Spatial Voting Models Research Articles

Discontinuity and non-existence of equilibrium in the probabilistic spatial voting model

This paper shows that in the simplest one-dimensional, two-candidate probabilistic spatial voting model (PSVM), a pure strategy Nash equilibrium may fail to exist. The existence problem studied here is the result of a discontinuity in the function mapping the candidates' platforms into their probabilities of winning. Proposition 1 of the paper shows that, whenever this probability of winning function satisfies a certain monotonicity property, it must be discontinuous on the diagonal. As an immediate consequence of the discontinuity in the probability of winning function, the candidates' objective functions are discontinuous as well. It is therefore impossible to invoke standard theorems guaranteeing the existence of a pure strategy equilibrium, and an example is developed in which in fact there is no pure strategy equilibrium. Finally, however, it is demonstrated that, for a large class of probability of winning functions, the PSVM satisfies all the conditions of a theorem of Dasgupta and Maskin (1986a) which guarantees that it will always have an equilibrium in mixed strategies.

Adaptive Models and Electoral Instability

The spatial model of voting is a benchmark in theories purporting to explain political behavior. Underlying the spatial model is the assumption that both voters and candidates possess complete information. Despite the fact that much of the survey literature fails to confirm this assumption, few formal theorists have modeled electorates where the voters and candidates lack full information. In large part, spatial theory's failure to illuminate problems of this kind stems from its reliance upon an unrealistic model of human cognition: substantive rationality. This paper returns to Downs' original statement of the importance of information costs in voter decision-making, and extends Down's analysis to include the constraints faced by candidates. In this paper, complexity theory provides the framework to construct computational experiments that explore the use of information by political actors. As in real elections, voters have a finite amount of attention they dedicate to political issues, and candidates possess a fixed budget with which to poll the electorate. The inclusion of the costs of acquiring information for both voters and candidates results in a broader set of electoral outcomes that challenge current formal results.

THE ROCHESTER SCHOOL: The Origins of Positive Political Theory

▪ Abstract The Rochester school of political science, led by William H Riker, pioneered the new method of positive political theory. Positive political theory, or rational choice theory, represents the attempt to build formal models of collective decision-making processes, often relying on the assumption of self-interested rational action. This method has been used to study such political processes as elections, legislative behavior, public goods, and treaty formation and diplomatic strategy in international relations. In this article, we provide a retrospective account of the Rochester school, which discusses Riker's theoretical synthesis and his institution building in the 1950s, 1960s, and 1970s. We discuss some of the most important Rochester school contributions related to spatial models of voting, agenda setting, structure-induced equilibria, heresthetics, game theory, and political theory. We also briefly situate positive political theory within the larger context of political science and economics.

Equilibrium in multicandidate probabilistic spatial voting

This paper presents a multicandidate spatial model of probabilistic voting in which voter utility functions contain a random element specific to each candidate. The model assumes no abstentions, sincere voting, and the maximization of expected vote by each candidate. We derive a sufficient condition for concavity of the candidate expected vote function with which the existence of equilibrium is related to the degree of voter uncertainty. We show that, under concavity, convergent equilibrium exists at a “minimum-sum point” at which total distances from all voter ideal points are minimized. We then discuss the location of convergent equilibrium for various measures of distance. In our examples, computer analysis indicates that non-convergent equilibria are only locally stable and disappear as voter uncertainty increases.

Opposition backlash and platform convergence in a spatial voting model with campaign contributions

This paper investigates the effects of campaign contributions on candidate behavior in elections. The particular focus is on how candidates choose their platforms when they know that the positions they take will influence the level of campaign contributions that they (and their opponents) receive from concerned interest groups. The analysis is carried out in the context of a simple one- dimensional spatial voting model with two candidates and two interest groups. Since the earliest Hotelling-Downs formulations, a central issue in the literature on spatial voting has been the degree to which, under various sets of assumptions, the candidates' platforms converge in equilibrium. This paper extends that literature by examining how the introduction of interest groups making campaign contributions affects the degree of platform convergence. The paper shows that when choosing their platforms, candidates face a trade-off between generataing increased support from opponents and provoking a backlash from the opposition. An example is developed to illustrate a surprising result that can occur because of the backlash effect: the introduction of two extremist interest groups may lead the candidates to moderate their platforms, resulting in a greater degree of platform convergence than would be observed in the absence of any campaign contributions.

