 If you limit yourself to simple mathematics, then voting is easy. Whoever gets the most votes wins. But this method of voting, called plurality, often fails to produce results people are happy with. So maybe we shouldn't just limit ourselves to simple mathematics. Maybe we should apply more mathematics and design a better system. One of the reasons that the process of mathematicianization is useful is that it forces us to ask what we want. So the question is, what do we want in a voting system? And the most important requirement of a voting system is probably that the same set of votes should produce the same results. Mathematically, we say that we want a social choice function. Now one potential problem in any voting system is that you may end up with a tie result. We say a system is resolute if it produces only one winner. This means we need to include some method of breaking ties. For example, the 2009 Town Council election for Cave Creek, Arizona ended in an exact tie, 660 to 660. Arizona law requires ties be broken by lot, in other words by chance. In this case, the candidates drew cards and so we have a voting system and we want to ask, is the system resolute and is it a social choice function? Since there is a way to resolve ties, it is a resolute voting system. But the same set of votes could produce a different result depending on which player, candidate, drew a better card. So this system of voting is not a social choice function. This leads us to the work of Condorcet. The Marquis de Condorcet was a French nobleman and mathematician who began to apply mathematics to societal questions. And he suggested the following approach. We should consider how voters feel about all the choices. And this leads to the idea of a preference ballot. A voter's preference ballot is a ranking of all the choices. It will be convenient if we don't allow tied rankings, but not necessary. Everything we're about to do can be done with tied rankings, it's just a little more complicated. For example, suppose you're deciding what to serve at a banquet and can choose between steak, s, chicken, c, or turkey, t. Our voters can rank the choices. So one voter might have preference s over c over t. They prefer steak over chicken and chicken over turkey. So if they rank the three choices, steak would be first, chicken would be second, and turkey would be third. So when they submitted their ballot, that would be one ballot with ranking s, c, t. Another voter might have preference t over s over c. They prefer turkey over steak and steak over chicken. So they'd rank turkey one, steak two, and chicken three. So their ballot would count as a t, s, c ballot. We can collect all the ballots together to produce the preference schedule. This shows how many ballots had a particular ranking. Now let's see if we can read this preference schedule. So I suppose the preference schedule for an election is as shown. We can use this to determine how many voters there were and how many preferred a over b. So this first column represents the voters whose rankings were a over b over c, and this says that zero voters submitted ballots like that. The second column corresponds to the ranking a over c over b, and we see there are 23 voters who submitted those rankings. The third column corresponds to the ranking b over a over c. There were two voters. And the last few columns correspond to the rankings b over c over a, c over a over b, and c over b over a. Now since these are all possible rankings of the three choices, then we know that everyone who voted submitted one of these six rankings. And so there were 60 votes all together. Now we also want to know how many preferred a over b. So remember the ranking is first place, second place, third place, and so we want to find those rankings that have a higher than b. And those are these. And so zero plus 23 plus 10 voters felt that they preferred a over b. And now we can introduce a formal definition of what we're talking about. A social choice function takes the preference schedule and returns a societal rank for the choices. We can interpret any voting system in terms of a social choice function, and even better, we can decide what we want our voting system to do, then create a social choice function that meets our requirements. We'll take a look at that next.