 Welcome back for our review to exam three for math 1030 contemporary mathematics for students at Southern Intel University. As usual, be your professor today, Dr. Andrew Mistledine. So this is our third exam of the semester. If you're watching this video, that means you've probably already taken exams one and two. And so a lot of the policies and structures about the exams for this course don't need to be said again in this video. Some things, of course, I should mention is that with regard to time place and manner of the exam, I can't put those in this video because it changes from semester to semester. So please take a look at the exam syllabus for those specific details. What we are going to talk about in this video are the topics and types of questions that we're going to see on exam three. So the thing to know about exam three is it's going to cover lessons 17 through 28. And there were three main topics that we saw in those set of lessons there. So this is like our political exam here. So we have our unit on voting. We have our unit on weighted voting, which is related but not exactly the same thing. And then we also have what we most recently learned about apportionment. And so these topics which we saw in these lessons are going to be what we see on exam number three. Much like the structure of previous exams, there's going to be 15 questions. This exam is going to have 10 questions in the multiple choice section worth five points each. And then it's going to have five questions in the free response section where most of them are about 10 points, although question 11 is a little bit easier. And so it's going to be worth only eight points. So without further ado, let's talk about specific questions that you're going to see on this exam to help us prepare for we'll study for exam three here. So in the multiple choice section, like I said, there's 10 questions worth five points each. Like usual, a multiple choice answer is either all or nothing. You either select the correct answer and you get the full five points or you don't select the correct answer and you don't get the points there. Now on this exam, because we have a lot of questions about voting, you're going to see a lot of preference schedules like this illustrated for the questions. And in fact, you might see the exact same preference schedule use more than once. That's okay. The outcome of the election potentially could change depending on the method that you use. We've seen that before. And so don't don't fixate on that. If you get the same preference schedule, hallelujah, you know, maybe it makes the task easier for you. It isn't very important that you look very carefully at the instructions on these voting ones, because the prompt will look like if you look at two and three here will look identical with the exception of in the instructions that might specify the voting method that you need to use. So question one, I usually like to start the exam with a fairly simple question. So that's what we're going to see with question number one. You might be asked to compute the winner of an election using the pluralities method, which is perhaps the easiest one to implement. But I also want to throw in here like approval voting, which is also something we learned about. It's also pretty easy to run. I might swap out plurality for approval voting. Remember with plurality, you're going to look for the candidate who receives the most first place of votes. That's the winner. Nothing else is considered. With approval voting, it's very similar. The approval winner is going to be whoever receives the most approval votes. And the thing is a candidate can give more than one vote to different candidates. But you just count things up and see who gets the most. Very, very similar. Approval voting, let's see, we talked about in lesson 20. And then plurality was we started our unit with lesson 17 there. Question number two is going to be a question about weighted voting. So lesson 21 might be one of the better ones to take a look at there. Let me put these markers about lessons there. Plurality, excuse me, was in lesson 17. Approval was in lesson 20. And then the main topics of lesson 21 introduced us to this weighted voting system. So things we should be prepared to do is if we look at one of these rubrics here, one of these scorecards, you can think of for the weighted voting system, can we interpret that? Like, okay, we have here the quota. These are the number of votes of each of the players always written in descending order. So can you be given this information about a voting system and make some analysis to it? Like, this one just asks how many players are there? Just some basic literacy when it comes to these weighted voting cards, scorecards, like so. Question number three, like I mentioned before, is another question about voting. This time, it's going to ask you to determine the winner using the board account like we introduced in lesson 18. So you will be given a preference schedule, either like this one, which could be one you see here, it could be completely different. These things will be randomized, of course. Can you determine the winner using the board account? I might also ask you to determine who is the winner of the election using a modified board account for which I'd give you a point vector, something like, oh, first place votes get five points, second place gets two points, third place gets one point, last place gets zero points, something like that. I would describe it to you and then similar to the board account, you would compute who has the most number of points and that would then be the winner of the election. Question