Executive Committee Vote

Date: 01/21/97 at 21:56:03
From: Keith Cooke
Subject: Probability and Statistics?

124 Delegates attend an annual convention at which a new 13-member  
executive committee will be elected from a list of 26 candidates. Each 
of the delegates must vote for 10 candidates (i.e must place an X next  
to at least 10 names on the list of 26).

The question posed is: What is the lowest number of votes a candidate 
could get and be elected?

I'm not sure whether this is a combinatorial, statistical, or 
impossible question.  

Under worst (or best) case scenario, I guess you could be elected with 
just one vote, but the question then becomes, what is the probability 
of that happening? I think that a number of practical assumptions 
would have to be made (e.g., everyone doesn't vote for the same ten 
candidates - random lists, no spoiled ballots, etc.) in order to 
figure this out. 

Date: 01/26/97 at 22:36:18
From: Doctor Mitteldorf
Subject: Re: Probability and Statistics?

Dear Keith,

It seems to me that the "worst case" from the point of view of 
democracy would be if there were 10 candidates who were very popular, 
so that everyone voted for the same 10, and then there would be three 
more slots on the executive committee for which no candidates had any 
votes at all.

You're right, that if this scenario "almost" obtained, it would be 
possible for all the votes but one to be concentrated in the top 12 
candidates, so that the 13th prevailed with just one vote.

To calculate the probability of this happening, you need some extra
assumptions about human behavior.  A very unrealistic assumption, but
one that makes for an interesting and challenging statistics problem,
would be to assume that every vote is random - that each delegate is 
equally likely to cast a vote for each candidate.  Then there's a 
complicated combinatorial problem: how many ways can the votes be
distributed so that there's a candidate who gets elected with only one 
vote?  (The total number of ways the votes are distributed should go 
in the denominator, but that's relatively easy to compute.)

If I had to answer this question for some practical purpose, I might 
decide that the combinatorial problem is too difficult, and I would 
use a "Monte Carlo simulation."  I would program a computer to act 
like 124 voters selecting 10 candidates from a list of 26.  The 
computer could make random selections, and tally up the votes in a 
tiny fraction of a second, then repeat the entire election several 
million times, noticing how many of the times resulted in this skewed 
result where someone gets elected by one vote.  I'd go off and have 
lunch, and when I got back, the computer would give me an estimate of 
the probability that this might happen - under the very unrealistic
assumption that each delegate is equally likely to cast a vote for 
each candidate.

If we knew more about the voting behavior of real people in the
situation, we could make the Monte Carlo simulation more realistic. 
(For example, we could say that everyone who votes for candidate A has 
an 80 percent chance of also voting for B, but only a 10 percent 
chance of voting for Z.)

I can't tell if this is a practical problem for which you're seeking 
practical guidance, or an abstract mathematical amusement. If it's the 
latter, you might want to try formulating the problem in the way that 
makes it most interesting to think about.  Try smaller numbers, and 
try counting the possible distributions of votes, and see it a formula 
suggests itself that extends to the larger numbers.

But if it's a practical answer you're after, the Monte Carlo 
simulation is the way to go.
 
-Doctor Mitteldorf,  The Math Forum
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