Spatial Voting Research Articles

The probability of an undominated central voter in 2-dimensional spatial majority voting

Place a voter v in the plane and scatter additional voters at random about v. What are v's chances of being undominated? When the number of voters, n, is even, this probability is known to be nonzero [5]. Here we prove that the exact probability is 2−(n−2).

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.

Recognizing majority-rule equilibrium in spatial voting games

It is provably dicult (NP-complete) to determine whether a given point can be defeated in a majority-rule spatial voting game. Nevertheless, one can easily generate a point with the property that if any point cannot be defeated, then this point cannot be defeated. Our results suggest that majority-rule equilibrium can exist as a purely practical matter: when the number of voters and the dimension of the policy space are both large, it can be too dicult to find an alternative to defeat the status quo. It is also computationally dicult to determine the radius of the yolk or the Nakamura number of a weighted voting game.

The half-win set and the geometry of spatial voting games

In the spatial context, when preferences can be characterized by circular indifference curves, we show that we can derive all the information about the majority preference relationship in a space from the win-set of any single point. Furthermore, the size of win sets increases for points along any ray outward from a central point in the space, the point that is the center of the yolk. To prove these results we employ a useful new geometric construction, the half-win set. The implication of these results is that embedding choice in a continuous n-dimensional space imposes great constraints on the nature of the majority-preference relationship.

Stable outcomes in spatial voting games

The traditional representation of outcomes for an n-person game as vectors in n-space was superseded in the 1960s when it became clear that, for political and related applications, a lower-dimensional space was generally sufficient for the purpose. Thus, instead of vectors in utility space, vectors in policy space came to be used. Utility to the various decision-makers (usually voters or parliamentarians) could then be defined in terms of distance from given ideal points. A well-known result states that, when the underlying policy space is one-dimensional, a non-empty core (one or more undominated points) will exist. In two or more dimensions, it is known that, for decisive games, where a bare majority is sufficient to carry motions, the core will be non-empty only under very special circumstances (degenerate cases). Stability is then sought in the form of near-core outcomes. Among the best known of the near-core outcomes are (a) the Copeland winner, defined as that outcome which defeats the greatest proportion of alternatives; (b) the yolk center, defined as that alternative which comes closest to satisfying all possible winning coalitions of voters; and (c) the minimal response (finagle) point, defined as the point which comes closest to defeating any other alternative. We show that each of these near-core outcomes has a well-defined analogue for games with side payments (classical von Neumann-Morgenstern games). In particular, the Copeland winner's analogue is the Shapley value and the yolk center's analogue is the nucleolus. The side-payment analogue of the finagle point has not been previously studied; a straightforward adaptation, known also as the finagle point, is however possible.

Optimal location of candidates in ideological space

The strong point is the point which minimizes the probability that a candidate be defeated in a two-party election. The center of power is the weighted average of voters' positions in a spatial voting game, where the weights are given by Shapley's modified value. We show that, under very general conditions, the strong point and the center of power coincide.

Finagle's Law and the Finagle Point, a New Solution Concept for Two-Candidate Competition in Spatial Voting Games Without a Core

We investigate the geometry underlying Finagle's Law, which states, No matter what happens, you can come out ahead if you just know how to finagle, and we introduce a new solution concept for two-candidate sequential spatial voting games, the point. The finagle radius is the radius of a circle such that if a candidate locates at its center, some alternative in the circle can beat any alternative in the space. The finagle point is the point with minimum finagle radius-from it a candidate can, with only minuscule changes in his initial policy location, find a response to any challenger that will defeat that challenger. For each possible candidate location, we provide a geometric construction which gives an outer bound for its a measure of the attractiveness of that location to a finagling politician. For three-voter games without a core, we provide an analytic solution for the point with minimal finagle radius that guarantees that the maximum finagle needed to defeat an opponent will, in general, be quite small relative to the Pareto set. We also show how the construction used to generate the finagle point in the three-voter case can be extended to the n-voter case. The basic idea underlying the finagle point is that it is unnecessary (and indeed usually impossible) to find a position that will defeat all challengers, but it is possible to find a position that is virtually invulnerable to challenge, since any position that beats it can be countered by shifting to a position very close to the original location that will defeat the challenger. Moreover, even if not at the point with minimal finagle radius, in searching for positions to respond optimally to the location of the other candidate, a candidate will in general not find it necessary to move far from an initial point located near the finagle point, since points located near the finagle point will also have a small finagle radius.

