The elegant median voter result of the Downsian model of electoral competition—which suggests that in order to maximize votes, parties move towards the ideological center—does not entirely translate into multiparty systems. To begin with, as Charles Plott (“A Notion of Equilibrium and Its Possibility under Majority Rule,” American Economic Review 57 [no. 4, 1967]: 787–806) shows, an equilibrium generally does not exist in multidimensional spaces. Moreover, Gary Cox (“Centripetal and Centrifugal Incentives in Electoral Systems,” American Journal of Political Science 34 [no. 4, 1987]: 903–35) demonstrates that if an equilibrium does exist in multicandidate contests, then they must be noncentrist. But multiparty politics are even more complex, as one must consider which coalition will govern following the election and what policy it will implement. These considerations may in turn affect the voters' strategies and the parties' choice of platforms.

Norman Schofield and Itai Sened's contribution is to consider the effect of valence in a multiparty stochastic voting model. When candidates differ in valence, a low-valence candidate generally has an incentive to choose a platform away from candidates with high valence, as any voter will naturally prefer the high-valence candidate if the candidates offer identical platforms. Thus, in a local Nash equilibrium (a pure Nash equilibrium generally does not exist), high-valence candidates tend to locate closer to the center of the voter distribution, while low-valence candidates tend to adopt extremist positions. Candidate convergence can occur, but only under relatively strict conditions. It is interesting that if the conditions fail, even high-valence candidates can improve their vote share by moving slightly away from the center. In Schofield and Sened's model, it is the difference in the candidates' valence that is crucial, along with the dimensionality of the policy space and the variance of the electoral distribution that determines the party system centrifugal tendencies.

One of the nicest aspects of Multiparty Democracy is how closely the formal model and its empirical tests are tied together. The authors consider elections in four parliamentary systems and the United States. The voters' utility functions, with the addition of sociodemographic variables, form the basis of the empirical vote models. Once the parameters of the model have been estimated, they are fed back into the formal model to obtain the “convergence coefficient.” If the convergence coefficient is large, the parties are predicted to diverge from the electoral mean. This is, indeed, the case for most of the elections considered. The convergence coefficient for Italy in 1996, however, is sufficiently low to sustain a convergent equilibrium. Yet, there is little to suggest policy convergence at the electoral mean, but Schofield and Sened argue that the strong effects of sociodemographic factors (which are not a part of the formal model) on the support of the Northern League explain divergence in this case. However, policy divergence occurs in all of the cases considered in the book. One's confidence in the theory would undeniably be strengthened with the inclusion of a case in which convergence did occur and was predicted by the convergence coefficient.

The fact that parties' policy platforms diverge thus offers limited support for the theory, and so it is important to consider the theory's other main implication, that low-valence parties locate farther away from the center. As analytical solutions do not exist, the (multiple) local Nash equilibria are found computationally, using the empirically estimated coefficients. While there are no formal tests to evaluate the closeness of the predicted and observed locations, the results appear fairly convincing, especially with regard to the larger, higher-valence parties that tend to locate closer to the electoral mean. A minor complaint is that only limited information is provided about the computational methods and only examples of the simulated equilibria are shown.

The authors' central formal result assumes vote-maximizing parties, but they also consider variations of the model that assume parties maximize the probability of winning and coalition membership. Some postelectoral situations are characterized by the presence of a core party, which can be assumed to form either a minority or a coalition government. If the core is empty, coalition instability ensues. The postelectoral game may affect the parties' strategies. Some parties may adopt a platform with the aim of becoming the core party, while other parties may seek to prevent this possibility from happening and to extract maximum gains from the coalition bargaining. The authors consider party positioning in several elections on the basis of this logic, which provides for an interesting, even compelling, account of the parties' calculations, but it falls short of a formal test as the theory does not produce sufficiently precise predictions.

The anticipation of the outcomes of the coalition-bargaining stages is highly intriguing and certainly warrants further consideration by scholars. However, Schofield and Sened's voters are assumed to vote sincerely—might they, too, not act strategically in order to influence the postelectoral bargaining? Recent work has shown that voters factor coalition outcomes into their vote choice. One also wonders what “valence” represents. In the second half of Multiparty Democracy, valence appears to represent candidate quality or charisma. But party positions appear to be fairly consistent over time, with some parties constantly located on the “electoral extremes.” If valence determines party platforms' proximity to the electoral mean, this would seem to imply that these parties are consistently incapable of selecting competent or charismatic leaders, while other parties are almost infallible in this regard. Following from my previous point, might it not be the case that valence merely reflects voters' expectations about the likelihood that the different parties will be a part of the governing coalition? The results seem to indicate that valence does matter, but personally, I find the result somewhat unsatisfying because the concept is too vague.

I have mainly touched on what I consider the central results of the book with regard to multiparty parliamentary democracies. Schofield and Sened's treatment of how a party leader's valence makes him or her independent of party activists and allows him to adopt a moderate position is no less insightful. Multiparty Democracy is a rigorous and innovative study of elections and coalition formation. The translation of citizen preferences into policy outcomes in the context of multiparty competition is a complex process, and the authors have enriched our understanding of this process and provided us with important tools for further study.