The Plurality Method

in our previous video we introduced an example of a math club election that was kind of weird in that anyone could win the election depending upon the voting method that we introduced uh so in the election we have candidates we have voters who vote for the candidates but the election method itself has a huge impact on the outcome of the election and so because of this we have to study what are possible voting methods that we can use as we're very early on in this unit it makes sense to begin with the most common uh voting method at least historically that's that's used we will refer to this as the plurality method there's lots of different names for this method uh we'll just call it the plurality method the idea is this is the method where you vote for your favorite we saw this with our suu math club example as well you vote for your favorite and whoever your favorite is that you cast their name on the ballot and nothing else and whoever had the most votes the number of the most number of First Place votes wouldn't feel election plurality is a very simple voting method and so let's do some examples of this imagine we have example we have an election with four candidates and we will call them candidates a b c and d in which case voters will you will actually look at their complete preference schedule so that as we see all of the voters preferences and that helps us analyze things in this election here and so we get the following 14 people voted for candidate a as their Top Choice then B then C then D so the 14 people preferred alphabetical order then 10 people voted can it see as the Top Choice there then B then D then a eight people voted for D then C then B then a four people voted for Bravo then Delta then Charlie then Alpha and then one person voted Charlie Delta Bravo Alpha so Charlie was their first choice Alpha was their last Choice that's how you want to interpret this thing here so who is the outcome who's the winner of this election so all we have to do is we just have to count how many first place votes did everyone get so we only have to look at the first line of this voting uh of this preference schedule here so let's look at the candidates will go in alphabetical order here so candidate a received 14 first place votes only people in the First Column put a as their first choice they'll be at 14 there um if we look at candidate B looking only at the first one we have came up B right here was voted first place four times uh then candidates C there was two different types of voting profiles that place candidates see at the top 10 for the first one one for the next one so candidate C is going to end up with ten plus one which ends up being 11 people voted for Canadian c as their first choice and then for candidate d uh there was only eight people that put d as their first option so that's how we have it there and so then the winner of the election would then be candidate a uh who received the first uh the the most number of First Place votes but in fact we can do the following here we can do a complete ranking of this thing so the first place votes go to candidate a who got 14. uh second place would then go to candidate C who got 11 uh third place would go to Canada d who got eight points and then lastly we have fourth place goes to candidate B who ended up with four first place votes there so that would be the outcome of this election and again like I said candid a would be the first uh the first one to win there but there's sort of a weird thing going on in this election here things I want to note if you were to take the total number of votes here all right um 14 plus 10 plus 8. plus four plus one that adds that to be 37. there were 37 people who participated in this election okay for which case if you then look at the margin here um 14 divided by 37. you end up with 38 percent so Kenneth a wins this plurality vote here but they only had 38 of the voters support now conversely if 14 people put a as their first place vote that means 23 people put a as not their first place vote and that gives you 62 percent so the thing is we have that a majority of the people did not place a as their first place vote but yet a is the winner of this election and in fact I want to illustrate this even more look at this right here in this election everyone who didn't vote for a is their first choice put a as their last choice so 62 of the voters here put a as their last Choice a majority of the voters think of a is the worst of the four candidates but by plurality a would win because a has the most first place votes right that's a curious thing happening here with plurality it's possible that a candidate who has a majority of last place votes can still win the election because they happen to have the most first place votes still is that really a fair thing we'll talk about that another time uh let's look at another another example of an election this time we're gonna have five candidates a b c d e and we have three there's only three voting profiles that manifest themselves 49 people voted alphabetically a b c d e uh 48 people uh their profile was eight three uh 48 had b e d c than a and then three people put C then B then e then D then a okay so again as we as we work through these things um candidate a had 49 first place votes Canada B had 48 first place votes Kenneth C had three and then Canada d and e they never got any first place votes whatsoever all right it's kind of a curious thing there um so with regard to plurality uh we would get again that a is the winner there so with a complete ranking we get first place is a with 49 votes second place within go to B with 48 votes uh with c you only have that's third place with three votes then it comes to fourth place it turns out that we actually get a tie a a tie there as in two people got the same score not like a TIE fighter from Star Wars or anything like that uh it turns out B and E got a tie with zero there so this is a common thing that sometimes has to happen um in an election that if you have a tie you need some method of breaking the tie uh in order to get a ranking now if you only care about the first place vote that's not as much of a problem but of course it's possible that people can have a tie with regard to first place as well so tie Breakers are things you do have to incorporate in elections we're gonna We're Not Gonna worry about ties so much right now maybe we'll talk about this a little bit later but we saw the same problem that we saw before right uh less than a majority of people supported the first place candidates and in fact there's 100 voters total here 49 plus 48 plus 3 is 100 here um if 49 people did voted for a that means 49 of the electorate voted for a as their top choice but look at it again everyone who didn't vote for a in the first place voted for a in the last place and so in this situation you're going to end up with 51 of the population uh that voted against a you know so 49 liked them Ford 51 didn't so a majority of the people didn't like a again all right and so let me let me also point out one thing that happened here um when you look at these people uh these these candidates here we see that a lot of people supported a so I mean unlike the previous one there is a lot of support for a uh I mean it wasn't a majority of it's really close yet 48 people that voted for B so A and B are like neck and neck in this situation but then you have C right here C got three votes which compared to the O2 was completely insignificant but the margin of votes that c got if it were given to a different candidate would have tipped them into the majority situation let's say that c didn't run in the election um for whatever reason C dropped out before election day then if C performing the election these people who would voted for C their second choice would have been B so if C had dropped out of the election these people would have voted for b instead in which case