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Formal and Spatial Modeling


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Formal and Spatial Modeling

Donald M. Gooch

Arkansas Tech University
In Greek mythology, Hercules is tasked with twelve impossible labors in order to regain honor and thus ascend to Mount Olympus as a god. The job of explaining formal theory and spatial theory in a brief, non-technical essay is a labor of sufficient difficulty to make the search for the Golden Fleece pale in comparison. Given that this author has no transcendental gifts (though Hippolyte's Belt may be around here somewhere), aspirations, or pretentions, I have eschewed the impossible task of summarizing the entirety of formal and spatial theory. Instead, this essay settles for the daunting yet mortal goal of a thorough yet concise introduction to some of the classical and contemporary works of the formal and spatial theories on politics and the concepts, definitions, and models upon which those works rest (for a more complete treatment of formal theory and its contribution as a field of political inquiry see Ordeshook, 1992, Shepsle and Bonchek, 1997, and Morton, 1999). While Duncan Black may have understated the mathematical underpinnings of spatial theory as “simple arithmetic,” it is as true today as it was then that the fundamental assumptions, intuitions, and predictions of formal and spatial theory can be grasped with a relatively basic foundation in mathematics such as algebra and geometry (Black 1958). More advanced treatments of the subject requires grounding in integral calculus and other advanced mathematical tools, however, you do not need these to understand what formal theory is, what the foundational principles of formal theory are, and the gamut of its predictions and conclusions regarding political institutions and behavior. To the extent possible without compromising the material, I will keep the discussion here broad and descriptive and thus accessible to the undergraduate reader.

What is Formal Theory?

Formal theory is a field of inquiry which uses mathematical techniques to explicitly and precisely define theoretical concepts and the relationships between those concepts. Formal mathematics permits the systemizing of theory, which allows precise deductions and synthesis as well as enhancing the decidability of scientific propositions. While ‘formal theory’ is common parlance, it also goes by rational choice theory, public choice, positive political theory, political economy, the economic theory of politics, and a variety of other synonyms. Two of the primary branches of formal theory in political science are game theory and spatial theory. Game theory is concerned primarily with the strategic interaction of utility-maximizing actors in competitive and cooperative settings (see chapter on game theory for more information). Spatial theory, as we will see, examines the behavior of actors by representing beliefs, positions, choices, and institutional contexts in terms of spatial distance (most frequently on the Cartesian plane). While formal theory shares a foundation in mathematics and logic with quantitative methods, it is distinct from the traditional empirical inquiry of standard statistical methods. Formal theory seeks to define political concepts and derive logical implications from their interrelations, while traditional methods are for assessing the relationships between variables through direct statistical analysis. Non-formal theory underpins much of the empirical work in political science. A non-formal model suggests relationships between actors and institutions in the real world of politics using common linguistics. There may be an underlying formal model that, as Arrow noted, has not been expressed formally due to mathematical or linguistic limitations (Arrow 1968). A model is formalized when we use abstract and symbolic representations to explicitly state the assumptions of the model and from which can be derived equilibrium and comparative statics predictions (Morton 1999; Binamore 1990; Elster 1986).

For example, a non-formal voting model might entail: “voters vote for viable candidates that share their beliefs and positions on issues.” This seems to be a reasonable statement of the voting process. Yet, there is a great deal of ambiguity in this statement. What does it mean for a candidate to be ‘viable’? To what extent must a candidate share the voter’s beliefs and positions relative to the other candidates? How does the voter assess candidate positions and how do they relate them to their own beliefs? Furthermore, how important to the voter is the prospect that their vote will be decisive in the election? Our non-formal model is silent on this question. A formal model aims at explicitly defining the processes at work (in this case, the act of voting), the actors participating in the process (voters, candidates) and the gamut of alternative outcomes based on those choices (does the citizen vote or not). Riker and Ordeshook, operationalizing a spatial model of voting based on the classic median voter theorem developed by Downs, give us just such a formal model of voting.

