00:01
For this question, we're told that five candidates are running for office, a, b, c, d, and e.
00:06
Which candidate wins under the board account method and the pairwise comparison method? so let's run the board account method first.
00:14
So for the board account method, everyone that gets a first place vote gets five points, second plate gets a four points, four points, third place three, fourth place two, and fifth place one.
00:24
Okay.
00:24
And again, they get five points because there's five places.
00:28
So five, five, four, four.
00:36
Three and there's not coming out very well let me try this again five four three two and one okay so basically we need to go through and add up all the points and don't forget that these numbers along the top tells me how many ballots look like this right so let's go ahead and find that point values for a okay so for three ballots for three ballots a is going to get three points.
01:15
Plus, for four ballots, a is going to get four points.
01:27
Plus for four ballots, a is going to get two points.
01:36
Plus for six ballots, a is going to get three points.
01:45
Plus for two ballots, a is going to get three points.
01:53
Plus for one ballot, a is going to get four points.
02:00
Okay? so we just need to add all that up, and we get 61.
02:06
Right, and then we have to go through and do the same thing for b.
02:11
So again, for three ballots, for three ballots, b is going to get five points for first place.
02:20
Plus for four ballots, b is going to get two points for fourth place, plus for four.
02:30
Four ballots.
02:31
And again, notice that all of the numbers out in front of the parentheses are going to be the same, because they're always these numbers on the top here.
02:37
So for four ballots, b is going to get another five points, plus for six ballots, six ballots b is going to get one point, plus for two ballots, b is going to get another five points.
02:59
Plus for one ballot, b is going to get two points.
03:04
And again, we just need to add these up, and we get six.
03:10
61.
03:12
All right.
03:12
And then let's go along and do the same thing for c.
03:20
So now i'm going to just do this a little bit quicker here.
03:22
So three ballots, four ballots, four ballots, six ballots, two ballots, and then one ballot there.
03:40
So c gets four points, first set of ballots, and then five points, the next set of ballots, and then three points and then four points and then two points and then one point so again adding all these up and we get 73 continuing on now with d so i'll do the same thing here three ballots here four ballots next four ballots next six ballots two ballots and then there's one ballot left and so for ballot a is going to get three points sorry d sorry d i was going to say already did they so d is going to get two points there and then d is going to get three points and then d is going to get four points and then d is going to get five points and then d is going to get one point and then d is going to get three points and again we just have to add all this up and we get 69 and then and lastly, let's do e.
05:10
So again, we'll need three ballots here, four here, four here, six here, two here.
05:23
And then, of course, there's just the single ballot at the end.
05:26
So e is going to get one, and then e is going to get one, and then e is going to get one, and then e is going to get two, and then e is going to get four.
05:37
And then lastly, e gets five.
05:39
Right? so again, adding all these up, and we get 36.
05:45
So we can see here that with the total of 73, with a total of 73, our winner would be candidate c.
05:54
Okay, our winner would be candidate c.
05:57
So that is the board account method.
06:00
All right.
06:01
So now let's look at the pair -wise comparison method.
06:06
And so for pair -wise comparison, what i have to do is i have to take each pair of two candidates and compare them head to head.
06:13
And then we will then we'll just count up the total points and then see who wins.
06:23
All right.
06:24
So the first thing we need to do is we just need to add up this top row here.
06:27
And we see that there are 20 balance.
06:31
Okay.
06:31
So for one person to beat another person, they had to have been ahead of them for 11 or more times.
06:38
Okay.
06:38
So we're going to write all of our pairings.
06:40
So we're going to do a, compare a to b.
06:43
So i'm just going to go through and i'm going to count how many times a is over.
06:46
B and then i'm going to add those up and if it's more than 10 or 11 or more than a is going to win otherwise b is going to win okay so if i look at this first set here uh a i have b in first place and a in third place so a is not over b but in this set a is over b so that's four that's four a is not over b in this set a is over b in this set for a total of four point plus six, that's 10.
07:19
A is not over b here, but a is over b here for one more for 11.
07:25
Okay, so a was over b 11 times 11 out of 20, which is more than half.
07:30
So a wins head to head over b.
07:33
Okay, then let me do a to c.
07:38
So i'm going to go through and do the same thing.
07:40
I want to count how many times a is over c.
07:43
All right.
07:44
So in the first set, c is over a.
07:47
And the second set c is over a and the third set c is over a and the fourth set c is over a.
07:52
And the fifth set, a is over c for two times.
07:56
And a is over, that's not enough.
07:58
So c wins head to head over a.
08:02
Then i need to compare a to d.
08:05
So i'm going to go through and do the same thing.
08:08
So in this first set, a is over d.
08:11
So that's three.
08:13
And the second set, a is over d.
08:15
That's four.
08:17
Four more for seven.
08:19
And the third set, a is not over d.
08:22
In the fourth set, a is not over d...