 have a row up here called rank. I'm gonna go ahead and use that to start demonstrating the rank that each candidate finishes in. And we could use the system to elect any number of candidates three, five, seven, whatever. So I'm just going to go arbitrarily far and just show how the process works as we go. Not really concerned with exactly how many candidates we're actually going to elect. If I look here at the totals at the bottom, let's see green candidate number one has four percent, green candidate number two has seven percent, six percent, seven percent, et cetera. Pretty clearly right away, the candidate who's doing best in the first round is the is D three, the third Democrat. And as you can see, I've color coded the various parties here. So you have Republicans in red, Democrats in blue, libertarians, I just chose yellow, greens obviously green. Now this system is agnostic to party affiliation. I'm only using this to sort of show why people are voting along these factional lines. This could be, you know, just partisan factions or it could be in a nonpartisan type of election people with just various ideologies. Again, the system is agnostic to this. This is just for demonstration purposes only. So anyway, we see that 27% is the winning amount for D three here in the first round. So I'm going to put here a one meeting. The first winner was D three. So now we've done a reweighting as you can see. So now there's only 14% there. That just is a result of all these people now having their votes sort of half as powerful. So you can see these three groups of people who voted for D three over here. Now each of their votes only counts as half a vote. That's why you see 0.5, 0.5, 0.5, et cetera. So that means in the next round, the best candidate is Republican number three. And that makes sense. So the Democrats had their say in the first round. So in the second round, now the Republicans have more say. So now they get to elect a candidate. Pardon the motorcycle noise. I'm in Italy. Okay, so now we've reawaited again. And if I go across the results here, it looks like 21% is the best. So now the Democrats are going to elect another candidate D two. So I say that D two is the third finisher. And it looks now like 16% is probably the best we have from any candidate. So the Republicans are going to get to elect another candidate R two. So now we wait again. And it looks like the best anyone's doing who is not already elected is let's see here we have 10%. Is there anybody doing better than 10% Nope. So the Republicans get to elect a third candidate. So that's going to be R four. And that would be the fifth overall candidate. And again, these rankings here actually, they're just from my benefit. So we can see what order people finished in the way the formulas work. The only thing that matters is that there's some value in this row. And I'll come back to that in a bit here. So now if we were going to elect a sixth candidate, who would that be? Well, 5% from Republican 6% from a Democrat. Let's see here. Okay, so now the minor parties are getting to the point where they can have their say because the major parties have elected so many people that now they've been their voters have been reweighted. And now they're weaker, finally getting the minor parties and say in the selection process. So the libertarians have a candidate who gets 8% support, the greens the best they're doing a 7%. So now the libertarians get to elect someone. So the sixth finisher is going to be L two, the second libertarian candidate. And now we have a reweighting. So it looks like the best anyone's doing is 7%. I believe so the greens have 7%. Does anybody over here beat that? Nope. D four is still the best unelected major party candidate with 6%. So now the greens get to have their voice heard. And that's going to be the seventh candidate elected. And we can continue down this path. Maybe I'll do two more. So who would be the eighth candidate? Here we have a 5%. We have anybody better than 5%. We have a couple who are tied for 5%. Democrats have somebody with 6%. Looks like 6% is the best. So the eighth candidate elected would be Democrat number four. And I'll just do one more here. Let's say we have a 3%, a 4%, 5%, and anybody beat 5%. Now, because I didn't really plan this example all that well, and it's got a pretty small numbers that aren't very precise here, you get two people tied for 5%. And the real world ties would be pretty unlikely. And there'd be some arbitrary tie breaking algorithm. So in this case, I could pick either R1, Republican number one, or D1, Democrat number one, they are both tied with 5%. But that's basically how the system works. So how I did this in Google spreadsheets was I basically have these totals here, which are just the percent for each faction times the votes, right? So if you go and you clear all these numbers for the rankings, what happens here is like, for instance, this 4% is just this one vote for green one times this 3% faction, that's 3%. Then this other vote for green one, that's a 1% faction. So you just add those up. And so Google spreadsheets actually has a function called sum product, which lets you just multiply the values in some corresponding sets of rows or columns. And in this case, what I've done here is I've used this dollar sign in the formula, which means that those aren't relative. So as you copy the function over across all these rows, I'm going to multiply everything in each row, right? And that is going to move across, but it's still going to be multiplying by these first two rows in every case. So those stay fixed. And what that means is that we have multiply that faction's percentage times how strong their each of their votes are. So they start out all having a power of one, just a full vote strength. And here I've used a reciprocal of the weighting. Because normally I would divide by the weighting, but there's no function in Google spreadsheets, which lets you say, you know, divide by the value in a column or a row, you can only multiply. So I just do one over the weighting factor to give me something I can multiply by. So for instance, if the power here was if this faction was a weighted by two, I'm not going to put two here, I'm just going to put 0.5, which is one over two. And that way I can just multiply it, right? So basically, every one of these totals is just multiply the percentage of people in that faction times their vote power, times whether they voted for that candidate and use add those all up. And the way I then do these weightings here is I just I have this formula for each of these fields are under elected. So all these fields across here. And it just says, if the value above is greater than zero, if there's anything in it, where I have the rankings, then put a one here, otherwise put a zero, right? So if I put like, say I said that green one was the first candidate elected, and I enter that, then this becomes a one, meaning yes, that person was elected. And then this weighting formula is again, I'm using some product to just multiply each of these elected values by each of these values here. So say for instance, this this faction here, they've approved only two candidates, they've approved green one and green two. So you basically just multiplying each of those values by whether it was whether that candidate has been elected in a previous round. So at this point, you get for instance, this faction of 3% has voted for G1, and G1 was elected. So that's one, whereas here they approve G2, but G2 has not yet been elected, so that wouldn't add up to anything. And so what you get in this function then is just one over one plus this sum product, which in this case is just one. So it reduces down to one over two, which is just 0.5. If that's a little confusing, just stop the video and play this part over again, the math is pretty simple actually. And as you can see, once the formulas are plugged in, actually operating this is extremely simple, like I'm just going across here and I'm looking at the biggest value down in this row of results. And I'm just entering that in the rankings. So again, just do this one more time. 27% is the best. So I would say a ranking of one, enter that, now I get a reweighting of everything 26% here for R3 is now the best. So I say ranking of two. Now I go back down, I think 16% is the best for any candidate who's unelected. No, no, no, sorry, 21%. Right. So now it bounces back over to the Democrats, so ranking of three. Now I think it's going to go back over to the Republicans with an 11% being the best. Yes. So now R4. And then let's do one more this time. So the fifth person elected, we have a 10% here. And we have, I think, 10% is the best. So yes, that would be the fifth person elected. And so on. And so as you can see, proportional approval voting is actually much simpler than say, single transferable vote, which is used in Australia and has been used since the early 1900s. You would not be able to do, as far as I can tell, a system like single transferable vote, particularly the proportional variety in a simple Google spreadsheet like this. It's just not possible.