then B would have not gotten 48 votes B would have gotten 51 votes for which that would have been the majority and more points than a uh and so though in some respect the only reason that A1 was because C ran in the election this is an example of what we call the spoiler effect the spoiler effect is that the outcome of the election would have been different if a candidate known as a spoiler in the situation had not run if C hadn't had run then B would have probably won the election and not a so it's perhaps a possibility that c spoiled the election for B there that's sort of a curious thing it it reeks a sort of unfairness but again these are topics we're going to talk about in the future I want you to understand how the plurality method works but it's also a good a good opportunity to talk about why the plurality method might not be the most Fair method don't get me wrong it's not completely garbage but it does have some issues all right so as we end lesson uh 17 I do want to end us with a real life example so all the previous ones have been fictitious so far but I want to give you an example from your real life let's look at the 2010 Governor's race in the state of Maine why why would we look at that one it turns out there's some curious things going on here so in that race we had five candidates that you can see right here the two that I want you to pay attention to are going to be right here Elliott car uh Elliot Cutler and Paul LePage okay why what's so significant about these two candidates well one look at what percentage of the votes they got here so I mean these are the total number of votes they got um and then it was 38.2 percent went to LePage 36.5 percent went to Cutler and then the next one the Democratic candidate Libby Mitchell uh she received 19.3 percent of the vote and then we had two other Independents who received less than five point less five percentage each right so these are the top three candidates um Cutler and LePage were neck and neck in this election um it turns out it would actually go back to the voting data the early voting it was it was winning for Cutler but then um because cities often get votes in faster than rural areas uh oftentimes not always the case but that was the case in this situation as as more of the rural vote came in um LePage then pass Cutler and won the election so um LePage was the was the elected governor um from the 2010 election right there it was really really close um you'll notice that in this situation none of the candidates received a majority of the votes and Cutler and of LePage were really really close together now some things that happened here Cutler was declared as an independence but Independents still have to have some type of policy platform and if you were to analyze Cutler's platform it sounded very liberal and thus was very similar not the same but similar to the Democratic platform uh that Mitchell had ran under um LePage here as a republican of course supported conservative uh platform and policies and therefore people who might have voted for Cutler might have also voted for Mitchell um and therefore this 19 percent who voted for Mitchell if Mitchell hadn't run might have voted for Cutler but also goes the other way around um had Cutler not ran in the election these 36 of people would very likely have voted for Mitchell instead we don't exactly know because the data doesn't have a complete preference schedule like we did with our fictitious elections here but the thing is when you add 36 and a half percent with 19 and 0.3 percent that gives you a total that's bigger than 38.2 so if one of those two people hadn't run Cutler or Mitchell then it might have been the case that the other one the other liberal candidate would have won and had beaten the conservative candidates maybe again these are all hypotheticals it can't be certain but there was a lot of finger pointing here Mitchell was saying that Cutler was a spoiler but Cutler's like you're you're the uh the spoiler and so the outcome of the election might have been different if one of these candidates not had run but the reason the main reason I'm including this has to do with some things that Cutler had done and I don't mean about the arrest of Cutler in 2022 that I'm not going to talk about that I mean that in the 2010 election um Cutler ran among other things with the platform that maybe the election the electoral system in Maine should change that is instead they should use something called rank Choice voting ranked choice of voting this is a topic we will talk about much more detail in future lessons here ranked Choice voting uh without going into all the details of things it doesn't just look at your first place vote it actually uses all of your preferences to help determine the outcome of the election and one thing that ranked Choice voting does well is it avoids the spoiler effect that the plurality method we've been talking about is a victim to because people who think Cutler was the best choice can still give some they can still influence the outcome of the election for Mitchell but likewise people who had Mitchell as their first choice could still put support behind Cutler they could actually support both of their candidates they like that are similar simultaneously it doesn't have to be a like a death match between two similar candidates here all right and so it turned this election actually sort of was the start of the conversation behind ranked Choice voting in the state of Maine um in 2014 there was another election between Cutler and LePage I mean there's other people running as well um LePage had one he was the governor in 2010 he won re-election in 2014. um Cutler got nowhere near as much support in 2014 um as he did in 2010 but nonetheless the conversation of ranked Choice was still in in conversation and so it actually turns out in the year 2016 the state of Maine actually did pass um the the pass a law that then allows for ranked Choice voting in elections I don't think they're Allowed by their constitution to use ranked Choice voting for governor races so it still wouldn't have helped Cutler in that situation and then also Cutler's never running for office again but like I said we're not talking about the 2022 arrests or anything like that um so State Maine was the first state in the United States to pass ranked Choice voting and this election obviously had a huge impact on voters voting for this new electoral system um so why did they do it well they a lot of people felt like they're wasting their vote if they didn't vote for one of the top two um but that might not be that your sincere vote in a Democratic Republic it's important that voters Express their sincere preference not some strategic involved like oh little you know Libby doesn't stand a chance I'm going to vote for Elliot that's not how people should vote um we should vote by our hearts by our desires by our platforms by our policies uh or I should say candidates who support those things and it could be the case that the plurality method is obstructing people's desire to vote sincerely and Alternatives like ranked Choice might actually be better in that now I'm not going to Advocate that ranked Choice voting is the best voting method in the in the world but as we explore this over the next couple lessons we might start to see why ranked Choice voting is superior to this plurality method even if itself is not perfect it could be a step towards a more perfect union and so with that we're going to end lesson 17. we'll talk about some more voting next time if you did learn a lot about voting in these videos and the mathematics related to it please like these videos share them with friends or colleagues who might benefit from them as well subscribe to the channel to see more videos like this in the future and of course if you have any questions whatsoever please post them in the comments below and I'll be glad to answer them as soon as I can