Social choice theory examines group decision making from axiomatic and mathematical perspectives. It often produces results that have troubling implications for democracy. Consider Kenneth Arrow's general possibility theorem (see Social Choice and Individual Values, [1951] 1963). It shows that no social welfare function can simultaneously satisfy several apparently reasonable postulates involving rationality and ethical norms. When this theorem is applied to the study of politics, it challenges the legitimacy of all collective decision-making procedures. No voting system can guarantee rational social preference orderings through ethical means when there are more than two voters and more than two alternatives in the choice set. Majority rule, for example, has been subject to criticism because it cannot ensure rational outcomes. Rationality, in this case, is defined in terms of transitivity. When majority rule fails to produce transitive collective preference orderings—a condition that is commonly called cycling—the outcomes may be interpreted as arbitrary or incoherent.

An important and interesting body of literature has emerged in the wake of the social choice challenge to democratic decision making. The inspiration for much of this work can be credited to William Riker. His Liberalism Against Populism (1982) has been particularly influential. It takes advantage of social choice findings to launch a sustained attack on the concept of “populist democracy,” which assumes that the outcomes of voting identify something like a popular will. Perhaps the most comprehensive response to Riker and the social choice critique of democracy is Gerry Mackie's Democracy Defended (2004). Mackie provides both philosophical and empirical rejoinders, including refutations of virtually every real world example of cycling found in the literature. His defense of democracy has been called a “true tour de force” that “definitely returns the ball to his opponents' court” (Sven E. Wilson, Perspectives on Political Science 33 [Fall 2004]: 228–29).

Anthony McGann (The Logic of Democracy) and David van Mill (Deliberation, Social Choice, and Absolutist Democracy) are players in the same game as Riker and Mackie. The side of the net on which they stand, however, differs, depending on the issue. They disagree with Mackie, for example, about the implications of not finding empirical evidence of cycling. But they also reject Riker's unqualified acceptance of the axioms associated with Arrow's theorem. McGann and van Mill, therefore, are confronting the same opponents. Nonetheless, it is important to recognize that they do so for distinct reasons.

First, neither is particularly impressed nor surprised by the failure to find instances of cycling in practice. McGann contends that knowledge of the full preference orderings of individuals is usually required to directly observe cycling. This information is typically not available, hence the lack of empirical evidence. Still, the existence of cycles can be inferred from their effects. Pervasive legislative practices like logrolling and coalition negotiation require intransitive social preference orderings (McGann, 70–76). These practices, therefore, imply the presence of cycles. Van Mill offers a somewhat different but complementary analysis. He claims that cycles are usually not observed because voting procedures typically violate the moral axiom of universal domain, which is assumed by Arrow's theorem. Cycles, in other words, are being suppressed. Institutions are inducing transitive social preference orderings (van Mill, 68–69). McGann's and van Mill's responses to the lack of direct evidence for cycling are convincing. In this sense, they have successfully put the empirical ball back in Mackie's court.

Second, both defend the practice of majority rule by challenging the validity of one of the postulates in Arrow's theorem. McGann argues that the consequences of violating the rationality axiom are misconstrued. The effects of cycling are generally misrepresented and its benefits are widely ignored. Cycling, for example, is often assumed to produce chaotic outcomes. This characterization, however, is overstated because the results of majority rule are typically limited to the uncovered set of centrally located alternatives (McGann, 62–70). In addition, he claims that majority rule and the existence of cycles are extremely useful in promoting certain democratic values. Van Mill, in contrast, contends that the moral axiom of universal domain is normatively and practically undesirable. Limiting political participation is an unavoidable part of political life and necessary for producing stability. The task is to combine some type of absolutism and democracy in a way that secures the benefits of each. McGann's and van Mill's critiques of Arrow's theorem place them squarely on Mackie's side of the court. Thus, they join him in serving the normative ball to Riker.

McGann's and van Mill's divided sympathies reflect a desire to stake out a third position in the dispute between social choice theory and political philosophy. They consider Arrow's theorem and majority rule complementary, not antagonistic. They use the insights of Arrow's theorem as constructive critiques, helping them better understand the nature of democratic decision making. This approach not only is unique but also makes a valuable contribution to the study of both social choice and normative democratic theories. Sharing the same basic methodological perspective, however, does not preclude significant differences in McGann's and van Mill's political projects. They are both advocates of majoritarian models of democracy. Nonetheless, they justify majority rule very differently.

