 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 talked about in this course, this idea of voting theory. Over the next couple of 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 actually reflections upon mathematical problems that have arised in US history here. This is a topic I'm pretty passionate about because all of the topics you've learned about in this series have ramifications in real life. There's a reason why we call this class Contemporary Mathematics. Things we've learned about networking and scheduling and being fair. Again, these have effects on how we interact with people. We can use mathematics to make the world a better place. Again, one that I feel very passionate about, this idea of how can we make the world more democratic. It turns out mathematics does offer us a tool here. Now to introduce us to all of the topics we need to understand 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. This is an example of the SU Math Club. Every year, the SU Math Club has an election at the end of the year to elect the new Math Club president who will run the next year. Now, of course, as seniors graduate, they don't get to cast a vote because they're not going to be there next year. 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 Math Club president. Let's look at what that election looks like. One particular year, we have three candidates that ran for the Math Club presidency. This is Math Club after all. 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. 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. My bias doesn't come out here. I'm not pushing for one candidate over the other with this example here. We're going to talk about who won the election. In the Math Club, we do things a little bit different than perhaps traditional US elections. 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. With the three candidates, we would decide who's my first place choice, who's my second place choice, who's my third place choice. Now, again, that might seem like an odd thing to do. People in Math Club sometimes do appear odd to other people, but it turns out there's actually a lot of advantages of having the voters present their entire voting preference as opposed to just their first place vote. That's this example, illustrate that as we will see, not just in this example, but in future examples as well. Now, just so you're aware, if we have three candidates and we have to rank each and every one of them, you 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 can cast, six possible rankings of the three candidates. You could do a candidate 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 remind, I'm using these abbreviations, C1 for candidate one, C2 for candidate two, C3 for candidate three, but to interpret this here, these three people said they wanted to vote for the algebra candidate, then the trigonometric candidate, then the geometric candidate. We had two people who voted for first choice was algebra, second choice was geometry, third choice was trigonometry. 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, the trigonometry party and the geometry party are actually very similar platforms. Honestly, like trigonometry 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 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 placed the algebra candidate between the trigonometric and the geometric candidate. I wanna mention that one happened right here as well. No one places had the preference of geometry, then algebra, then trigonometry. The people who vote for geometry are typically likely gonna vote for trigonometry as their second choice. And those who voted trigonometric are most likely to vote geometric as their second choice. Cause 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. So a lot of people preferred algebra and then they have sort of a second choice between either trigonometry or geometry, a trigonometry or geometry. And then we had six people who then were much more geometric, maybe putting trigonometry first and then put algebra in the last. Again, given those branches of mathematics, that kind of makes sense. Anyways, we didn't 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 now see how they have voted and not just their first choices, we see all of their preferences. Who should win this election? Now, it turns out that can be a very difficult question. And with the present table, I'm gonna rewrite it to make it a little bit more readable, okay? So let's rewrite that table in the following way. We had, 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 third choice, right? And again, this is the phenomenon we saw before. 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 and geometry last or vice versa and no one's political ideology went in that direction. Then we look at the second candidate. This was the trigonometric candidate. Two people ranked C2 as the first candidate. Seven people put C2 as their second choice and 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 but not as their first or last. That kind of makes sense because trigonometry is sort of like in the middle between algebra and geometry. 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 was algebraic, if their top choice was algebra, their second choice is probably more likely to be trigonometry over pure geometry. But conversely, if someone's first choice was geometry, it makes sense that their second choice is probably gonna be more likely trigonometry than over pure algebra. Again, they kind of sit in the middle a little bit that trigonometry party does, okay? And then we had four people who voted for geometry as their first choice, then four people for geometry as their second choice and three people for geometry as their last choice there. So this is the voters' preferences here. This is the so-called preference schedule for this election. So again, I asked the question, who is 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 US election, at least at the timing of this video, the way that election is determined is by counting the votes using the method vote for your favorite. So with 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 as your first place preference. 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, 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 most first place votes, in which case that would then make algebra the winner of this election because they got the most first place votes. That's fair, right? Well, is it really? Let's analyze that just a little bit longer here. Well, sure, see one did get 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. 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 majority in this situation? Well, the idea is you would take half of 11, okay? That's gonna give you 5.5. People can't cast a half a vote. So what we have to do is we have to round this thing up. 