Voting Theory

welcome back to our lecture series math 1030 contemporary mathematics for students at Southern Utah University as usual I'll be a professor today Dr Andrew misseldine in lesson 17 we're going to begin a new chapter and a new unit about political mathematics I should say mathematical applications in political and social sciences this first unit right here this first chapter I should say is about one of my favorite topics that we talk about this course this idea of voting Theory so over the next couple lessons we're going to talk about the mathematics of Elections the mathematics of representation the mathematics of apportionment that is a lot of the mathematics at the heart of how democratic governments work particularly the United States democratic government a lot of the principles that we will learn are actually Reflections upon mathematical problems that have arised in U.S history here so this is a topic I'm pretty passionate about because I mean all of the topics we've learned on this series have ramifications in real life I mean there's a reason why we call this class contemporary mathematics things we've learned about networking and scheduling and being fair these again these have effects on how we interact with people I mean we can use mathematics to make the world a better place and again one that I feel very passionate about this idea of how can we make the world more democratic and it turns out mathematics does offer us a tool here now to introduce us to all of the topics we need to understand uh with regard to mathematical voting Theory the terminology the vocabulary what are the moving Parts in these election problems I'm going to present you an example um and so this is an example of the suu math club okay so every year the suu math club has an election at the end of the year to elect the new math club president who will run the next year uh now of course as seniors graduate they don't get to cast a vote because they're not going to be there next year and so let's suppose that in a particular year there are 11 members of the math club who were not graduating but who cast a vote for next year's map Club president all right let's look at what that election looks like and so one particular year we had three candidates that ran for the math club presidency and this is math club after all um their political parties are the following we had one who belonged to the algebra party one who belonged to the trigonometry party and one who belonged to the geometry party okay just so you're aware full disclosure here I'm a registered member of the algebra party and so I do want you to be aware as I'm going through this example you're not uh you know my bias doesn't come out here I'm not pushing for one candidate over the other with this sample here so we're going to talk about who won the election okay now in the math club we do things a little bit different than perhaps traditional U.S elections um when uh when we cast our ballots we don't just cast our ballots for who our first choice is we actually rank all of our candidates in the order of preference so with the three candidates we would we would decide who's my first place Choice who's my second place Choice who's my third place Choice all right now again that might seem like an odd thing to do you know people in math club sometimes do appear on to other people but it turns out there's actually a lot of advantages of having having the voters present their entire voting preference as opposed to just their first place votes and that's this example illustrate that as we'll see not just in this example but in future examples as well now just so you're aware okay if we have three candidates if we have to rank each and every one of them your base basically have three options for who your first choice is you're gonna have two options for who your second choice is and you'll have one option for your last choice so if you take three times two times one that gives you what we call three factorial which in this case is six there are six possible ballots you could test six possible rankings of the three candidates you could do uh can it one two then three you could do candidate one three then two candidate two one and three candidate two three and one candidate three one two or candidate three two one those are the three possible choices you could make and so of our 11 non-graduate members of the math club these were the votes that they said there were three voters who placed and and remind I'm using these abbreviations C1 for Canada one C2 for Canada two C3 for Canada three but to interpret this here these uh this these three people said they wanted to vote for the algebra candidate then the trigonometric candidate then the geometric candidate uh we had two people who voted for First Choice was algebra second choice was geometry third choice was trigonometry um we had no one who voted for trigonometry first then algebra then geometry now that might seem surprising at first but if you think about the political platforms of these parties um the trigonometry party in the geometry party are actually very similar platforms honestly like trigonometry uh is a is a subset of geometry so it turns out politically a lot of people who feel geometric are actually inclined to think trigonometric as well and vice versa and so this profile right here would be like oh you've you value the trigonometric principles first then algebra then geometry well there's not a lot of people and in this case no one this year plays the algebra candidate between the trigonometric and the geometric candidate I want to mention that one happened right here as well no one places had had the preference of geometry thin algebra than trigonometry the people who vote for geometry are typically likely going to vote for trigonometry as their second choice and those who voted trigonometric are most likely to vote geometric as their second choice because again the platforms are very similar between those parties But continuing with our list here we had two voters who did trigonometry then geometry then algebra and we had four voters who went geometry then trigonometry then algebra okay and that kind of what we saw here uh so a lot of people preferred algebra and then they have sort of a second choice between either trigonometry or geometry a trigonometry geometry and then we had six people who then were much more geometric uh maybe putting trigonometry first and then put algebra in the last again given those branches of mathematics that kind of makes sense anyways we then have to ask the fundamental question here who is the winner of this election and if you want to like pause the