[EM] "population paradox" using largest remainder with small # of voters?

[Adam Tarr]

Hi folks,

A small request for help.  I belong to a sports league that has several 
sectional tournaments, with a certain number of teams from each section 
advancing to the regional tournament.  The sections have varying numbers of 
teams, and the number of bids each section gets to the regional competition 
is dependent on how many teams are in the section.  The idea is to make it 
proportional - every section gets a proportional number of teams at the 
regional tournament to their size.

So basically, this is equivalent to the apportionment problem common to 
closed party list.  It's even easier to see the analogy if you look at it 
as apportionment of seats in the house of representatives.  Basically, the 
teams in each section are the voters, and the seats in the house are the 
bids to the regional tournament.

OK, so here's the problem.  Currently, the apportionment is done using a 
slightly perverted version of largest remainder, aka Hamilton's method.  I 
am trying to convince the powers that be to switch to Webster's 
method.  Being able to show the "population paradox", where a section could 
lose teams but gain an additional bid, or vice versa, would help a lot in 
convincing people.  The problem is, I'm having a very hard time coming up 
with an example given the very small number of "voters" (i.e. teams) that 
are involved here.

The parameters are as follows: the regional tournament has 16 teams - this 
is fixed.  Realistically, most sections will have between 3 and 20 teams, 
with some approaching 25 or so.  6-14 or so is be the most common 
range.  Every section is guaranteed at least one bid to the regional 
tournament, provided they have at least one team, so no example should 
conclude that a section gets zero bids.  Every region has either three or 
four sections.

Given these parameters (three or four "parties", between three and twenty 
"voters" in each party, sixteen "seats" in the house), can anyone come up 
with an example of the population paradox playing out?  Barring the 
presence of such an example, all I can do is show a case where the results 
differ from Webster's method, and try to argue that it's less proportional 
as a result.  But this is a lot less convincing.

Thanks for any help,
  Adam


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