An individual will decide to vote if and only if (Riker and Ordeshook 1968):


 Equation 1

Where, for each voter:



P = the probability that this person’s vote will affect the outcome of the election

NCD = perceived net benefits of one candidate over another (net candidate differential)

D = individual’s sense of civic duty

C = costs associated with the act of voting (opportunity costs, driving time, gas, etc.)
This cost-benefit of analysis by the voter hinges the act of voting on the difference among the candidates between the perceived spatial distance of the candidates’ position and that of the potential voter’s own preferences, conditioned by the probability that the individual’s vote will be decisive in the election. The difference between candidates is conditioned on the probability of a decisive vote, because if the voter’s vote is not decisive then the candidate differential is essentially irrelevant to the outcome of the election from the perspective of the potential voter. This formal theory of voting uses mathematical notation to precisely relate the costs of voting to the benefits the voter receives from voting, and in so doing provides a non-obvious expected outcome that tells us something interesting about the rational voter. As the probability of a decisive vote goes to zero, the differences between the candidates on issues is eliminated from the calculus of the vote decision. This fact led scholars to predict that citizens wouldn’t collect costly information on politics such as the policy positions of specific candidates or parties. Also, while many scholars have wrestled with the puzzle of low voter turnout in the United States, the Downsian (and by extension, Riker-Ordeshook) voting model suggests the real puzzle is that anyone votes at all.

Formal Theory, Quantitative Methods and Empirical Inquiry

One way to think about the difference between formal theory and quantitative methods employed for empirical inquiry, given that both use the language of mathematics, is in terms of the scientific method. The scientific method as applied in the social sciences and as traditionally stated, involves a research question of some importance to our understanding of social phenomena, developing theories as to the processes, actors, and interactions within the social context and stating hypotheses derived from these theories for empirical testing, the use of techniques to test these hypotheses against real world data, and the reporting of the results of those tests. Formal theory in political science is oriented towards developing precise theories with specifically defined assumptions and the derivation of their implications (the ‘front end’ of scientific inquiry) while statistical methodology applies mathematical rigor to the testing of theories and hypotheses (the ‘back end’ of scientific inquiry). This dichotomy, like most dichotomies, is somewhat problematic. While it is true that the foci of formal theory and quantitative methods are distinct and have been historically pursed separately in political science, it is incorrect to assert that empiricists are unconcerned with precise theorizing and formal theorists are indifferent to empirical testing. Both formal theory and quantitative methods are effective tools to employ in the study of political phenomenon and in combination can produce significant contributions to the knowledge of politics (Barry 1978).

The increasing role of formal theory in political science is not without its critics. The behavioral revolution in political science that drew the discipline away from informal normative theories and descriptive analysis inspired greater and greater attention to developing strong empirical measures of political phenomena, and many see formal theory as a distraction from ‘real’ politics and important empirical analysis of politics. Yet, there is merit in assessing pure theory on its own right. Among the meritorious reasons are its contributions to mathematics, formal theory’s revelation of surprising and counter-intuitive yet logical expectations and outcomes, its service as a precursor to the development of empirically testable models, and the provision of insights into political phenomenon that are not currently reducible to testable data. It is certainly not the case that any and every question regarding politics lends itself to formal theorizing, as we are nowhere close to a ‘complete’ model of politics (the natural sciences are closer, but they have also failed to obtain the prize). Many of the first principles from which formal theories are derived are either undiscovered or only partially described and understood. There is room in the discipline for both forms of inquiry. The ambition of most formal theorists is to provide pieces of the puzzles of politics with increasingly better developed and more rigorously tested models of behavior.

There is a host of valuable empirical contributions to our understanding of politics that do not employ formal theoretics. However, that is not to suggest that formal theorizing is superfluous. Where possible it is best to precisely define both our theoretical expectations and our empirical tests of those expectations. It is difficult to test theories that lack precision or clear implications and the ambiguity of these non-formal theories can result in conflicting and mutually exclusive tests. The usefulness of precise theories is lessened without ways to test them against reality, and theories that wander too far away from the real world of politics makes the discipline less relevant both to policy makers and students of practical politics. The combination of the two methods, where we use quantitative methodology to assess the predictions and comparative statics of formal models on empirical data (often referred to as EITM: Empirical Implications of Theoretical Models), is one of the more significant modern trends in political science and is an active field of inquiry in the discipline coexisting alongside the more traditional behavioral and pure theoretic approaches. Whether through the investigation of the empirical implications of formal models or the mind experiments of pure formal theory, these models have much to contribute to the study of politics today. Why do two parties from plurality electoral systems, how do two major parties in first-past-the-post electoral system respond to the threat of entry by third parties, why do voters turn out, how many seats should a party seek to control in a legislature, can we get irrational aggregate social outcomes when society is composed of rational individuals, why and how do procedural rules in institutions such as legislatures matter, and why do individuals choose to join interest groups? These questions and more lend themselves to formal analysis (Palfrey 1989; Ordeshook 1992; Olson 1965; Riker and Ordeshook 1968; Downs 1957).