McGann's aim is to show that democratic procedures, defined in terms of popular sovereignty and political equality, entail proportional representation and majority rule. The liberal conception of political equality is the key to his analysis (Chapter 2). He claims that the logical requirements of political equality can be satisfied only by proportional electoral systems (Chapter 3). Moreover, he maintains that majority rule is the only procedure capable of treating all voters and alternatives equally (Chapter 4). The supposed cost of using majority rule is its susceptibility to cyclical outcomes. In an innovative twist, McGann convincingly argues that cycles actually advance values associated with political equality, including the protection of minorities (Chapter 5) and reasonable deliberation (Chapter 6). He concludes that cycling is not problematic or fatal for democracy. On the contrary, it is an important part of the democratic decision-making process (Chapter 9).

Thus, McGann finds a way to effectively reconcile cycling and majority rule. He contends that cycles, instead of undermining majority rule, actually make it a more normatively attractive decision-making procedure. In essence, his response to the social choice critique is to transform cycles from a liability to an asset. He accepts the findings of social choice theory and, at the same time, buttresses the argument for majority rule. His solution is elegant and very compelling.

The Logic of Democracy is a truly remarkable and important work. It is an exemplar of intellectual rigor and creativity. McGann confidently engages and integrates the literature on social choice theory, normative political philosophy, and comparative institutions. He obviously has a solid understanding of each field and uses his knowledge effectively in presenting a unified theory of democracy. Particularly impressive is his ability to discuss sophisticated concepts in a very accessible manner. Consequently, the value of this book is not limited to specialists. It demands the attention of anyone interested in democratic decision making.

Van Mill's project is to argue for what he terms “absolutist democracy.” His book is divided into two parts. First, he considers the dispute between social choice theory and deliberative democracy. He observes that Arrow's theorem and Jürgen Habermas's theory of deliberation share very similar assumptions about what constitutes a fair decision-making process. Arrow predicts chaos in the form of cycling, while Habermas expects consensus through preference transformation. Van Mill concludes that deliberation cannot overcome the problem of cycling (Chapter 2). Consequently, one of Arrow's moral axioms must be violated if political instability is to be avoided. Van Mill contends that democracy should abandon the condition of universal domain (Chapter 3). Though seemingly unpalatable, restricting participation in the decision-making process is useful because it can ensure political stability. The task, therefore, is to find an acceptable way to reconcile freedom, equality, and coercion.

The second part of van Mill's project is to defend the violation of universal domain and the concept of majority rule. He begins by using Thomas Hobbes's idea of sovereignty to maintain that absolutism is a necessary feature of political decision making (Chapter 5). In the process, he persuasively argues that Hobbes's understanding of absolutism is compatible with democracy and does not necessarily result in tyranny. Van Mill turns next to a defense of democratic absolutism in the form of majority rule against liberal constitutionalism (Chapter 6). He concludes that democracy is best served by giving absolute political power to the people, as opposed to trusting elites or ostensibly limiting the power by institutions.

Deliberation, Social Choice, and Absolutist Democracy provides a competent analysis of the dispute between social choice theory and deliberative democracy, along with a provocative discussion of Hobbes's political theory. One of most valuable aspects of van Mill's book is its emphasis on the limits of democratic decision making. Social choice theory and deliberative democracy offer idealized understandings of democracy. In practice, however, ideals have to be compromised to achieve political stability. Absolutism, in particular, cannot be avoided. Consequently, democratic theory must find a way to “walk the tightrope between freedom and stability” (p. 72).

The concept of majority rule has been challenged most notably by theories of guardianship, constitutionalism, and, more recently, social choice. In the first two cases, majoritarian advocates have already provided forceful if not persuasive responses. Previous rejoinders to the social choice critique of majority rule have typically tried to overcome the problem of cycling through deliberation or by dismissing its practical relevance. In contrast, both McGann and van Mill accept the findings of Arrow's theorem and simultaneously defend majority rule. This approach is both innovative and valuable. It also demonstrates, once again, the robustness of majoritarianism.