[EM] Plurality has 4 times the inequality of voting power

Michael Ossipoff mikeo2106 at msn.com
Thu Mar 1 02:49:29 PST 2007

I’ve told why Approval doesn’t give more voting power to people who mark 
more candidates.

But what if we define a “voting power” that can vary between voters? What 
quantity would make sense as that voting power? How about the voter’s 
opportunity to increase his/her expectation in the election?

Say we don’t know which two candidates will be the frontrunners. If there’s 
a 2-way tie, we don’t know which two candidates the tie will be between.

And say, for simplicity, the candidates’ merit, or “utility” for the voters 
is only two-valued. The voters’ sincere personal ratings of the candidates 
have only two levels or values.

So the amount by which you can increase your expectation can be measured by 
the number of pairs of candidates of unequal merit that you’re voting 
between, by voting for the better one and not for the worse one.

By what factor can two voters’ expectation-increasing opportunity differ?

In Plurality:

The most fortunate voter  rates one candidate 1 and all the rest 0. S/he is 
voting between N-1 meaningful pairs when s/he votes for that one.

The least fortunate voter rates  one candidate 0 and all the rest 1. S/he 
can vote between only 1 meaningful pair.

So, with Plurality, the maximum ratio between voters’ voting power is N-1

In Approval:

The most fortunate voter rates N/2 candidates 1 and rates N/2 candidates 0. 
S/he is voting between  (N/2)^2 meaningful pairs. Of course that can be 
written N^2/4

The least fortunate voter either rates 1 candidate 1 and N-1 candidates 0, 
or rates N candidates 1 and rates N-1 candidatres 0. S/he can vote among N-1 
meaningful pairs.

The ratio is (N^2/4)/(N-1)

Divide Plurality’s ratio by that of Approval:

(N-1) divided bv (N^2/4)/(N-1)  =  (N-1)^2/(N^2/4)

If N is large, N-1 is close to N, and so, as N increases, that expression 
approaches, as a limit, the value of 4.

When N is large, Plurality can be 4 times as unfair as Approval.

What about smaller N?

To find the point at which Approval’s and Plurality’s ratios are equal, set 
the expressions for their ratios equal, and solve for N.  The answer is 
close to 3. But 3 is the smallest number of candidates for which it makes 
any difference what the method is.

So, Plurality ranges from 1 to 4 times more unfair than Approval, depending 
on how many candidates there are. When there are many candidates , Plurality 
is 4 times more unfair than Approval.

Mike Ossipoff

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