Rational Choice and its Foundational Assumptions

Formal theory is a deductive form of inquiry, deriving implications and relationships from established first principles. One of the foundations of formal and spatial theory is the rationality assumption (hence why it is often referred to as ‘rational choice’). Most formal theories of politics adopt some variation of the rationality assumption. Rationality, as it is generally conceived in formal modeling, is the assumption that individuals have a set of preferences and beliefs and that they act intentionally in constraining real world contexts consistent with those preferences and beliefs. Rationality, in this context, is not a person doing what you think they should do if you were in their shoes, such as staying home and studying rather than going out to a party before the Big Test because you would value getting a good grade on the exam more than having fun on a Friday night. It doesn’t mean having super human knowledge or being brilliant decision-makers. Individuals order their complete preferences as they see fit, and they make choices aimed at getting the best possible outcome according to those preferences.

There are three important principles at work here. The first is that of completeness or comparability. If I am to choose among possible alternatives, I have to know what all the alternatives are, and be capable of comparing them to one another. The second is the mathematical principle of transitivity (if A > B and B > C then A > C). In order to make a rational choice, you have to be able to order your preferences consistently. The transitive principle permits a rational choice because the interrelation between all of my choices makes sense. If I prefer pizza to waffles, and I prefer waffles to apples, then it isn’t ‘rational’ to prefer apples to pizza. Third, rational choice models assume that individual actors are self-interested, in that they attempt to get the best outcome possible for themselves. This is also called “utility-maximizing” where utility is just a quantifying term for a benefit to the individual and maximizing means that the individual seeks to get the largest benefit possible. Now, this isn’t to say that all potential choices meet the comparability and transitivity and maximizing requirements. Indeed, many do not. However, behavior that is intentional, self-interested, and maximizing across comparable and transitive preference orderings, which is true of many political choices and decisions, yields itself to rational choice analysis (Riker and Ordeshook 1973).

This definition of rationality reveals another primary aspect of formal theory: methodological individualism. Most formal theories employ the individual as the fundamental unit of analysis.1 An individual can have preferences and beliefs while group, firms, states, etc. cannot. Consider again the Riker-Ordeshook model of voting. Note that the model defines the individual citizen’s calculus in deciding whether or not to vote, reflecting its assumption of methodological individualism. The Riker-Ordeshook model is also a good example of an application of the rationality assumption. They presume that the voter will assess both the internal factors (preferences ordered across candidates) and external factors (the probability that the voter’s vote will be decisive) in making a ‘rational’ cost-benefit decision whether or not to vote.

The assumption of rationality is one of the more controversial aspects of formal theory. Many critics argue that human beings lack the capacity, the evolutionary development, and the necessary information to make rational decisions as conceived by formal models. While this may be the case, it does not necessarily mean that rationality is a useless or even pernicious assumption in formal theory. Assumptions can be both unrealistic and useful in terms of either identifying puzzles (if X is the rational choice, why do most individuals choose Y?) or by reflecting an important aspect of decision-making, even if it does not accurately represent many individual decision-making processes. All models are inaccurate to some degree. Models are crude, stylized approximations of the real world, intended to reflect some important aspect of politics rather than every aspect. Model airplanes fall short of the ‘realism’ ideal in terms of material composition, scale, and functionality. Yet model airplanes have contributed in numerous ways to our understanding of flight and the design of aircraft. Furthermore, there is a tradeoff in loss of parsimony when we make our models more complex in pursuit of realism. The measure of a model of politics is not whether it perfectly approximates the real world, but rather its usefulness in contributing to our understanding of politics (Morton 1999; Ordeshook 1992; Shepsle and Bonchek 1997).