number four is going to be a question about apportionment for which, much like the weighted voting and voting tables that we see before, you'll be given some table about the apportionment here. So in this case, we have our six states, A, B, C, D, E, F. There is some description about buses being distributed, but that's not going to be too much of a concern to us here. We have our six states, we have the populations of the six states, we see how many seats have to be apportioned amongst them, and then you're going to be asked some questions about like, what is the standard divisor? What would be the surplus if we distribute everyone based upon their lower quota? That's how this one's phrased here. What would be the excess if we were to distribute by their upper quotas? Things like that. So question number four is not going to ask you to solve an apportionment problem, but it's going to ask you some of the important ingredients in that problem, like the states, the populations, the standard divisors, the seats, etc. Okay, so be prepared to answer things like that. So this is going to be something very basic about apportionment that we introduced in lesson 24 when we first started talking about apportionment. Question number five returns to that of weighted voting, for which we're going to make things a little bit harder here. So consider the weighted voting system, which we're given here. So we have a quota of five, we have four players whose weights are exactly three, two, one, one. Okay. And so we have some winning coalition players, one, three, and four. And you can actually determine that, sure enough, this is a winning coalition. So then it might ask you, okay, given this winning coalition, which of the players were critical players? So critical players was a big deal when we talked about the bonds off power distribution in lesson 22. Critical players were those players that if we didn't have them in the coalition, it would be a losing coalition. So which of these players is a critical coalition? Maybe you think it's only player four, maybe it's player one and three, you would select that accordingly. A variant of this question, instead of asking you for the critical players, it might ask you who are the pivotal players, like we talked about in lesson 23. In lesson 23, we're interested in the pivotal players. So like if you put them in order in a sequence, so this one really wouldn't work for a pivotal player because for those, you need to have a coalition that's ordered. So like there's a first, a second, third, typically with the coalition, we don't care about the order, but for pivotal, it does matter. You need a sequence of players in that situation. Regardless, the pivotal player is the person in the sequence that tips it over. And so there could only be one pivotal player. So you might be asked who is the pivotal player in that situation. So be prepared to determine like who is the critical players, who is the pivotal players, given various coalitions of voters there. So let's move on to the next page here. Question number six. This would be a question about apportionment and specifically this is coming from lesson 26. This question is just going to ask you to compute the geometric mean. Particularly, it wants you to calculate how you would round this number because this is a skill we use with the Huntington Hill method, of course. So let's suppose that we got a quota of 11.499. And so therefore, if we were to use the geometric mean for this thing, that is, if you're going to round that number, 11.499, you'll notice like 0.499, that's sort of like in the danger zone. It's like, I don't know off the top of my head whether it's going to round up or round down. Yikes. So the first thing you do is you're going to calculate the geometric mean between 11 and 12. Remember that if you have two numbers a and b, their geometric mean is the square root of a times b. So that's the very first thing you're going to calculate. You see a number here, 11.5, 11.478, 11.489. So some of these numbers are the geometric means. Some of them are just made up. In case you're wondering, the other two numbers that aren't the geometric mean are probably the arithmetic and harmonic means, but who cares? Whatever the number is, one of those is the correct geometric mean. So that's the first thing you have to do. The second thing is they're going to ask you, based upon the geometric mean, do you round up or do you round down? Like if you look at this number here, let's say that you think the geometric mean is 11.5. Well, if the geometric mean is 11.5, this number is less than the geometric mean, so you would round it down. Therefore, choice A would be the correct one. But on the other hand, if you believed that the geometric mean turned out to be 11.489, then our Q is larger than 11.489. So we've round up based upon geometric rounding. And so you have, there's two parts to this question. What is the geometric mean? And based upon the geometric mean, are you going to round your quota up or down? And again, this is a skill we use when we do the Huntington Hill method of apportionment. Alright, question number seven, we're going to return to just basic voting here. So can we determine the winner of an election using the pairwise comparison, Copeland's method, which we look at all the pairwise comparisons. So there are four of the four candidates here, A, B, C, D. So we look at all the possible pairings, there's going to be six possible pairs there, we determine one on one, who is the winner of each, you get a point, if it's a tie, you get half a point. And then whoever has the most pairwise victories is in the winner