In this chapter, we explore the debate over voting methods that marked the birth of social choice theory as an academic field. Let us briefly recall the history and point of contention. On one side, we have the Chevalier Jean Charles de Borda, a mathematician and military scientist. Borda proposed his voting method in a lecture to the French Academy of Sciences urging the body to adopt his voting method for member elections. Borda's method, as we know, converts voter preferences into a single ranking, the social welfare. While ties do occur, they are not so common. In 1780, Borda published the first mathematical proof of fairness. He proved that the Borda Count can never elect a candidate ranked last by a majority of the voters (a Majority Loser). We proved this result ourselves in Theorem 12.15. In fact, Borda proved a stronger theorem. Answering the open question from Chapter 12, we state the following theorem:

Theorem 21.1 Borda's Theorem The Borda Count satisfies the Condorcet Loser Criterion.

On the other side of the debate is the Marquis Nicolas de Condorcet, a prominent intellectual of the Enlightenment period in France. Condorcet also had mathematical evidence supporting his political ideals for elections. Condorcet had a proof that the best chance for achieving the “correct” result in an election is to obey the majority opinion. Condorcet's Jury Theorem (Theorem 7.10) was stated in Chapter 7. For Condorcet, the majority opinion is expressed through head-to-head Simple Majority elections. As we have seen, the Borda Count does not always select the majority winner as the social choice and so violates a basic principle of fairness.

It was Condorcet himself who pointed out the principal obstacle to his own majoritarian ideals. When there are three or more candidates, the majority opinion, as expressed through head-to-head elections, can be fundamentally conflicted. We can have a Condorcet Paradox. Moving ahead almost 200 years to 1951, Kenneth Arrow used the Condorcet Paradox to prove his Impossibility Theorem, beginning the modern era of social choice theory.