video for a second contemplate that question for a second of these 11 voters we've now see how they have voted and not just how not just diverse choices we see all of their preferences who's should win this election now it turns out that can be a very difficult question and with the present table I'm going to rewrite it to make it a little bit more readable okay so let's rewrite that table in the following way we had uh if we just focus on the candidates now and not the voters there were five people that put candidate one the algebra party as their first choice no one put candidate one as their second choice and six people put candidate one as their uh as their third choice right and again this is the phenomenon we saw before um you were either leaning algebraic or you were leaning geometric algebra was never in the middle because if it was in the middle that would mean that you put trigonometry first in Geometry last or vice versa and no one no one's political ideology went in that direction then when you look at the second candidate this was the trigonometric candidate two people ranked C2 as the first candidate seven people put C2 as the second that's their second choice and three or two people put trigonometry as their second choice there so I want you to notice here this is sort of like the lukewarm candidate right A lot of people put trigonometry as their second choice uh but not as a lot as their firsters left that kind of makes sense because trigonometry is sort of like in the middle between algebra and geometry um trigonometry is definitely within the scope of geometry but trigonometry also involves solving lots of algebraic equations and graphs stuff you do in an algebra class as well so it makes sense that if someone's was algebraic if their Top Choice was algebra their second choice is probably more likely to be trigonometry over purely or pure geometry free but conversely if someone's First Choice was geometry it makes sense that their second choice is probably going to be more likely trigonometry than over pure algebra again they kind of sit in the middle a little bit the trigonometry party does okay um and then we had four people who voted for uh geometry as their first choice than four people for geometries or second choice and three people for geometry as their last Choice there okay so those are the this is the voters preferences here this is the so-called preference schedule for this election so again I asked the question who's the winner and it turns out that even though we have all of the information about the preferences of the voters we actually can't answer this question without more information about how the election works now in a typical U.S election at least at the timing of this video um the way that election is determined is by counting the votes using the method vote for your favorite So within a typical election whether you're voting for the president or the governor or Senator or mayor or School Board whoever you typically vote for who is your favorite and whoever gets the most first place votes that is because you vote your favorite person is your first place reference whoever gets the most first place preferences is then determined the winner here and so looking at this election here if you're voting for your favorite you only have to look at people's first place choices there for which algebra had five first place votes uh trigonometry had two first place votes and geometry had four first place votes and so with that if you record that down here candidate one would receive the fur of the most first place votes in which case that would then make algebra the winner of this election because they got the first the most first place votes that's fair right well is it really um let's analyze that just a little bit longer here well sure C1 did get the the number of the most number of First Place votes but I want you to actually let's look at that table again I want you to analyze some things here um we have 11 people in this election that is 11 voters in the election three candidates here and if we were to consider like what's a what's a majority in this situation well the idea is you would take half of 11. okay that's going to give you 5.5 um people can't cast a half a vote so what we have to do is we have to round this thing up um so we end up with a majority being six if someone receives six votes or more then they've received the majority of the votes and yeah if someone has a majority of votes you would think that should then be the person who should win the election now in this situation because there are three candidates there isn't a actual majority winner that's the possibility if you have only two candidates then one candidate must have received uh at least 50 percent of the votes it could be 50 50 because it's a tie but someone got at least 50 if you have two candidates amongst three candidates the only thing you can guarantee is that someone got at least a third of the votes but a third is less than half right someone might have not got a majority no one did now candidate uh candidate one was really close uh algebra got five votes which is one short of six but the geometry party also got four it's only two less than six but here's a very curious observation the algebra party got six last place votes what that means is that a majority of the voters think the algebra party was the worst of the three candidates but yet if we only look at the first place votes it would look like algebra is the most popular candidate right that seems weird like how can the most popular candidate be actually the most disliked candidate the algebra added the majority of last place votes here um and honestly what's happening here has a lot to do with candidates two and three um that is we have this so-called spoiler effect I mentioned this earlier that the out that the trigonometry and geometry parties are very similar from their um political platform right geometry has trigonometric as a subset so these trigonometry people are just very specialized geometry people if you look between the two parties there their two platforms are so similar and again everyone who liked geometry preferred trigonometry over algebra and everyone who like trigonometry preferred geometry over algebra if you were to put the geometry and trigonometry vote together that would give you six votes which is a majority so a majority of the voters in math club preferred the geometric ticket but the the sort of independent here the trigonometry candidate acted as a spoiler if the trigonometry person didn't run in this election right if if they didn't the trigonometry person didn't vote in this election those two