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?

- 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.)
- 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.)
- 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.
- 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?

## 9 comments

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March 20, 2011 at 10:36 pm

AndyAs you’ve revealed you’re the sort of person who loves to say ‘duality’, I’m surprised you didn’t also use the phrase ‘single-crossing property.’

March 20, 2011 at 11:24 pm

itovertakesme#3 would be interesting to think about using a fixed point theorem…

#4 since races are mainly designed for the spectators, i would think time-based races work best when you want to spin it as a “last-man standing” sort of thing, like in the t.v. show “survivor”. somehow it feels more like a test of endurance. also good for when you don’t want to put a limit on how far the contestants can go, like who can lose the most weight in 30 days. it would be really boring to watch “who can lose 30 pounds fastest”. although if you were going to do an online contest, you might want one based on quantity rather than duration because it would make people check back to see if the race is over yet. so i guess it depends whether you get more suspense in having duration unknown or quantity unknown. i will think about this more.

ultimately though for things like sprinting/running, there are so many lengths available, that assuming ppl choose their race depending on how well they can do versus their opponents (i realize this can get a little messy because it could make for cycles… another possible research topic), then doesn’t this problem get solved? E.g. Sprinters like Mr. Green could compete in the 100 M and those like Mr. Black could compete in the 200 M.

March 21, 2011 at 12:12 am

DphThis is one of the reasons track cycling can be so interesting. Lots of non traditional methods of ranking can mean many interesting strategies.

March 21, 2011 at 6:22 am

Joshua GansYou know this is a very interesting taxonomy of races. It is interesting that the X-prize and NetFlix competitions were mostly about fixing the time and seeing how far you can get but the Space Race was about being first.

March 21, 2011 at 6:50 am

EmreI would speculate that fixed distance races are more common because it is easy to identify who crosses a fixed line first. Unless you want to time the contestants, you do not even need a chronometer. With a fixed time race, measuring the distance covered by a given contestant in a fixed amount of time is not quite so easy. You will at least need a chronometer. Plus you will have to have multiple judges to identify the exact spot the contestant is when the time is up. So my guess is that the format of the contest is affected by ease of measurement.

March 21, 2011 at 9:36 am

Evan RobertsThe one hour track race used to be more common in the 1970s and 1980s, but is rarely run anywhere these days. It is the only timed distance the IAAF maintains records for. You need a lap counter for every person, and when I’ve seen this done at amateur events it’s “have lap counter or don’t run”.

The rarity of the race is indicated by the fact the “area” (~continental) records for the distance are mostly very old by athletics standards, and don’t exist at all for some areas.

http://www.iaaf.org/statistics/records/inout=o/discType=5/disc=HOUR/detail.html

March 21, 2011 at 9:20 am

FranzThese graphs give the minimum amount of time needed to run a certain distance and vice versa. Those are not however the resulting chart of distance against time, since runners are conserving their energy early on. This is a problem in case of an elimination race, because a given maximum time for certain distance (or vice versa) might interfere with the energy conservation. The information we have is not enough, as runners choose the time path that keeps them in front of the pace car as long as possible. This time path might not be optimal for any distance.

If the runners are however to overtake a car, the race can be analyzed with the information and the diagram given.

March 21, 2011 at 12:14 pm

Jim SFranz’s comment, for Dummies: the interpretation given in #2 is flawed. (This may be what Jeff was getting at with the parenthetical comment about needing an assumption. A sufficient assumption would be that “always running as fast as you can” is the optimal strategy in any race.)

You can do what Franz said, and deduce the outcome of an elimination race given the graphs. But you can also keep Jeff’s definition of a race based on a curve in the graph, but reinterpret it as running an infinite number of races, and counting the outcomes of only some of them.

March 22, 2011 at 6:56 am

conchis“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.”

Actually, this isn’t limited to either rays drawn from the origin, or crossings from below. You could have a race where the ordering is determined by when the racers catch up to a moving target, which you’d potentially want to start some distance ahead at t=0. (I’m imagining something like the lure in greyhound racing, but where the dogs can actually catch it).