This question occurs to me every time I go jogging. I suspect every runner probabilist in the world must have thought of it (though I'm no probabilist), but I could not specifically find it online. I hope some MO readers will find it worth thinking about. Here's the basic set up:
Assume all runners go along a loop of
length $L$ in the same direction.
Assume that the distances $D_i$ ($i$
denoting a given individual) of the
runners' runs are i.i.d. with
distribution $P_d$, the runners'
speeds $S_i$ are i.i.d. with
distribution $P_s$, the starting point
of each runner is uniform over the
loop and the starting gaps between
starting times has distribution $P_t$.
Given your pace, the total distance of
your run and the # of times you were
passed and that you passed others,
what can you say about $P_s$?
Specifically, what can you infer about where your pace lies wrt the population of runners on the loop? Are you in the top 5%? Are you above average? Etc.
Of course, you'd need to make assumptions about many of these things, fixing $P_d$ for instance. And as a modeling exercise there are several interesting elaborations, such as letting the random variables $D_i$ and $S_i$ be correlated.
There obviously isn't one right answer, hence the 'soft' tag. But I'm interested in hearing if others have thought about it, how they might set up the problem and what sorts of assumptions would make the conclusions most interesting. As you take $L$ smaller and smaller so that you have to contend with the possibility of lapping people and getting lapped, things become harder, and in different ways depending on whether or not we allow registering getting passed by the same person. We could also let people go around clockwise or counterclockwise with some probability.
It seems so obvious that this information imparts qualitative information about your relative fitness, but quantifying it isn't straight forward. The data is well defined, the question is pretty easy to ask, but the modeling part leaves a lot of flexibility. I'm interested in hearing how the creative brains here at MO would set up the problem to interpret the evidence (or if there is a fun paper on this sort of problem somewhere).
Best Answer
We know what happens if you are slowest or fastest. If you are dead center then you would not have a reason (would you?) to expect one event to happen more often then the other. Not complicating things without cause, I wonder: have you ruled out the (too?) obvious answer a that if you have passed j people and been passed by k then the best estimate (absent other information) is that you are faster than $\frac{j}{j+k}$ of the other runners and slower than $\frac{k}{j+k}$. With assumptions on the distributions maybe one could say more.
Under some assumptions you could infer things by how long it has been since any passings happened, or by how quickly the relative frequency converges, but I'll assume that you run blindfolded and at some stage are stopped and told "over the course of you run you passed j and were passed by k.
I wonder if it helps conceptually to renormalize relative to your speed and say that you are a stationary observer next to a track, people have been scattered on it in random positions, some go clockwise and some counterclockwise according to some distribution which maybe be biased in one direction (and maybe with various speeds).Given that you observe j going counterclockwise and k going clockwise...