"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!

I've been ambivalent about the best way to apply this present value concept. Just elect the candidate with the highest Present Value of first preferences? Or repeatedly eliminate all candidates with present value less than 50%? That kind of elimination would require re-voting after each elimination stage ... or use of RCV ballots for an instant version ... which is getting away from the simplicity I was hoping for. For now I'm going to let it simmer on the back burner ... then perhaps revive it in a multi-winner context some day. -Forest On Tue, Apr 18, 2023, 8:41 AM Forest Simmons <forest.simmons21@gmail.com> wrote: > "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! > > > >