Competition and the Logistic Distribution

In my recent post on the other bell curves, I wondered if there was some familiar processes underlying the logistic distribution that would parallel coin-flipping for the normal distribution.  I noted there that logistic distribution works better for the ELO rating system than the normal distribution did.

I also noted that the curve that describes the logistic distribution arises from population growth models that account for competition for resources.  The ELO rating system is also used in the context of competition, but for games.  I wondered whether there was something to this.  Was it a mere linguistic coincidence, or was there a deeper connection?

Well, I still don't have a complete answer, but I can see at least one connection between the two scenarios.  In both cases, there is competition for a finite resource.  For the population growth model it is food supply, and for games, it is wins. 

The parallel is not exact, however.  In particular, the logistic model is used to describe the size of a population, but for a game, the logistic distribution is only used to compute the probable outcomes within a set of competitors, not the number of competitors. So the connection is not complete.