using Copeland's method. Question number eight, very similar, you're going to be asked to compute the winner of the election using the plurality with elimination method, what we often call IRV instant runoff of voting. Remember, this is the method, which if a candidate has a majority of votes, they are declared the winner. If no candidate has a majority of votes, then we remove the candidate with the least amount of first place votes, then those like let's say D was the one who was eliminated. I mean, I don't think it's actually D, but you know, let's say it was D, then you've removed D from consideration, then these people would then shift their votes to player or to candidates A and C, you would then calculate, okay, who has the most first place votes in that situation, if a candidate has a majority of the first place votes, they're declared the winner. If not, you then remove the next player who has the least amount of votes, maybe C, and then you would calculate there. Of course, once you get down to two candidates, one of them must have the majority of votes, so it'll end. So with this one is there's only four candidates in the election, at most you do two eliminations in order to find the winner. Of course, there could already be a majority candidate, maybe the process stops immediately. So be able to compute the IRV winner or the Copeland's method. Both of these came from less than 19. Both methods were introduced there. Be comfortable computing the winner of election using either of those two methods. Moving on to question number nine, this one's more of an oddball type thing because in the lessons, we learned about things like fairness. So with regard to voting, we learned about fairness in lesson 20, and with regard to fairness of apportionment, we also learned about that in lesson 27. So we learned things like, okay, there's the monotonicity criteria, there's the majority criteria, the condorsay criterion, the irrele... I always forget this one, the irrelevance of alternatives. We... these are these fairest conditions with regard to voting. With regard to apportionment, we have some that you can see on the screen right now, the new state's paradox, the Alabama paradox, the population paradox, the lower quota violation, the upper quota violation, the quota rule itself. And so as we've studied problems to do with voting in apportionment, I guess I should mention there also are some fairness concerns that show up with weighted voting, but question 10, we'll deal with that one. So we'll talk about that in just a second. But with these fairness situations that appear with voting and apportionment, there's a lot of things that could happen. So question number nine, what's going to happen is they're going to present you a situation, either an apportionment situation like you see here on question number nine, or I'll give you a voting situation. So maybe some information about a voting preference schedule or something. And then a situation is going to be described. And then you then have to describe which paradox is being demonstrated, which violation is being demonstrated, which fairness criterion is being violated with the situation. So you do need to know those fairness conditions, those paradoxes from those lessons and be able to correctly identify what violation is happening in that situation. So go back to lessons 20 and 27 to see some more like that. Like I said, number 10 is going to be similar, but it's going to be focused on weighted voting. So you're given a weighted voting system like so, and it asks you like, what value should the quota be to make P1 a dictator? And since this is a multiple choice, you can actually check. Does Q equal 10 make P1 a dictator? Does 18 make P1 a dictator? Does 12, you can go through it and see which one in fact does it. And this is one about being a dictator. You also could change the question to be something that like, which one will give, which one, which quota will make players so-and-so have veto power? Which one would make a player so-and-so be a dummy? So these are also issues like when we talk about like dictators and veto and dummies, this has something to do with fairness, right? Because if a player is a dummy, that means their vote never makes a difference. And that doesn't seem very fair. And if a player is a dictator, then that means that their vote is always the determining vote for a coalition. That's not very fair. Because again, when you look at this election right here, sure, player one has 11 votes, but these other players three, three and two have votes. So it should mean something. They shouldn't be dummies. Player one should not be a dictator. That doesn't seem super fair. But of course, with so many votes that player one has, like you'll notice 11 is, its sum is greater than the sum of the other ones. Three, three, two, that gives us eight. So clearly player one is going to need to have veto power. That does seem very fair since player one has so many votes. But we don't want to make these votes meaningless. And therefore there's some fairness issues going on there with dictators and vetoes, etc. And so that brings us to the end of the multiple choice section. Again, you can see more information about weighted voting. These issues were mostly talked about in lesson 21, but of course you can go to 22, 23 also to learn about the power distributions. And these topics we're talking about there as well. A dictator would have, a dictator has 100% of the power. A dummy has 0% of the power. So now let's move to the free response section of this exam. There's five questions. Most of them are worth 10 points, but question 11 is actually worth