The Manipulability of Voting Systems

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. 1. INTRODUCTION. When one speaks of a mathematical analysis of voting, two results spring to the forefront: the voting paradox of Condorcet [7] and Arrow's Impossibility Theorem [1]. In fact, most mathematicians-although perhaps unable to state either precisely-have heard of both, and these two results are finding their way into more and more undergraduate textbooks for non-majors; see [6], [28], or [29]. But Condorcet's and Arrow's contributions are, we feel, only the first two parts in a natural progression that is a trilogy-ending with the remarkable Gibbard-SatterthWaite Manipulability Theorem [17], [25]-or perhaps (as we might argue) a tetralogy, culminating in the striking generalization recently proved by Duggan and Schwartz [9], [10]. The basic voting-theoretic context in which we work has ballots that are lists (sometimes allowing ties, sometimes not) and elections whose outcome is a non-empty set of alternatives (again, sometimes allowing ties for the win, and sometimes not). A ballot in which there are no ties is called a linear ballot. Following standard terminology in the field, a sequence P of ballots is called a profile. If P is a profile, then the set of winners, according to some specified voting system V, is denoted by V(P). Most people are aware of several examples of voting systems in this context. Plu-rality, for example, is the system in which the winner is the alternative with the most first-place votes. Scoring systems, on the other hand, assign points to alternatives based on where they appear on a ballot; the special case in which a first-place vote is worth n-1 points, a second-place vote is worth n-2, etc. is known as the Borda count. The Hare system (respectively, the Coombs method) proceeds by iteratively deleting the alternatives with the fewest first-place votes (respectively, the most last-place votes). All of these voting systems can produce ties for the win. There are other voting systems that are less trivial mathematically, but not as well known. For example, assume for simplicity that we have n voters and …