John Roemer sets out to slaughter a golden calf of positive political theory: the Downsian model of electoral competition and its famous implication, the median voter theorem. Roemer contends that not only is the premise of the Downsian model—political parties seeking election without policy objectives of their own—historically incorrect, but it also leads to implausible predictions. Throughout political history, he postulates, parties have represented disparate interests and have advocated divergent policies; thus, a model that is premised on the opposite and predicts convergence as the only equilibrium—provided an equilibrium even exists—cannot be the right tool for analyzing politics.

Roemer presents a broad set of models illustrating that the one-dimensional Downsian model is by far not the only game in town and should not be adopted as if by default; the right model needs to be carefully chosen on the basis of context and the phenomenon to be explained. The presentation of the models follows a typology that represents the eight permutations of three independent dichotomies: office-motivated (Downs model) versus policy-motivated parties (Wittman model), perfect information versus uncertainty about the electorate, and one versus multiple policy dimensions.

This typology reflects a choice, as other aspects of the electoral competition are held constant throughout. Specifically, parties in the models can commit to the policies they adopt; the models are static, concerned only with one-shot elections without primaries; there is no uncertainty about the parties' goals or preferences; and voters turn out and vote without being strategic (this behavior is optimal provided the election is not repeated, voters are fully informed, and voting is costless).

The first four permutations considered do not hold many surprises: So long as policy resides on a single dimension, parties convergence in equilibrium both in the Downs and in the Wittman model under complete information. With uncertainty, parties continue to converge in the Downs model but diverge in the Wittman model. In an example, Roemer shows how dramatically the predictions of the Downs and Wittman models can differ: In a model of purely redistributive taxation, policy-oriented parties may diverge completely and adopt extreme policies, while of course Downsian parties would converge.

The presentation throughout the book is rigorous and not for the mathematically faint of heart. But Roemer also illustrates the nature of the results and the differences between the models with a wealth of examples that are instructive in their own right. These models are either solved explicitly or parametrized—even parametrized to reflect the U.S. political economy—and solved numerically. The examples examine the theme of class struggle over the tax to finance public goods or one that is purely redistributive.

In some of the most interesting models, party preferences are determined endogenously. Roemer calls a polity a “purely representative democracy” if parties adopt the preferences of their (voting) constituency. In the simplest case, the party's preference is a weighted average of its supporters' policy preference. In the more compelling case, which Roemer terms a Condorcet-Nash equilibrium, the party selects via majority rule a citizen-candidate among its base who represents the party in Wittman-style electoral competition; that is, the candidate competes by maximizing his or her expected utility from policy. In the Condorcet-Nash equilibrium, the median voter of each party chooses the candidate who best represents him or her. Interesting, and reflecting the theme of equilibrium policy divergence, is the possibility that each party's candidate may be more extreme than the party's median voter.

A radical departure from the standard modeling approach is introduced with the “party unanimity Nash equilibrium” (PUNE) concept, which is Roemer's most original contribution in the book. Substantively, the PUNE concept stipulates that parties are composed of heterogeneous factions that differ not only in their policy preference but also with regard to their “attitudes” about political strategy: Militants within the party derive utility from the policy for which the party stands, irrespective of whether or not this policy is apt to win an election; reformists are Wittman-style voters who want the party to maximize their expected utility from policy; and finally, opportunists are Downsian voters who want their party to win regardless of the policy it adopts.

The PUNE concept deliberately leaves the intraparty bargaining among the three factions undetermined, postulating instead that no coalition of factions other than the grand coalition can change the party's platform (hence, the name). This has the technical consequence that any given policy qualifies as the party's best response unless there is a policy available that is (weakly) preferred by all three factions. Since some policies are always Pareto undominated, an equilibrium easily exists. This is Roemer's stroke that cuts the Gordian knot of nonexistence of Nash equilibrium in multiple dimensions.

One may wonder whether the PUNE concept does not impose too much stability by causing parties to respond too sluggishly and leading to large equilibrium sets. In fact, in the one-dimensional setting, with uncertainty any policy pair between the two parties' ideal points constitutes a PUNE, and consequently the concept has virtually no predictive power. In the multidimensional setting, however, the size of the PUNE-set depends on the specific model, and in several extensive examples examining redistributive policy and class politics, Roemer shows that PUNEs can predict and illustrate relevant phenomena.

Political Competition makes its mark along several dimensions. First, it offers a coherent framework for thinking about a large class of spatial models, their strengths and weaknesses, and their relationships with one another. Through numerous detailed examples, it demonstrates how each one of these models may be put to work in a specific context. There could not be a more explicit reminder that the one-dimensional Downs model is but one in a large class of models of political competition, and that its usefulness in a specific context needs to be compared to that of these other models. Second, the book offers a real alternative for modeling spatial competition in multiple dimensions, an enterprise usually stymied by the lack of equilibria in the absence of aggregate uncertainty. Third, the book provides a thorough analysis of the political economy of taxation and redistribution, ranging from a simple one-dimensional analysis of a public goods model to the historic class struggle in interwar Europe.