The mathematical analysis of voting methods on the basis of fairness traces back to a debate between the Chevalier de Borda and the Marquis de Condorcet, two prominent eighteenth-century French intellectuals. Borda introduced the voting method we now call the Borda Count for elections with multiple candidates. He argued for the superiority of his method over the Plurality Method using a fairness criterion. Soon after Borda's work was published, Condorcet published a paper revealing a basic violation of fairness by the Borda Count. Condorcet observed that it is possible for a candidate to have the support of the majority of the voters and still lose the Borda Count, a violation of the Majority Winner Criterion. In this chapter, we introduce several fairness criteria: the Majority Winner, Condorcet Winner, Majority Loser, and Condorcet Loser Criteria. We examine the voting methods introduced in Chapter 5 using these criteria.

The study of voting methods by fairness criteria provides an excellent introduction to the dichotomy between proofs and counterexamples in mathematics, as discussed in Chapter 10. The fairness criteria that we present here are each universal statements about voting methods. The question of whether the voting method satisfies or violates the given fairness criterion becomes the prototypical mathematics question: is there a proof or is there a counterexample?

Fairness Criteria

To deserve the name, a fairness criterion should identify a property that corresponds to a “fair” outcome of an election. Devising fairness criteria is part art and part science. Fairness is something of a subjective question, and yet we need a precise, mathematical description of the criteria if we expect to use them to analyze voting methods. The fairness criteria we present will be structured as follows:

Definition 12.1

A fairness criterion is a universal statement of the following form: for all preference tables of a particular type, a voting method produces a social choice with a given property.

We say that a given voting method satisfies the fairness criterion if the universal statement is true for this voting method.