 So, remember, the preference schedule is a record of how all voters feel about all choices. We can define a social choice function that takes the preference schedule and ranks all the choices. Now, many voting systems can be identified with a social choice function, but as we saw, plurality fails the Condorcet winning condition, and in fact, it often elects the Condorcet loser. So can we try to do better? So one possibility is something called a Borda count. In a Borda count, named after Condorcet's contemporary, Jean-Charles Borda, the choices are given points based on how a voter ranks them. A last place vote gives zero points, the next to last place gets one point, and so on, and the winner is the choice with the most total points. So let's see how that works out. Suppose an election has three candidates A, B, and C, and a preference schedule that looks like this. If the election is run on using a Borda count, which candidate wins? Since there's a first, second, and third place, a last place, third place, vote is worth zero points, a second place is worth one, and a first place vote is worth two points. So let's find those point totals. It's probably easiest to go candidate by candidate. We see that A wins three plus seven ten first place votes, and that gets candidate A two times ten twenty points. A also picks up some second place votes here and here, and that's one plus two three second place votes for three more points, and so A wins a total of twenty three points. Let's take a look at B. So B wins these first place votes, and that's one plus eight nine first place votes, and that gets B eighteen points, and then these voters and these voters gave B a second place vote, so that gets B an additional nine second place votes for an additional nine points, and altogether B wins twenty seven points. Finally C wins first place votes from these voters, two plus six eight first place votes for sixteen points, and these voters here and here gave C second place votes, that's fifteen second place votes for fifteen more points, and so altogether C wins thirty one points, and the winner of the board account is the candidate which scores the most points, and that's going to be C, and so C wins if we use the board account. Another common possibility is something called instant runoff voting. In instant runoff voting if any choice wins a majority of first place vote, that choice wins the election. Of course if there's three or more candidates it's possible that nobody gets a majority, so if no choice has a majority, the choice with the fewest first place votes is eliminated, and the corresponding ballots are re-ranked, and we'll repeat this process until done. Now this might sound a little complicated, but actually it's pretty easy to apply in practice, so let's consider the same preference table, and first of all let's note how many total votes we have. So there are twenty seven votes all together, so a majority is more than half, so that would be fourteen votes. So we'll look at the first place votes only, and we see that A wins three plus seven ten first place votes, B wins nine first place votes, and C wins eight first place votes, and so no candidate has a majority. Now you might recall that plurality takes our preference schedule and just counts the first place votes, so our first step here would actually determine the plurality winner if this were plurality A would win the election. So again since no candidate has a majority of first place votes, but C has the fewest first place votes, so they are eliminated, and the ballots re-ranked. So for these two sets of preference ballots, C was in the last place, so eliminating C doesn't change anything. But in the remaining ballots, the elimination of C will cause the lower ranked candidates to move up one place. Now in the real political universe, this could have been accomplished by having a runoff election between A and B, but that would require running a second election, and so the reason that this is called instant runoff is that we collect enough information to be able to run that second election instantly. And again, we'll count only the first place votes. If we run that runoff election, then we see that A wins 12 first place votes, and B wins 15 first place votes. And what this means is that B wins if we use instant runoff. So notice that when our preference schedule was this, the winner depended on how we voted, using plurality A1, using the board account C1, and using instant runoff B1. So what's the best system? Well, obviously it's a system that elects the choice that you want, but we're supposed to choose the system before we run the election. So the real question is, what do you want the system to do?