The Spatial Theory of Voting and the Presidential Election of 1824

Theory: We extend research on the spatial theory of voting and the electoral connection by exploring how U.S. House members responded to Electoral College gridlock in the presidential election of 1824. We analyze whether John Quincy Adams's victory was consistent with a spatial-theoretic, ideological model of voting or more closely followed the standard historical account that emphasizes Adams's role in a vote-buying corrupt bargain. Hypotheses: The corrupt-bargain thesis predicts that several states won by Adams in the House ballot swung on votes cast by MCs closer to Andrew Jackson than to Adams. The sincere voting model predicts MCs to vote for the closest candidate. Prediction errors should be most frequent near the cutting line between Adams and his closest opponent. Methods: We use a spatial model of sincere voting as a baseline against which to test the corrupt-bargain hypothesis. We use logistic regression to further test implications from the spatial model. Legislator and candidate preferences are based on transformed Nominate scores (Poole and Rosenthal 1997; Poole 1998), which locate ideal points for House members and the 1824 presidential candidates in a common space. Results: Adams's victory and profile of support were consistent with the sincere voting model and did not support the corrupt-bargain hypothesis. Voting errors involving Adams lie systematically closer to the cutting line than did correct votes. Additional evidence from an examination of lame-duck MCs' subsequent careers and an analysis of the congressional elections of 1826 support our findings.

Nash equilibrium in multiparty competitionwith “stochastic” voters

Theoretical spatial models of electoral voting tend to predict either convergence to an electoral mean (when voting is probabilistic) or chaos (when voting is deterministic). Here, we construct an empirical model of voting for the Israeli Knesset in 1992 (based on a large electoral sample and on analysis of party declarations). The probabilistic voting model so estimated fits the known election results. We then use the same model to simulate the effect of expected vote maximization by the parties. Contrary to the usual results, there is no unique convergent Nash equilibrium under this objective function. We do infer, however, that the two large parties are “Downsian”, in the sense that they maximize expected vote (up to the margin of error of the model). We suggest that the empirical results are compatible with a hybrid model of utility maximization, where each party computes the effects of its policy declaration both in terms of electoral response and of post-election coalition negotations.

Landscape formation in a spatial voting model

In spatial models of political competition, parties select multidimensional platforms with the objective of garnering votes. If parties have limited information and are constrained to move locally, i.e. if they are adaptive, then the slope of the multidimensional landscape on which they choose positions becomes important. In this paper, we formally derive a relationship between voters' preferences and the slope of the electronal landscape.

The proximity and the directional theories of issue voting: Comparative results for the USA and Germany

Abstract In several recent studies George Rabinowitz and his co–authors challenge the ‘classical’ spatial model of issue voting, the proximity model, by introducing a directional model. In this article we examine whether different measurement of perceived issue positions of candidates or parties leads to diverging judgments about the predictive power of the directional model (which is claimed to be empirically superior), as compared to the proximity model, using data from the USA and Germany. The results demonstrate that the measurement preferred by Rabinowitz et al. tends to bias empirical findings in favour of directional theory. If we use a more plausible operational definition of issue positions of candidates and parties the directional model in both countries fails to turn out superior.

An analysis of voter predictive dimensions and recovery of the underlying issue space

Based on the spatial voting model posited by Enelow and Hinich, this paper presents a means of determining the nature of the issue dimensions used by voters. This method allows for calculation of the extent to which the predictive dimensions recover the underlying issues space. Of particular interest is the extent to which a single predictive dimension can recover the issue space. The method also suggests that Enelow and Hinich are correct in their hypothesis that the social liberal-conservative axis and the economic liberal-conservative axis are the dimensions most used by the electorate.

Partisan views on the economy

textabstractIn this paper it is argued that political parties may have incentives to adopt a partisan view on the working of the economic system. Our approach is based on a dynamical spatial voting model in which political parties are policy oriented. This model revolves around two interrelated issues x and y. The policy maker sets x directly. There exist two views on the relationship between x and y. Model uncertainty confronts policy makers with the problem of the selection of a model to base their actions on. We show that if voters have imperfect information about the working of the economic system that model selection contains a strategic element. Policy makers are inclined to adopt a view on the working of the economic system which fits in with their preferences. There is no inherent logic that places monetarists to the right of New Economists. They have different models of economic mechanism, but they need not have different political values. A conservative can be a Keynesian and a liberal a monetarist. These combinations are in fact surprisingly rare. James Tobin, 1974,The New Economics One Decade Older, p. 62. I am greatly indebted to Peter Broer, Ben Heydra, Jos Jansen and Wilko Letterie for many helpful suggestions. Furthermore, I would like to thank an anonymous referee for his comments.

Incomplete Information and Ideological Explanations of Platform Divergence

One of the paradoxes of formal spatial voting models is the robustness of the theoretical result that candidates will converge toward centrists positions and the empirical observation of persistent policy divergence of candidates. A solution is that candidates are ideological (have policy preferences). When candidates have policy preferences and incomplete information about voter preferences, then platform divergence is theoretically predicted. Experimental tests of the ideological model are presented. It is shown that platform divergence is significant when candidates are ideological and have incomplete information about voter preferences. However, candidate positions are more convergent, on average, than the theory predicts, suggesting that subjects value winning independently of the expected payment.

