Weighted choices

[Jason Friedman]

>> 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.

PIcking half to play one game and half to play the other is not an
option.  For one, there really isn't a concept of half when a variable
number of players are available on a given Saturday.  For two, we
could have an unexpected absence, or during the game an injury.  For
three, the parents paid to have their child play in the league, and
the league runs until the end of October, not long enough for that
strategy to work out.