eight points. It's kind of a Popory type problem. So let me explain what's going to happen here. So there's a lot of versions of question 11 you could see. The version that you see in front of you right now is a question about fairness. Kind of a continuation of what we were seeing in questions nine and 10 in the multiple choice section. So fairness, in this case, I mean fairness of elections, so like lesson 20 stuff, but by all means you could get some fairness with regard to apportionment. So be prepared for something like that as well. Are some fairness issues with weighted voting also come up from lesson 21? Okay, so in this situation you see the problem here. You're supposed to explain or provide an example of why the plurality method might violate the conversation criterion, which remember the conversation criterion says that a conversation winner always is the winner of the election. That is you'd have to come up with a election for which there is one candidate that beats every pair-wise comparison, but doesn't have the most first place votes. So you could come up with an example to demonstrate that, or if coming up with an example is a little bit challenging, you can just kind of explain the general idea of how that would happen. So just kind of explain it here. And again, there's space provided here. This is the free response. You need to make sure in addition to answering a question, you provide any supporting work. So there is some space provided so you can go into details about these things. So the question could be phrased like this, in which case it could change the voting method and the fairness criteria, be like, oh, why does the board account violate such and such? You should be prepared to think about some questions in that regard. But again, it could also be a question about apportionment, like why does the Jefferson method, why might it violate the quota rule? And you either provide an example or at least give an explanation that explains correctly why that happens, even if a counter example, you won't have to pull up with one really quickly. Similar thing with weighted voting. Could you explain how the weighted voting works? Why is so-and-so a dictator or things like that? So fairness is a big thing that's going to come up with this. But also might say another question I'd like to throw in here is like the idea of a tie, because this is something we've talked about very shortly in the lessons there. But how do you deal with a tie? Well, what if I give you a voting method and tell you like, okay, I want you to determine the winner of the voting with respect to this method. But if a tie happens, then use this technique to break the tie and then give me a complete ranking of all of the candidates there, which in case you could probably expect there's going to be a tie, because why would I introduce the tie breaker if there isn't a tie? So some things like that. And so I should also mention that on this exam that we need to know all of the voting methods that we've learned about. So like the big four, plurality, board account, IRV, Copeland's method, you don't know all of those ones. Of course, approval voting, you should also know about that one as well. But I also might on this exam introduce a new voting method. Like I might explain it to you like, oh, here's a new voting method that's a variation of one we've already done. Like tie breaking is one that does exactly that. Those voting methods we've talked about don't actually have an inherent tie breaker built into it. So if some new method or a variation of the voting methods we've learned was introduced, could you determine the winner of the elections? Could you determine the complete ranking? It's not just who's the winner, but who's second place, who's third place, et cetera, et cetera. So one of those types of questions is going to probably appear in number 11, for which, again, you might be asked about some fairness issues, or you might be introduced to some type of like new novel voting concept and asked to be able to use it correctly on the exam right here. All right. So then with that said, let's move to the more straightforward questions in the free response. A lot of them have to do with apportionment as you might imagine. Question number 12 is such a one for which you are going to be asked given the information here. So we have six states, we have populations, we have, where's the number of seats, 200 seats to be apportioned. I want you then to complete the table here that is find the apportionment using, in this case it says Hamilton's method, might also switch this out with Laundice's method for which on question 12, it's going to be one of the two quota methods. We know Hamilton's method or Laundice's method are both quota methods. So use that to fill out this table. So there is work here, there's blank space provided for you to show any work that you might need. But if you fill out the table for this one, that would be considered full credit. Of course, if you fill out correctly, right? I do want to see all of the standard quotas and I need to see all the apportionments. Of course, with Hamilton's method and other quota methods, there might be some surplus to talk about. So again, there might be some things to write down here, but you definitely need to fill out the table in order to get full points on this one. Both of these quota methods were introduced into lesson 24. So feel free to go back there if you need to study them a little bit more. Question number 13 will be about computing a power distribution. So you'll be given a weighted voting system, a quota with the votes of the players and things, and