The Scotland—England Divide: Politics and Locality in Britain

Growing territorial cleavages within Britain have attracted considerable interest from social scientists in recent years. New spatial models of voting, accounting for the declining homogeneity of electoral behaviour,1 have drawn attention to localised patterns of political support, as well as the emergence of regional cleavages across Britain as a whole.2 Talk of a North–South split has become the common currency of political commentators to account for regional disparities between Conservative and Labour support.3 Alongside these local and regional divisions in voting patterns, there is a new interest in broader aspects of the political differences between the nations of the UK, notably between England on the one hand, and Scotland and Wales on the other.4

Adaptive Parties in Spatial Elections

We develop a model of two-party spatial elections that departs from the standard model in three respects: parties' information about voters' preferences is limited to polls; parties can be either office-seeking or ideological; and parties are not perfect optimizers, that is, they are modelled as boundedly rational adaptive actors. We employ computer search algorithms to model the adaptive behavior of parties and show that three distinct search algorithms lead to similar results. Our findings suggest that convergence in spatial voting models is robust to variations in the intelligence of parties. We also find that an adaptive party in a complex issue space may not be able to defeat a well-positioned incumbent.

The aggregated excess demand function and other aggregation procedures

Two theorems are given; the first extends the Sonnenschein-Mantel-Debreu theorem characterizing aggregate demand functions from the set ofn≧2 commodities to all of the 2 n −(n+1) subsets of two or more commodities. The second theorem concerns spatial voting models for k≧2 candidates over a space of n≧2 issues. The theorem characterizes the sincere elecion rankings of thek candidates over all of the 2 n −1 subsets of one or more issues. Both theorems have the same kind of conclusion; anything can happen. By demonstrating the mathematical reasons for these conclusions and by recalling related, recent results from statistics, voting, and economics, it is argued that this “anything can happen” conclusion is the type one must anticipate for aggregation procedures; particularly for the processes commonly used in economic models where the procedure is responsive to changes in agents' preferences, changes in data, etc.

Evaluating dimensionality in spatial voting models

Spatial models of voting are widely used in Economics and Political Science. Spatial theory is an attractive framework to analyze choice because Euclidean geometry is easy to visualize and the language of politics is full of spatial references. Empirical tests have generally supported the theory but the estimation methods employed to produce the spatial representations of voters have raised serious statistical issues which have not been fully resolved. One of these issues is determining the number of dimensions. This is a difficult problem because the number of estimated parameters increases with the number of dimensions. We show a solution for this problem in this paper. We assume a uniform distribution of voters through an N-dimensional unit hypersphere with perfect spatial voting and equally salient dimensions. We then solve for the projection of this perfect voting onto one dimension and the resultant classification error. We derive the probability density function of the classification errors which can be used to calculate the likelihood that a sample of votes was drawn from an N-dimensional hypersphere. Our Monte-Carlo investigation into the properties of this likelihood function shows that it can be used reliably for low N with small sample sizes.

A Spatial Model of Roll Call Voting: Senators, Constituents, Presidents, and Interest Groups in Supreme Court Confirmations

We test a spatial model of Supreme Court confirmation votes that examines the effects of (1) the ideological distance between senators' constituents and nominees, (2) the personal ideologies of senators, (3) the qualifications of the nominee, (4) the strength of the president, and (5) the mobilization for and against nominees by interest groups. The data consist of the 1,475 individual confirmation votes from the 1955 nomination of John Harlan until the 1987-88 nomination of Anthony Kennedy (voice votes excluded). All of the above factors significantly affect confirmation voting. The model explains 78% of the variance in senators' decisions, predicts 92% of the individual votes correctly, and predicts all of the aggregate outcomes correctly.

Limiting median lines do not suffice to determine the yolk

The yolk, an important concept in the spatial model of voting, is defined in two dimensions as the smallest circle intersecting all median lines. In the literature one finds the assumption that the limiting median lines, i.e., those that pass through two voter ideal points, suffice to determine the yolk. Counterex-amples are given here which invalidate this assumption.

Limiting median lines frequently determine the yolk

In two-dimensional Euclidean spatial voting models the yolk is the smallest circle which intersects all median lines. It has been assumed that limiting median lines, which intersect two voter ideal points, are sufficient to determine the yolk but recently counter examples have been reported. This note describes the impact of those findings on the results of computation of the location and size of yolk.

ON BUYING LEGISLATURES*

This paper analyzes a simple spatial voting model that includes lobbyists who are able to buy votes on bills to change the status quo. The key results are: (i) if lobbyists can discriminate across legislators when buying votes, then they will pay the largest bribes to legislators who are slightly opposed to the proposed change, rather than to legislators who strongly support or strongly oppose the change; (ii) equilibrium policies exist, and with quadratic preferences these equilibria always lie between the average of the lobbyists' ideal points and the median of the legislators' ideal points; and (iii) “moderate” lobbyists, whose positions on a policy issue are close to the median of the legislators' ideal points, will prefer the issue to be salient, while more extreme lobbyists will generally prefer the issue to be obscure.