (1 + (N * 2.5))<o:p></o:p>

And in general a race with Y boats and N (N > 1) races the expected winning score should be:<o:p></o:p>

<o:p> </o:p>

     1+(N*((sum of points for an individual race)/(Y)))<o:p></o:p>

<o:p> </o:p>

In general, with most yacht clubs, mine included (MCYC), you try to get in as many races as possible.<o:p></o:p>

<o:p> </o:p>

<o:p> </o:p>

Posted by mickey on 2006-01-29 21:58:26