[EM] "Present values of first place votes

[Forest Simmons]

"Present Value" is a tool for comparing different investments at different
times by taking into account the effect of compound interest.  It works by
designating a reference moment in time as "Present" and evolving all of the
various investments, forward or backward as the case may be, from their
known or assumed values at specified times to that common moment ... for
comparison.

The same identical mathematics allows us to compare first place votes at
various stages of an elimination, for example.

In particular, suppose that the fraction of voters that prefer X over any
of the more favorable candidates on the agenda is given by fV(X). If there
are 99 more favorable candidates than X, and fV(X)=1/20 ... how does X
compare with Y, given that there are only fifteen candidates on the agenda
more favorable than Y, and fV(Y)=3/10?

I'm going to derive the formula from scratch assuming the same kind of
exponential law that governs compound interest. If you just want result
without the derivation, you may skip ahead several paragraphs to the
formula.

We need the base of the exponential law in question. In the financial
context the base is (1+r) where r is the interest rate per compounding
period ... whether year, quarter, month, day, minute, or instant.

In case of "nstantanteous" or continuous compounding, the base is e^r,
which is the limit of (1+r/N)^N as N approaches infinity.

We find our base with the aid of a thought experiment:

Suppose there are n candidates and they all have N/n first place votes ...
so that they are perfectly tied. In the case of two candidates each has
half of the first place votes.

This consideration tells us that having 1/n of the first place votes when
you are worst on the agenda is like having 1/2 of the votes having advanced
from last place to second place on the agenda as other candidates were
eliminated.

So moving ahead (n-1) steps changes our share of the votes from 1/n to 1/2.

In our proposed model of exponential growth of value of top votes, this
means that

(1/n)*b^(n-1)=(1/2) ... which we solve for the base b to get

b=(n/2)^(1/(n-1))

Once we have this base b we are practically home free.

Suppose twenty percent of the voters share your opinion that candidate X,
tenth from the best on the agenda is actually the best candidate of all.
What is the present value of that twenty percent vote?

According to our model, the present value PV(X) is 0.20*b^9,  because X has
to move nine steps to go from nine ahead of X to none ahead of X.

Suppose some other candidate Y sixth from best on the agenda is the
favorite of 25 percent of the voters. How does Y compare to X in present
value?

PV(Y)=0.25b^5.

The ratio of PV(X) to PV(Y) is
(20÷25)b^(9-5) ... or .8*b^4

If the total number of candidates is 101, then b=50.5^.01 .... so the ratio
is
.8*50.5^.04 which is about .94.

So PV(X) is only about 94 percent of PV(Y).

In general, PV(X)=fV(X)b^k where fV(X) is the fraction of voters that
prefer X over any of the sgenda's more favorable alternatives, and k is the
number of those more favorable alternatives.

To declone this formula we rewrite k as (k/n)*n and replace the k/n factor
with the fraction of first place votes held by those k more favorable
candidates:

PV(X)=fV(X)*b^(n*sum{fV(Y)|Y more favorable than X})

That's all I have time for today!
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