Limits on agenda control in spatial voting games

A theorem due to McKelvey implies that, if a single agent controls the agenda of a spatial voting game, he can almost always design an agenda that yields whatever voting outcome he wishes. Here we make use of a geometrical construct called the “yolk” to demonstrate the existence of significant limits on such agenda control. We show that the feasibility of agenda control is inversely related to the size of the yolk. In general, there are strong centripetal forces in spatial voting games, which make it much easier to move voting processes in a centrist direction than in non-centrist one. Thus, outcomes of plausible agenda processes will probably be found in the central area of the space.

A model of candidate convergence under uncertainty about voter preferences

The paper considers two-candidate spatial voting games without a core where candidates have only imperfect knowledge of voters' preferences. We prove that a candidate maximizes his chances of victory by choosing a position as close as possible to that of the candidate who committed to his position first. We also determine the optimal position for this first candidate.

Centripetal forces in spatial voting: On the size of the Yolk

The yolk, the smallest circle which intersects all median lines, has been shown to be an important tool in understanding the nature of majority voting in a spatial voting context. The center of the yolk is a natural ‘center’ of the set of voter ideal points. The radius of the yolk can be used to provide bounds on the size of the feasible set of outcomes of sophisticated voting under standard amendment procedure, and on the limits of agenda manipulation and cycling when voting is sincere. We show that under many plausible conditions the yolk can be expected to be small. Thus, majority rule processes in spatial voting games will be far better behaved than has commonly been supposed, and the possible outcomes of agenda manipulations will be generally constrained. This result was first conjectured by Tullock (1967).

Stable outcomes in spatial voting games: Guillermo Owen, Department of Mathematics, Naval Postgraduate School, Monterey, CA 93943, U.S.A., and School of Social Sciences, University of California, Irvine, CA 92717, U.S.A.

Stable outcomes in spatial voting games: Guillermo Owen, Department of Mathematics, Naval Postgraduate School, Monterey, CA 93943, U.S.A., and School of Social Sciences, University of California, Irvine, CA 92717, U.S.A.

The Core and the Stability of Group Choice in Spatial Voting Games

The core of a voting game is the set of undominated outcomes, that is, those that once in place cannot be overturned. For spatial voting games, a core is structurally stable if it remains in existence even if there are small perturbations in the location of voter ideal points. While for simple majority rule a core will exist in games with more than one dimension only under extremely restrictive symmetry conditions, we show that, for certain supramajorities, a core must exist. We also provide conditions under which it is possible to construct a structurally stable core. If there are only a few dimensions, our results demonstrate the stability properties of such frequently used rules as two-thirds and three-fourths. We further explore the implications of our results for the nature of political stability by looking at outcomes in experimental spatial voting games and at Belgian cabinet formation in the late 1970s.

Necessary and Sufficient Conditions for a Majority Winner in n-Dimensional Spatial Voting Games: An Intuitive Geometric Approach

Necessary and sufficient conditions for there to be a majority winner in n-dimensional spatial voting games are well known, but they are customarily stated in symbolic terms in a fashion which is virtually incomprehensible to those without some reasonable degree of mathematical training, and the proofs of the basic results are even less accessible to the nonmathematically sophisticated reader. We offer proofs of the key results in this area restricted to the important case where voter preferences are a simple function of distance, that is, where, in a choice between any two alternatives, voters prefer the alternative that is closer to their ideal outcome. Our proofs, unlike those customary in the literature, can be understood by anyone who can remember high school geometry. The nature of our proofs is such as to show how the basic n-dimensional results, including the Plott (1967) conditions, the Kramer (1973) sequential voting theorem, the McKelvey (1976, 1979) agenda manipulation result, the Shepsle (1979) germaneness restriction result, and the McCubbins and Schwartz (1985) budget constraint result, can all be derived as reasonably straightforward extensions of Duncan Black's famous median voter result in the one-dimensional case.

