The basic principle of fairness for a game is easy to articulate: Each player should have an equal opportunity to win. For a social choice, the meaning of fairness is a far richer question. The criteria introduced in Chapter 12 focus on the suitability or unsuitability of certain candidates to be the social choice. In this chapter, we introduce various new fairness criteria that focus on the effect of changes in the preference table. We begin with a notion called Monotonicity concerning the invariance of election outcomes when voters increase their preferences for the winning candidate. We then introduce two criteria, Anonymity and Neutrality, that capture two fundamental democratic principles. These criteria focus on the existence of election biases, either for particular voters or for candidates. We prove May's Theorem (Theorem 18.8) to the effect that the only voting method for two-candidate elections satisfying these three fairness criteria is the Simple Majority Method.

We then turn to the notion of an impossibility theorem, an idea made famous by Arrow's thesis. We introduce Arrow's criterion of Independence and prove a simplified version of his impossibility theorem for social choice methods (Theorem 18.12) proved by Alfred Mackay.

Arrow's original theorem focuses on social welfare methods as opposed to social choice methods. We introduce Arrow's Independence Criterion for social welfare methods as well as the Pareto Criterion in this setting. We then state Arrow's Impossibility Theorem (Theorem 18.19). We conclude by proving an impossibility theorem formulated by Amartya Sen that reveals a basic incompatibility between individual liberties and collective rights.

Before beginning, we recall that we do not allow ties in either social choice or social welfare outcomes. We follow Convention 5.9 and use alphabetical order to break all ties arising in the final outcome of an election. The problem of ties will be a more substantive issue in this chapter. Indeed, we will see that our convention leads directly to certain violations of fairness criteria.