Kenneth Arrow's 1951 doctoral thesis launched a new wave of research on the mathematics of social choice. As with Nash's Equilibrium Theorem in game theory, Arrow's Impossibility Theorem has become the centerpiece of modern social choice theory, inspiring countless extensions, variants, alternate proofs, and responses.

We prove Arrow's Theorem (Theorem 18.19) in this chapter following the elegant argument given by Amayarta Sen. We begin by reviewing the basic ingredients of the statement (social welfare methods and their fairness criterion). Next, we introduce the idea of a dictating set and characterize the Pareto Criterion in terms of this notion. We then state Arrow's Principle, which asserts that dictating power passes from a set of voters to some proper subset. The Impossibility Theorem is then deduced as a direct consequence of Arrow's Principle. Our main work consists of proving Arrow's Principle. We prove this result using two further principles called the Power Contraction and Power Expansion Principles.

We conclude with some developments in social choice theory appearing after Arrow's thesis. We prove a theorem of Sen, called the Sen Coherence Theorem, that gives conditions on a preference table that ensure that the social preference relations are transitive. When the voters have Sen-coherent preferences, Condorcet's ideal of majority rule can be realized. We also introduce two voting methods that have attracted recent interest, Approval Voting and Majority Judgment. These methods do not fit neatly into our framework for voting methods. We briefly discuss the fairness properties of these methods.

Social Welfare and Fairness

Recall that a social welfare method is a mechanism that takes a preference table and produces a ranking of the candidates called the social welfare. We assume that ties are broken (see Convention 5.9) so that the ranking is strict, first to last. Recall that each of our social choice methods extend to a social welfare methods.