Weighted choices

[Peter Otten]

Jason Friedman wrote:

[You may have sent this in private mail by accident, so I take the freedom 
to bring this back to the mailing list]

> I'm realizing with your question 

which was roughly: Do picks from a pool influence probability of subsequent 
picks, i. e. weather the pool is finite or infinite.

> about which I want, though, leads me
> to state the actual problem, which takes longer to explain.

Giving the actual problem is always a good idea as it allows answers that 
aren't limited by your grasp of the problem.

> I coach a flag football team of 11-year-olds.  A stated goal of the
> league is that every player should get nearly equal playing time and
> that winning is of secondary importance.  That said, some players just
> can't throw the ball at all, and having a quarterback who cannot throw
> is no fun for anyone.  The other coach and I will pick two players who
> will be the quarterback for any given snap. The other players can play
> any position (center, receiver, runner) other than quarterback.
> 
> The game is 5-on-5 and our roster contains ten players, and we can
> expect a random player or two missing on any given week.  I'm trying
> to come up with a system that causes at least one of those
> quarterbacks to be on the offensive field on every play.  Further, to
> compensate for the fact that the quarterback gets to touch the ball on
> every play, we want the quarterbacks to appear at the non-quarterback
> positions somewhat less than the other players.
> 
> This is all quite challenging.  11-year-olds are exquisitely tuned to
> unfairness and most have a keen idea of the play that will surely
> score a touchdown if only the clueless coach would let them run it.
> Oh, and the originator of this play will be the key
> quarterback/runner/receiver for the play.  Oh, and in case the coach
> forgets, they will remind him.

OK, you're well inside the "finite" domain. Also, you probably want less 
than the "natural" randomness. I'd probably shuffle the potential 
quarterbacks and the others in independent lists, and then pick one half of 
each to form a team. The other half would play in the next game. 
Additionally you can keep track of every player's total number of games and 
games not played in a row, and apply a correction if either exceeds a limit 
acceptable for a kid. 

As for the actual position in the team, that a least should be left for the 
coach to decide. 

Personally I would prefer to have some discussion between the kids with the 
coach having an eye on the shy or not-so-popular ones -- and if that leads 
to more lost games than necessary, so be it.