 In this video, we provide the solution to question number seven for practice exam three for math 1030, in which case we are given an election preference schedule with four candidates A, B, C, D, and we're asked to determine who's the winner of the election using Copeland's method, that is, pairwise comparisons. So there are six comparisons we have to do between the four candidates there, so let's do them one by one. So there's the comparison between A and B. So if we're looking at A, A wins this one, B is preferred here, A is preferred here, A is preferred here, and B is preferred there. So B is going to win 30 votes, as opposed to A, which would get 65. So A is the preferred candidate in that matchup, so we give A the win. Then with regard to A versus C, similar thing, A is preferred here, C is preferred here, C is preferred here, A is preferred in this one. And C is preferred in that one. So A is going to get 25 plus 30, so that's 55 votes. C is going to get 20 plus 10 plus 10, that's only 40 votes. So A is the preferred candidate in that matchup as well. And then if we do A versus D, we see that A is preferred here, A is preferred here, and here. D is preferred here. So in this situation, D is preferred 40 times, 40 of the votes. But for A here, you're going to end up with 25 plus 20, which is 45 plus 10, which is 55. 55 is bigger than 40, so A is preferred there as well. So honestly, we can stop at this moment, because since A has won all three matchups with all of its possible pairings, and we only have to find the winner here. A is a condorcet winner, and condorcet winners always win in Copeland's method. You can't get more than three points in pairwise comparison if there's only four candidates. So it turns out that A is in fact the winner of the election. Now for the sake of it, I am going to include the other comparisons just for the sake of practice. But to be aware, we don't actually need this information to determine the winner here. It's going to be A. No one can get more votes than A. So if it comes to B versus C, C is preferred here. We'll do underlined. C is preferred here. B is preferred. C is preferred. B is preferred. C is preferred. B is going to get 50 votes. 20 plus 30 is 50. And then C is going to end up with 45. So B looks like they won that one. If you do B versus D, so D is preferred. Then here's B is preferred. D is preferred. D, D. So D is definitely the winner there. We only got 20 votes in preference. So D won that one. And then finally, if you do C versus D, like so, D is preferred. C is preferred. C is preferred. D and D is preferred. So D got 30 preferences. D got, well, way more than 30. You got 30 just right here. So D is the winner there. And so we notice in this situation that C actually turned out to be a condor say loser. It didn't win any of the pair wise comparisons. So first place was A like we saw before. Second place would actually be D. Third place went to B. And then fourth place went to C. Now, of course, this question only wants the first place winner, but just for the sake of completeness, I provided the complete ordering of the candidates.