Almost every kind of race works like this:  we agree on a distance and we see who can complete that distance in the shortest time.  But that is not the only way to test who is the fastest.  The most obvious alternative is to switch the roles of the two variables:  fix a time and see who can go the farthest in that span of time.

Once you think of that the next question to ask is, does it matter?  That is, if the purpose of the race is to generate a ranking of the contestants (first place, second place, etc) then are there rankings that can be generated using a fixed-time race that cannot be replicated using an appropriately chosen fixed-distance race?

I thought about this and here is a simple way to formalize the question.  Below I have represented three racers.  A racer is characterized by a curve which shows for every distance how long it takes him to complete that distance.

Now a race can be represented in the same diagram.  For example, a standard fixed-distance race looks like this.

The vertical line indicates the distance and we can see that Green completes that distance in the shortest time, followed by Black and then Blue.  So this race generates the ranking Green>Black>Blue.  A fixed-time race looks like a horizontal line:

To determine the ranking generated by a fixed-time race we move from right to left along the horizontal line.  In this time span, Black runs the farthest followed by Green and then Blue.

(You may wonder if we can use the same curve  for a fixed-time race. After all, if the racers are trying to go as far as possible in a given length of time they would adjust their strategies accordingly. But in fact the exact same curve applies.  To see this suppose that Blue finishes a d-distance race in t seconds. Then d must be the farthest he can run in t seconds.  Because if he could run any farther than d, then it would follow that he can complete d in less time than t seconds.  This is known as duality by the people who love to use the word duality.)

OK, now we ask the question.  Take an arbitrary fixed-time race, i.e. a horizontal line, and the ordering it generates.  Can we find a fixed-distance race, i.e. a vertical line that generates the same ordering?  And it is easy to see that, with 3 racers, this is always possible.  Look at this picture:

To find the fixed-distance race that would generate the same ordering as a given fixed-time race, we go to the racer who would take second place (here that is Black) and we find the distance he completes in our fixed-time race.  A race to complete that distance in the shortest time will generate exactly the same ordering of the contestants.  This is illustrated for a specific race in the diagram but it is easy to see that this method always works.

However, it turns out that these two varieties of races are no longer equivalent once we have more than 3 racers.  For example, suppose we add the Red racer below.


And consider the fixed-time race shown by the horizontal line in the picture.  This race generates the ordering Black>Green>Blue>Red.  If you study the picture you will see that it is impossible to generate that ordering by any vertical line. Indeed, at any distance where Blue comes out ahead of Red, the Green racer will be the overall winner.

Likewise, the ordering Green>Black>Red>Blue which is generated by the fixed-distance race in the picture cannot be generated by any fixed-time race.

So, what does this mean?

  1. The choice of race format is not innocuous.  The possible outcomes of the race are partially predetermined what would appear to be just arbitrary units of measurement.  (Indeed I would be a world class sprinter if not for the blind adherence to fixed-distance racing.)
  2. There are even more types of races to consider.  For example, consider a ray (or any curve) drawn from the origin.  That defines a race if we order the racers by the first point they cross the curve from below. One way to interpret such a race is that there is a pace car on the track with the racers and a racer is eliminated as soon as he is passed by the pace car.  If you play around with it you will see that these races can also generate new orderings that cannot be duplicated. (We may need an assumption here because duality by itself may not be enough, I don’t know.)
  3. That raises a question which is possibly even a publishable research project: What is a minimal set of races that spans all possible races? That is, find a minimal set of races such that if there is any group of contestants and any race (inside or outside the minimal set) that generates some ordering of those contestants then there is a race in the set which generates the same ordering.
  4. There are of course contests that are time based rather than quantity based.  For example, hot dog eating contests.  So another question is, if you have to pick a format, then which kinds of feats better lend themselves to quantity competition and which to duration competition?
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