That said there have been significant efforts to incorporate more realistic (but still rational) assumptions regarding individual behavior in formal models. One important modification is a move away from deterministic models to probabilistic models of choice. We have noted that utility maximization is a key component of rational choice models, where we assign utilities to the outcomes of choices and the rational individual chooses the highest utility outcome. When an individual is highly confident that X action will lead to Y outcome, we say that individual is operating under the condition of certainty. However, in many contexts, an individual is uncertain as to what actions lead to what outcome. Rather, they make choices that may or may not lead to a particular outcome. In such instances, the individual is uncertain about what happens when they make a particular choice. When the individual has a good sense of the likelihood of certain outcomes (say, 50 percent chance of Y and a 50 percent chance of Z), we say that individual is operating under the condition of risk. When they have no idea what will happen or what is likely to happen, they are faced with the condition of uncertainty (Dawes 1988). Under probabilistic conditions, it is particularly useful to assign numbers to outcomes, which formal theory defines as utility. Quantifying the outcomes permits one to incorporate probabilistic decisions into models of behavior. Now rather than choosing acts that necessarily produce a particular outcome, the individual chooses among lotteries where the utility from outcomes is conditioned on the probability of that outcome occurring. This is expected utility theory, and is an important innovation in modeling behavior. I may value being named King of the World very highly, and thus that outcome has a high utility, however given that the probability of that outcome approaches zero, my expected utility from taking actions aimed at having me crowned King of the World is actually quite low. We can assign a single number to the action-lottery and the individual can rationally choose the lottery that he or she can expect to yield the highest utility (Shepsle and Bonchek 1997).

While probabilistic models may be more realistic, they still assume that individuals are rational utility-maximizers. Other theorists have relaxed the assumption of rationality itself. While I cannot give them full treatment here, nonlinear expected utility, prospect theory, bounded rationality, learning, and evolutionary models use near or quasi-rational models of behavior (Morton 1999). Bounded rationality incorporates decision makers with incomplete information and furthermore they have cognitive limitations and emotions that prevent or complicate utility-maximizing based on the limited information they do have (Jones 2001; Simon 1957). Herbert Spencer, an early developer of boundedly rational models, calls it ‘satisfycing’ rather than satisfying a preference ordering. An individual who satisfyces doesn’t consider all possible alternatives or delay decision-making until all the information is in, but rather searches among the limited number of readily available alternatives and chooses the best of those (Simon 1957). Prospect theory is a psychological theory of decision making where individuals evaluate losses and gains differently. Loss aversion, where I fear losses more than I value gains, is an example of applying prospect theory and it generated different predicted behavior than traditional expected utility theory (Kahneman and Tversky 1979).

Solving and Testing Formal Models: Equilibriums, Point Predictions, & Comparative Statics

After a formal model has been developed, the model is ‘solved’ for predictions presented as theorems or results. The implications (or solution) of the model are deducted axiomatically from the assumptions and structure of the model itself. In most formal models relevant to political science, the researcher seeks to solve the model analytically, which involves the search for equilibria (stable outcomes). Where analytical solutions are not feasible or possible, obtaining numerical solutions through computer simulation is an option. If the formal model is game theoretic, then the interactions between the ‘players’ are strategic. A common solution concept in this form of model is the Nash equilibrium, where each player’s choice is optimal given the choices of other players and thus no player has an incentive to change strategies within the game. Solving for the decision of our potential voter in the Riker-Ordeshook model of turnout in a two candidate election yields the instrumental solution that this individual should only vote for her preferred candidate if the probability of her vote being decisive exceeds twice the cost of voting2. It furthermore yields the nontrivial result that, even if the cost of voting is very small, she should only vote if the probability of her breaking a tie exceeds 2/1000. Given a large n election such as a presidential election, the instrumental part of the R-O equation yields a prediction is that the individual chooses not to vote.

In evaluating a formal model empirically (relating the model to data from the real world: e.g. election results, legislative votes, presidential vetoes, etc.), we can evaluate assumptions, predictions, and alternative models. The evaluation of assumptions is a validation of the relevance of the formal model. If an assumption of a model is violated frequently in the real world, then the scope of the applicability of that model is less. In evaluating predictions, we can look at point predictions, which are the value of the variables in the model when in equilibrium (if the model predicts one or multiple equilibria). Another method of evaluation is comparative statics, where changes in the endogenous variables of the model in equilibrium (dependent variable) change with the values of an exogenous variable (independent variable). Finally, we can assess models by looking at them in competition with other contrary formulations of the political phenomenon (Morton 1999).