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 at least 50% of the votes. It could be 50-50 because it's a tie, but someone got at least 50% of you up 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 one was really close. Algebra got five votes, which is one short of six. But the Geometry Party also got four. That'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 could the most popular candidate be actually the most disliked candidate? The Algebra had the majority of last-place votes here. And honestly, what's happening here has a lot to do with candidates two and three. That is, we have this so-called spoiler effect. I mentioned this earlier that the Trigonometry and Geometry Parties are very similar from their political platform, right? Geometry has a subset. So these Trigonometry people are just very specialized Geometry people. If you look between the two parties, their two platforms are so similar. And again, everyone who liked Geometry preferred Trigonometry over Algebra. And everyone who liked Trigonometry preferred Geometry over Algebra. If you were to put the Geometry and Trigonometry votes 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 sort of independent here, the Trigonometry candidate acted as a spoiler. If the Trigonometry person didn't run in this election, right? If the Trigonometry person didn't vote in this election, those two people who preferred 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 would have beaten 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 gonna turn this thing on its head, right? Okay, Algebra was the least liked candidate. Maybe we do 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 pick for anyone other than that. Sadly, in many US presidential elections, that's how the people were speaking in the 2020 election. There was a lot of voter dissatisfaction between the two major candidates of Trump and Biden. And so a lot of people were saying things like, oh my goodness, I have to choose between 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 have to choose between who I hate less than the other. Basically, you're picking who do you hate the most. 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'd 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? Be that way, whoever is the most wins, okay? In this situation, Algebra has a majority of last place votes. So we're gonna give the Algebra Party negative six points. Candidate two, the Trignology Party will get negative two points and Candidate three would get negative three points in that situation. So the people hated Algebra the most in this election and they hated Trigonometry the least. And after all, you only have two last place votes. So in this situation, if the voting strategy, if 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 outcome, right? Where we only look at people's last place votes? Well, you definitely don't vote in the person who has a majority of last place votes, that's nice. But 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 candidate. No one really likes it, but no one really hates it. So we saw 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 choice there. So maybe we can make a more fair outcome if we were to look at everyone's 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, then one point for their second. So this is not, and then zero for the last, right? I don't wanna 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're gonna do something like the following. For the algebra party, you're gonna take five times two, which gives 10 points, then zero times one, which gives zero points, and then zero times six, which gives zero points there. 10 plus zero plus zero gives candidate one 10 points there. If you do that for the Trigonometry party, you're gonna get two times two, which is four, plus one times seven, which is seven, and then you just, yeah, nothing there, because your last doesn't earn you anything there. So you're gonna get four plus seven, 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 the algebra party there. But then look at the geometry party here. You get four times two, which is eight points, plus four points, plus no points, right? You get eight 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 ballot, you get the geometry candidate that wins there. I want you to 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, Dr. Donald Sarri here, which 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 mattery. He does a lot of work with mathematical physics as well. But Dr. Sarri also does a lot of research on mathematical voting. And so his quote, I think it's 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 of an 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 gonna 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 gonna be chosen, all right? Now these could be people and typically it could be people who are gonna 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, but it's also possible that we want to rank them. Like we choose who's the winner, who's second place, who's third place, who's fourth place, that has more to do with the method, but the candidates are the people we're going to choose. Then the next thing to pay attention to are the voters. The voters are 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. And when it comes to elections, we're for the moment gonna focus on elections where all voters have the same vote here. So one voter, one vote type of thing. 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 candidate is chosen by the preference of the voters. How do the voters share their preference? Well, they fill out a ballot. 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 tell the commission who is your favorite choice. 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 their first, second, third, fourth, fifth, sixth, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, ooh, it's exhausting 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. And so that doesn't give us all of the preference, but it might be enough to help us determine who's the winner of that election there. The types of ballots that we are gonna use in this chapter are gonna 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. Cause again, the candidate pool might just be so large that it's too much of a burden 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 ballots in our conversations here. We also are interested in the, well, the tables that we saw earlier is known as a preference schedule. It's a table that indicates what were the preferences of all of the voters there, okay? That was the very first table we showed where I showed you how many people ranked the candidates this way, this way, and this way. These preference schedules will be just the tabular representation of the ballot information. You get from the votes, 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 gather the ballots, we interpret 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, a small group of people wins, and they're not ranked like, oh, all three of you are gonna be on the school bar now. Or there could be a ranked list for which is a first place, second place, third place, fourth place. Sports tournaments, athletic tournaments are very much like democratic elections 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 in 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 as you study election theory, it's very similar to how one might study sports and athletics there. There's a lot of interesting parallels there. All right, now as we discovered 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 schedule, a different voting method can produce a different outcome. And that then leads to the notion of fairness. How fairly does the voting method determine, 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 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 it. 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 that 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.