In this chapter, we delve deeper into the idea of an election as a game of strategy (Definition 2.9). The implications of this definition are somewhat unsettling. Is a popular election a game to be won? In a democracy, we expect social choice to be a faithful accounting of the true preferences of the people rather than an outcome arrived at by the strategic cleverness of certain of the voters. Unfortunately, the ideal of a strategy-proof, democratic election is an impossible dream. We state the Gibbard-Satterthwaite Theorem (Theorem 6.4), which asserts that with three or more candidates, every nondictatorial voting method is vulnerable to strategic manipulation. We verify the theorem for each of the democratic voting methods introduced in Chapter 5. We also show how strategic considerations in social choice can elect a third-party candidate in a popular election. The latter result is an example of the Paradox of the Chair. Finally, we give an example of the No-Show Paradox, a scenario in which voters in an election better serve their own interests by staying home as opposed to participating in the election.

The Condorcet Paradox

The developments in this chapter will all flow out of arguably the central example in social choice theory. We introduce this important example by means of the following simple puzzle: