Its obvious right?  Ok but before you read on, say the answer to yourself.

Is it because he is the most able to make up any lost time by the earlier teammates?  Because in the anchor leg you know exactly what needs to be done?  Now what about this argument:  The total time is just the sum of the individual times.  So it doesn’t matter what order they swim in.

That would be true if everyone was swimming (running, potato-sacking, etc.) as fast as they could.  But it is universally accepted strategy to put the fastest last.  If you advocate this strategy you are assuming that not everyone is swimming as fast as they can.

For example, take the argument that in the anchor leg you know exactly what needs to be done.  Inherent in this argument is the view that swimmers swim just fast enough to get the job done.

(That tends to sound wrong because we don’t think of competitive athletes as shirkers.  But don’t get drawn in by the framing.  If you like, say it this way:  when the competition demands it, they “rise to the occasion.”  Whichever way you say it, they put in more or less effort depending on the competition.  And one does not have to interpret this as a cold calculation trading off performance versus effort.  Call it race psychology, competitive spirit, whatever.  It amounts to the same thing:  you swim faster when you need to and therefore slower when you don’t.)

But even so its not obvious why this by itself is an argument for putting the fastest last.  So let’s think it through.  Suppose the relay has two legs.  The guy who goes first knows how much of an advantage the opposing team has in the anchor leg and therefore doesn’t he know the amount by which he has to beat the opponent in the opening leg?

No, for two reasons.  First, at best he can know the average gap he needs to finish with.  But the anchor leg opponent might have an unusually good swim (or the anchor teammate might have a bad one.) Without knowing how that will turn out, the opening leg swimmer trades off additional effort in return for winning against better and better (correspondingly less and less likely) possible performance by the anchor opponent.  He correctly discounts the unlikely event that the anchor opponent has a very good race, but if he knew that was going to happen he would swim faster.

The anchor swimmer gets to see when that happens.  So the anchor swimmer knows when to swim faster.  (Again this would be irrelevant if they were always swimming at top speed.)

The other reason is similar.  You can’t see behind you (or at least your rear-ward view is severely limited.)  The opening leg swimmer can only know that he is ahead of his opponent, but not by how much.  If his goal is to beat the opening leg opponent by a certain distance, he can only hope to do this on average.  He would like to swim faster when the opening leg opponent is behind but doing better than average.  The anchor swimmer sees the gap when he takes over.  Again he has more information.

There is still one step missing in the argument.  Why is it the fastest swimmer who makes best use of the information?  Because he can swim faster right?  It’s not that simple and indeed we need an assumption about what is implied by being “the fastest.”  Consider a couple more examples.

Suppose the team consists of one swimmer who has only one speed and it is very fast and another swimmer who has two speeds, both slower than his teammate.  In this case you want the slower swimmer to swim with more information.  Because in this case the faster swimmer can make no use of it.

For another example, suppose that the two teammates have the same two speeds but the first teammate finds it takes less effort to jump into the higher gear.  Then here again you want the second swimmer to anchor.  But this time it is because he gets the greater incentive boost.  You just tell the first swimmer to swim at top speed and you rely on the “spirit of competition” to kick the second swimmer into high gear when he’s behind.

More generally, in order for it to be optimal to put the fastest swimmer in the anchor leg it must be that faster also means a greater range of speeds and correspondingly more effort to reach the upper end of that range.  The anchor swimmer should be the team’s top under-achiever.

Exercises:

  1. What happens in a running-backwards relay race?  Or a backstroke relay (which I don’t think exists.)
  2. In a swimming relay with 4 teammates why is it conventional strategy to put the slowest swimmer third?
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