people who prefer trigonometry would have been voted for the geometry party instead in which case the geometry candidate C3 would have got six votes which is a majority and not and would have beaten uh the algebra party who only got five right so the fact that the trigonometry party had a candidate in the election spoiled the outcome for the geometry party and gave it to the algebra party so even though algebra got the most votes again algebra was the least preferred and maybe the only reason algebra won was because the trigonometry party spoiled the outcome of the election maybe counting only your first place votes is not a fair outcome to the election so I'm going to turn this thing on its head right okay algebra was the least liked candidate maybe we do re redo the election where you vote for your least favorite that is you tell me who you hate the most and you're like I'll prefer anyone other than that um sadly in many U.S presidential elections that's how the people were speaking um in the 2020 election there was a lot of voter dissatisfaction um between the two major candidates of uh of trump and Biden um and so a lot of people were saying things like oh my goodness I have to choose between the the better of two evils right in 2016 you had Trump versus Clinton people were saying the same thing back then this happens a lot where people are like oh I'm just I have to choose between who I who I hate less than the other basically you're picking who do you hate the most um you're not picking them to win you're picking them to lose so what if we don't focus on picking on the winner what if we focus on picking on the loser what happens in that situation okay so in that case we would only look at people's third place votes okay and so then every time someone votes you for third place let's give you a negative point right that way Whoever has the most wins okay in this situation algebra has the has a majority of last place votes so we're going to get the algebra party negative six points um Canada two the trigonometry party will get negative two points and candidate three would get uh would get negative three points in that situation so if people hated algebra the most in this election and they hated trigonometry the least and after all um you only have two last place votes so in this situation if the voting strategy of the voting method is to vote for the least favorite then it turns out that the trigonometry party would win there now again is that a fair a fair outcome right where we only look at people's last place votes um well you definitely don't vote in the person who has a majority of last place votes that's that's nice but what about what about C2 here sure they have the fewest last place votes but they also have the fewest first place votes not a lot of people hate trigonometry but not a lot of people like it either this trigonometry candidate is definitely the lukewarm candidates no one really likes it but no one really hates it so we sold a lot of unfairness if we only looked at people's first place choice and that same type of unfairness seems to come out if we only look at their last place voice uh last place of choice there so maybe we can make a more fair outcome if we were to look at everyone's um preference their first last and their middle one well how can we do that well maybe I mean because my first place Choice shouldn't count the same as my second place Choice which definitely shouldn't count the same as my last place choice so maybe we do some type of like point system where I give like two points for People's First Choice um then one point for the second so this is not and then a zero for the last right I don't want to give a point to someone I don't like there maybe we do something like that that way my first place Choice gets the stronger thing but you're also still considering my second choice maybe that gives a more fair outcome here for which if that's the case then we do something like the following for for the algebra party you're going to take five times two which gives ten points then zero times one which gives zero points and then zero times six which gives zero points there ten plus zero plus zero gives candidate one ten points there um if you do that for the trigonometry party you're gonna get two times two which is four um plus one times seven which is seven and then you just yeah nothing there because the last doesn't earn you anything that there so you're going to get 4 plus 7 which is 11. so in this case if I look at the second choice because trigonometry is valued so much as the second choice that actually makes it a better candidate than um than the algebra party there but then look at the geometry party here you get 4 times 2 which is eight points plus four points plus no points right you get a plus four which is 12. 12 is actually the most points there so using this like this weighted point system where I'm using the entire preference palette you get the geometry candidate that wins there I want you to point notice what has just happened with one election method we've got algebra to win using a different election method we got trigonometry to win and using this third method we got geometry to win so notice I didn't change the preferences of the voters the voters voted the same way in all of those three attempts but by slightly changing the rules of the election I got a different outcome and in this case that's because the three candidates are all really close to each other but this does illustrate a very important point I like to introduce a quote from a very famous mathematician who studies these type of voting Theory problems uh Dr Donald sorry here which I actually attended a talk of his once many years ago when I was a graduate student at Brigham Young University at the time he actually was given a talk about dark matter he does a lot of work with mathematical physics as well but Dr sorry also uh does a lot of research on mathematical voting and so his quote I think is just perfect for this example here rather than reflecting the views of the voters it is entirely possible for an election outcome to more accurately reflect the choice and election procedure as we change the election procedure using the same voting preference schedule for our 11 voters here we had three different outcomes and as such it then begs the question which was the right outcome which is the best outcome what is the right outcome what is the one that's most fair and these are topics that we're going to dive into as we explore these ideas of mathematical voting Theory inside of this chapter here now to conclude this video I want to then list all of the important terms that came up