you're going to be asked to compute the power distribution for that weighted system. And there, of course, there's two power distributions we've learned about. There is the Bonsoff power distribution, like this question is asking about, that was introduced in lesson 22. There's also the Schaplers-Schubick power distribution, that was introduced in lesson 23. The Bonsoff distribution has to do with the critical players. So you want to identify all the critical counts of each of the players. The Schaplers-Schubick method has to do with the pivotal players. So you have to determine what's the pivotal count for each of the players and then go from there. The two answers are not identical. So if it asks for the Bonsoff and you try to do Schaplers-Schubick, that's not going to get you the credit you're looking for. And there's a lot to keep track here. So plenty of space to work out and compute that power distribution. That's question number 13 will be worth 10 points. Question number 14 will be another apportionment type question. You'll be given some table with information about states and populations. You'll be asked to apportion some number of seats, in this case, 137. For question number 14, this is going to be a question that involves a divisor method. So things like Jefferson's method and Adam's method, Webster's method. The Huntington Hill method is itself a divisor method. But since it's a little bit different and got its own lecture, we're going to do that in the very next question. So this one's going to be Jefferson, Adams, or Webster's compute to the answer. Now, again, to get full credit, much like with the Hamilton question from before, while there is space provided, what I do need is that this table is filled out completely. So you definitely need to tell me what are the apportionments, the number of seats to go into each of the state. I also want to see what are the modified divisors that's showing your work. Now with the Jefferson, Adams method and Webster's method, since these are divisor methods, they're based upon using modified divisors. And so there's a lot of guess and check for which that works really well on a spreadsheet or some computer program, but can be very time consuming if you're doing it all by hand. So for the sake of the exam, I actually am going to provide to you a successful modified divisor to avoid all the guessing and checking. And so with that modified divisor, I do want to see the correct apportionment of the seats using the various methods. This one says Jefferson's method. So remember, there's like this different business about rounding up, rounding down, the different quota method, excuse me, the different divisor methods have those. So these methods were introduced into lesson 25. So feel free to go back there to study them more if you need. And then this gets us to the last question on this exam question number 15, which will be asking you to do apportionment using the Huntington Hill method. That was the only method introduced in lesson 25. And therefore, you do need to use that one. Same setup as others. You have some number of states with some number of population. You have some number of seats to apportion. But since the Huntington Hill method is a divisor method, and so there's a lot of guessing and checking, I provide to you a successful modified divisor. So use that one. Using that modified divisor, you're going to compute any of the modified quotas. And then there's going to be some rounding in here. Huntington Hill, of course, uses the geometric mean. So determine how many seats need to be apportioned based upon that. And again, there's going to be some calculations you have to do that don't fit in the table. What is the geometric mean of some of these numbers? So fill that in the blanks provided. And now I'll give you the full credit there. All of those three apportionment problems are worth 10 points. The power distribution problem is also worth 10 points. And that then brings us to the end of the exam. One thing I should mention, before we end this video, if there was a topic in our lessons, 17th through 28th, I didn't say anything about. Lesson 28 was about gerrymandering, about how you can take a region and break up into districts in such a way that it has some partisan advantage helping one political party versus another. And perhaps those districts are no longer representative of the political distribution of the region. Can you manipulate the districts to help help one party over another? While that did show up in this unit on political mathematics, for the sake of the exam, because this is actually our biggest unit, there's 12 lessons covered on this exam as opposed to all other exams, which only cover about eight lessons. So this exam is much bigger in terms of topics. And so I did make the executive decision that gerrymandering as a topic will not be covered on exam three. It won't be covered on exam four either. It just won't be covered on the exam. Of course, if you haven't yet done so, do complete that homework assignment associated to gerrymandering. That is part of your graded assignments, but you don't need to study for it as you prepare for this exam. So with that said, we're now at the end of this exam review. Hopefully you found this helpful. In addition to this review video, hopefully you're engaging in the other resources, like you've read the exam syllabus, take a look at the practice exam, etc. If you have any questions, do reach out to me, as I'm glad to try to answer any questions you have for this exam. Best of luck. This is not your first rodeo, and I'm sure you're just fine. Of course, you study for this appropriately.