 We can analyze voting mathematically by having each voter rank the choices best to worst, compiling the rankings into a preference schedule, then applying a social choice function to determine a winner. Now we want our social choice function to be resolute so that we have a way of breaking ties, but what else do we want? One seemingly desirable feature is the majoritarian principle. If the majority, that's more than 50%, supports a choice, that choice should win. The majoritarian principle gives us a way to decide the winner in a two-candidate election, and this suggests the following. Suppose a candidate wins against all other candidates in a one-on-one election. That candidate, known as the Condorcet winner, should be the winner of the election. For example, suppose the preference schedule for an election is as shown. If there is a Condorcet winner, what is it? So we need to consider the possible one-on-one elections. There are three choices, S, C, and T, so we need to compare S to C, S to T, and C to T. Let's take a look at that first comparison, S to C. If a voter preferred S over C, then S would be higher on their preference form than C. This corresponds to the voters who submitted ballots S over C over T, S over T over C, and T over S over C. And the number of voters in each of these three cases, so 3 plus 1 plus 3, 7 voters preferred S over C. Meanwhile, the remaining 3 plus 1 plus 2, 6 voters preferred C over S. So S wins the majority, and in a one-on-one election, S versus C, S wins. We also want to run S versus T, so the voters who preferred S over T were those who submitted the ballots with ranking S over C over T, S over T over C, and C over S over T. So 7 voters preferred S over T, and the remaining 6 voters preferred T over S, and once again S wins the majority. So in the contest S versus T, S also wins. Now we'll run that third election C versus T in a moment, but at this point the important thing to notice is that since S won against the other candidates, S versus C, S versus T, then S is the Condorcet winner. Let's go ahead and see the results of that third race just to be complete. The third race is C versus T, and again those who preferred C over T submitted the ballots. And so there were 7 voters who preferred C over T, 6 who felt the opposite, and C wins the majority. So in the contest C versus T, C wins. Now a Condorcet winner is a candidate that wins every one-on-one race against another candidate, and we can extend this idea to have a Condorcet loser who loses every one-on-one race against another candidate. To suggest a Condorcet losing condition, the Condorcet loser should never be the winner of the election. So let's take a look at our preference schedule. Do we have a Condorcet loser? And we already found these results when we found the Condorcet winner. The thing to recognize here is that of our 3 choices, T never appears as a winner. We lost every one-on-one election, so T is a Condorcet loser. The Condorcet winner would seem to be the best way to make a decision, except a Condorcet winner doesn't always exist. Even worse, we can't always translate Pairway's preferences into a meaningful result. So Condorcet continues the preference schedule and observes the following. Let's consider how many preferred A over B, how many preferred B over C, and how many preferred C over A. So those who preferred A over B, that's these 3 groups of voters, 33 all together, that's a majority. For those who preferred B over C, that's these 3 groups of voters, and again A majority preferred B over C. A majority of voters prefer A to B, a majority of voters prefer B to C. So you'd expect a majority of voters to prefer A over C. Well the voters who prefer A over C are this set, but that's only 25 of the voters, and this is not a majority of the voters. And we say that preferences aren't transitive. Now this is usually called a paradox because it's an unexpected result, but it actually isn't. We see something like this happen in individual preferences as well. So given a choice between steak and chicken, you might choose steak. But if we're out of steak and you now have a choice between chicken and ice cream, you might choose chicken. But if we try to choose between steak and ice cream, you might choose ice cream even though you prefer steak to chicken and chicken to ice cream. Let's take a familiar voting system and see what happens with it. So let's first of all find the social choice function associated with plurality. So remember how plurality works. Everybody votes for one choice, and the choice with the most votes wins. So let's consider our preference schedule. Suppose you have a ranking of three choices, A over B over C, but you could only vote for one of them. Then you would vote for, well for now let's keep it simple and take the obvious choice if A is our first choice and the only get to vote for one will probably vote for A our first choice and so plurality is the social choice function where only the first place votes are relevant as the choice with the most first place votes wins and the important thing here is we can take our preference schedule and see who wins in plurality or any other voting system. So in plurality we only count the first place votes. We see that S has 3 plus 1, 4 first place votes. C has 3 plus 1, 4 first place votes. And T has 3 plus 2, 5 first place votes. And since T has the most votes, T wins a plurality vote. So is this a good system? Well certainly for T it's a great system. It's the best system in the world. Can't possibly be better. I take full credit for this system because I invented it. But let's consider it from an objective point of view. While T won the plurality vote, remember we found that S was the condorcet winner and T was in fact the condorcet loser. And what this means is that plurality fails the condorcet winning condition and plurality also fails the condorcet losing condition. And so the question we might ask is, can we do better?