模倣子 Settling the Votes in a Ranked Choice Vote

Previous Post on Prop 1 - Macromemetic Index - Math of Elections 

Introduction

I wanted to take a closer look at how the votes of the defeated parties get rolled up into the winning parties' totals.

How Ballots Look

Let's say we have the Republican, Democrat, Farmer's, and Green parties running after having made it through the primaries as the top three parties. A ballot has a first pick, a second, a third, and a fourth choice, so a shorthand for a ballot could look like RDFG, where Republican is first choice, then Democrats, then Farmers, then Green. Every ballot looks like this.

There are six kinds of ballots where the Republicans are the first pick: RDFG, RDGF, RFDG, RFGD, RGDF, and RGFD.

How a Vote Might Look

There are 24 different kinds of ways to cast a ballot. Here's a sample vote below. I've marked in Blue and Red the ballots that go to the winning party (Republicans) and the runner-up party (Democrats) when the other two parties are taken out.

Rep	RDFG 1%	RDGF 1%	RFDG 18%	RFGD 12%	RGDF 3%	RGFD 5%	Total 40%
Dem	DRFG 5%	DRGF 5%	DFRG 2%	DFGR 3%	DGRF 10%	DGFR 5%	Total 30%
Farm	FRDG 5%	FRGD 8%	FDRG 1%	FDGR 1%	FGRD 3%	FGDR 2%	Total 20%
Green	GRDF 1%	GRFD 0%	GDRF 3%	GDFR 3%	GFRD 1%	GFDR 2%	Total 10%

So the Republicans get their 40% plus 16% from the Farmers and 2% from the Greens, for 58% and a win. The Democrats pick up 4% from the Farmers and 8% from the Greens, for 42%. 

Is it possible for the Democrats to win with the initial 40%-30% first choice vote, if the other parties votes go differently?

Coming Back from a Bad First Choice Vote

If the Greens all go to the Democrats, and the Farmers split both ways, the Democrats can actually win.

Rep	RDFG 1%	RDGF 1%	RFDG 18%	RFGD 12%	RGDF 3%	RGFD 5%	Total 40%
Dem	DRFG 5%	DRGF 5%	DFRG 2%	DFGR 3%	DGRF 10%	DGFR 5%	Total 30%
Farm	FRDG 4%	FRGD 3%	FDRG 5%	FDGR 2%	FGRD 2%	FGDR 4%	Total 20%
Green	GRDF 0%	GRFD 0%	GDRF 3%	GDFR 4%	GFRD 0%	GFDR 3%	Total 10%

Here the Republicans pick up 9% of the Farmers vote, for a total of 49%. The Democrats pick up the whole 10% of the Greens, and then 11% of the Farmers split vote for 51%.

So even if one is not most peoples' first choice, or even if one is, one can be the second or third choice after more minor parties, and pick up those ballots when those smaller parties drop out. This can only really happen if said smaller parties have the amount of votes making the difference between winning and losing. If they are smaller than that, and the gap is wide, then they can't make much of a difference.