The uncovered set in spatial voting games

In a majority rule voting game, the uncovered set is the set of alternatives each of which can defeat every other alternative in the space either directly or indirectly at one remove. Research has suggested that outcomes under most reasonable agenda processes (both sincere and sophisticated) will be confined to the uncovered set. Most research on the uncovered set has been done in the context of voting games with a finite number of alternatives and relatively little is known about the properties of the uncovered set in spatial voting games.

Stability and Centrality of Legislative Choice in the Spatial Context

Majority-rule spatial voting games lacking a core still always present a “near-core” outcome, more commonly known as the Copeland winner. This is the alternative that defeats or ties the greatest number of alternatives in the space. Previous research has not tested the Copeland winner as a solution concept for spatial voting games without a core, lacking a way to calculate where the Copeland winner was with an infinite number of alternatives. We provide a straightforward algorithm to find the Copeland winner and show that it corresponds well to experimental outcomes in an important set of experimental legislative voting games. We also provide an intuitive motivation for why legislative outcomes in the spatial context may be expected to lie close to the Copeland winner. Finally, we show a connection between the Copeland winner and the Shapley value and provide a simple but powerful algorithm to calculate the Copeland scores of all points in the space in terms of the (modified) power values of each of the voters and their locations in the space.

The Spatial Theory of Voting: An Introduction.

Preface 1. Spatial voting models: the behavioural assumptions 2. The unidimensional spatial voting model 3. A two-dimensional spatial model 4. A general spatial model of candidate competition 5. The influence of candidate characteristics and abstention on election outcomes 6. Voting on budgets 7. Models of voter uncertainty 8. Institutions 9. Empirical testing of the spatial theory of elections 10. Concluding observations References Answers to selected problems Index.

Congressional rationality and spatial voting

It is very difficult to demonstrate that representatives employ sophisticated cost-benefit calculations in evaluating constituent benefits when making many legislative decisions. This is because most modern American legislation is ambiguous about which constituencies will receive particularized benefits. This paper examines a series of locational rollcall votes in which the benefits going to constituents were obvious. In such cases representatives balance increases in voter support within their congressional districts and potential increases in their political power within Congress to be gained by vote trading. If constituency benefits are great, representatives overwhelmingly support legislation to provide such benefits. As the benefits decline, vote trading increases.

Pareto optimality in spatial voting models

This paper studies the Pareto optimality properties of policy proposals that are made byk(k≧2) strategic candidates that face uncertainty about the choices that the voters will make. Our first theorem shows that, under very general conditions, any proposal that is a best reply for a candidate is necessarily Pareto optimal. This theorem, in turn, implies that, under slightly stronger conditions, all candidate proposals that are made in a Nash equilibrium or sequentially are necessarily Pareto optimal. Our second theorem shows that, when these conditions are themselves slightly strengthened, any proposal outside of the Pareto set is strictly dominated by at least one proposal inside the Pareto set.

U.S. Presidential Elections 1968-80: A Spatial Analysis

A methodology is presented for empirical estimation of spatial models of voting in mass elections. The basic choice model posits utility functions that depend on spatial distance and a random error term. The parameters of the utility functions are estimated by a polytomous logit model, which is applied to spatial maps of voters and candidates for all presidential elections since 1968. These maps indicate that presidential candidates take positions at the periphery of the distribution of voters. Results based on these maps support the hypothesis of sincere, spatial voting in two-candidate elections. In three-candidate elections, the coefficients of the voter utility function adjust to proxy for the lesser viability of the minor party candidate. In contrast to choice, turnout shows only weak spatial effects. The data are inconsistent with the proposition that turnout is low because voters are not offered distinct alternatives. While the utility function used by voters appears stable through time and candidate and voter positions appear stable in the last few months prior to an election, voter and candidate positions exhibit substantial change over longer time periods.