in our example here so what are the important ingredients of a voting problem well first we have the candidates sometimes they're called the Alternatives because as people vote they're then choosing who they want and these are the options in front of them so the candidates are the things that are going to be chosen all right now these could be people and and typically it could be people who are going to be delegates or representatives in some type of government but we could potentially elect anything for anything it's about making a social choice we have a group of people who have to make a decision and we have to decide now the candidates might be we might have to choose a winner uh but it's also possible that we want to rank them like we have a we choose of who's the winner who's second place who's third place who's fourth place who's that has more to do with the method but the candidates are the people we're going to choose um then the next thing to pay attention to are the voters the voters or the people who get to make the choice the voters their preference is what decides which candidate will win the election so the voters get to say who wins and who loses based upon their preferences um and when it comes to elections uh weird for the moment gonna focus on elections where all voters have the same vote here so one voter one vote type of thing uh we will actually talk about the notion of weighted voting in the future where it's asymmetrical where some voters have more say in the election than others but that's a topic for another day now the the candidate is chosen by the preference of the voters how do the voters share their preference well they fill out a ballot of the ballot is some type of data that's collected about the voters preferences now there's a lot of different types of ballots out there there might be like a winners only ballot where you only um where you only tell the the the the the commission who is your favorite choice and in the previous math club election we had a full ranking so a ranking a fully ranked preference ballot but you might also have like a partial ranking of some kind like if you have 17 candidates it might be difficult for the voter to decide who is the first second third floor fifth sixth seventh eight nine ten eleven twelve thirteen fourteen fifteen sixteen Seventeen oh it's exhausted just saying it but all those decisions might be difficult so it might just be like oh list your top three list your top five um and so that doesn't give us all of the preference but it might be enough to help us determine who's the who's the winner of that election there um the types of ballots that we are going to use in this chapter are going to be these preference ballots where the voter gives their entire rankings of preference of all of the candidates for practical reasons that might not happen in real life because again the Canada pool might just be so large that it's it's too much of a burden um on the voter but it also might be a burden on like the commissioner who is counting the votes as well so while that might not always be the case in real life for the sake of this mathematical exploration we will only use preference ballots full preference balance in our conversations here we also are interested in the well the the tables that we saw earlier is known as a preference schedule it's a table that that indicates what were the preferences of all of the voters there okay uh that was the very first table we showed where I showed you how many people ranked the candidates this way this way in this way these preference schedules will be just be a tabular representation of the ballot information we get from the voters voters because they give us a full ranking of the candidates then we have the outcome who is the winner of the election that's important right so we have candidates who voters vote for they put their information on they put their preferences on a ballot we get gather the ballots we interpret the their preferences and then some outcome is determined and like I mentioned earlier that outcome could be a single person or it could be a couple people and those people could be ranked you have elections where only one person wins you have elections where two people three people small group of people wins and they're not ranked like oh all three of you are going to be on the school board now or there could be a ranked list for which is a first place second place third place fourth place um Sports tournaments athletic tournaments are very much like Democratic elections um for which we have to determine an outcome we rank the candidates in that situation the only difference between a democratic election and an athletic tournament is that an election the preference of the voters determines the outcome as opposed to an athletic tournament where it's the skill of the athletes that determine the outcome or I should say it's more about the points they earn in the games but that is correlated to the skill of the athletes and things like that but that but as you study election Theory it's very similar to how one might study Sports and Athletics there um there's a lot of interesting parallels there all right now as we discover in this example there's also different voting methods that we can do the voting method is the method that takes the voter's preference and determines the outcome of the election so even if you have the same preference the same preference schedule a different voting method can produce a different outcome and that then leads the notion of fairness how fairly does the voting method determine um how fair does the voting method take the voter's preference and determine the outcome because the math club could have a very easy election the algebra party always wins that could be the that could be the voting method is that very fair I'd probably say not because it doesn't take any reflection on the voter's preference whatsoever you could also be like modified okay the winner of the election will always be the algebra party unless there's unanimous support for a different candidate well that is more fair because then it takes some consideration of the voter's preference but still it's a pretty high burden um that in someone to beat the algebra party like why is the algebra party being treated differently than the other parties it seems biased in that regard and so our major goal in this unit is to talk about what are things we can do to quantitatively increase the fairness in an election and it turns out the standard election method used in places like the United States might not actually be the most fair but we'll learn about that of course in the next several videos