Public Choice: Democratic Theory, Institutions & Voting Paradoxes

Between 1950 and 1965 the seminal and foundational works in formal and spatial theory were published. Among them were Arrow’s Social Choice and Individual Values, Black’s The Theory of Committees and Elections, Buchanan and Tullock’s The Calculus of Consent: Logical Foundations of Constitutional Democracy, Riker’s The Theory of Political Coalitions, Olson’s The Logic of Collective Action and the Theory of Public Goods, and Anthony Downs’ Economic Theory of Democracy. Each represents an important contribution to formal modeling and identified important paradoxes or puzzles of logical political behavior, collective action, choice mechanisms and democratic theory that continue to be the subject of innovative research today (Arrow 1963; Black 1958; Buchanan and Tullock 1962; Riker 1962; Olson 1965; Downs 1957).

However, the study of choice mechanisms using mathematics actually got its start in the eighteenth century. Procedural problems in electoral systems lead Condorcet and Borda to investigate the problem analytically, while Pareto went on to apply mathematics in attempting to understand social phenomenon as early as the 1920’s (Pareto 1927; Shepsle and Bonchek 1997). It is Pareto’s efficiency concept that underlies Buchanan and Tullock’s calculus of consent. These thinkers paved the way for the explosion of formal and spatial political theory in the 1960’s. The works of these (mostly) economists formed the pillars upon which modern public choice theory was built. Public choice theory (also called social choice) has primarily focused on macro-institutional factors and how the structure of government interacts and often conflicts with the aggregate preferences of the public. Olson’s work on the collective action problems inherent to group formation, Downs’ conclusions regarding the rational ignorance of voters, Riker’s theory on minimum winning coalitions and Buchanan and Tullock’s treatise on the political organization of a free society are all significant contributions worthy of attention, but here I will focus on only one aspect of public choice theory where formal models generally and spatial models specifically have been at the forefront: electoral competition, i.e. voting behavior in relation to choice mechanisms.

Condorcet was one of the first to apply mathematical modeling to the problem of making a collective decision among a group of individuals with preference diversity (not everyone wants the same thing). One of the common themes in public choice theory is the normative principle that choice mechanisms should reflect democratic values. One such mechanism intended to reflect a democratic choice is first preference majority rule, where the top preference of the largest number of individuals is given effect as the ‘decision’ on behalf of the collective. But there can be multiple ‘majorities’ and what choice is made can be dependent upon what order the alternatives are presented, particularly in pair-wise comparisons. If there is one choice that defeats all others in a pair-wise vote, it is said to be a Condorcet winner. Condorcet identified a problem with majority rule when group preferences are intransitive. Remember we discussed the assumptions of rationality with one being that individuals have transitive preference orderings. The rationality of individuals, however, does not require that group preferences be transitive. When group preferences are intransitive (Condorcet’s Paradox), the possibility for cycling exists. When A defeats B and B defeats C and C defeats A there is not one majority rule election that will produce the ‘group’ or democratic preference, as no such preference exists (McLean and Urken 1993).

Condorcet examined just the special case of majority rule, though Condorcet’s Paradox is not trivial as the probability of it existing in a natural group increases with the size of the group and the number of alternatives. Arrow looked at the problem more generally. He made a few basic, minimal assumptions about what a democratic process would entail (all preference orderings are possible including those with indifference, Pareto optimality, the independence of irrelevant alternatives, and non-dictatorship) and asked whether it was possible to construct a method that would aggregate those preferences in such a way as to satisfy these conditions. Arrow’s theorem asserts that there is no such possible choice mechanism. What this means practically is that there is a tradeoff between having a system that is rational in translating preferences to policy and the concentration of political power. Put another way, dictators are good for consistency. Which isn’t to say that all social aggregation is unfair or intransitive, but rather that there is no mechanism that can guarantee such an outcome given any context. Arrow’s result shows that democratic processes yielding socially coherent policy is a much more difficult proposition than we had thought (Arrow 1963).

The Spatial Theory of Voting: the Median Voter Theorem and Theoretical Modifications

As noted earlier, one of the major innovations of formal theory was the use of geometric space to represent political choices. Let’s set out some of the basics of spatial theory here. The standard spatial model depicts voting with Euclidean preferences in a one or two dimensional space. This means that political choice is represented as choice of some point on a line or a two-dimensional space, over which all the actors have preferences. Specifically, each actor j has an ideal point (top preference) on the line or space, prefers a point closer to this ideal point to one more distant from it, and is indifferent between two equally distant points. In the two dimensional case, an actor’s indifference curves are concentric circles centered on his ideal point. Actor j’s preference set Pj(x) is the set of points j prefers to x. Furthermore, in most spatial models, preference orderings are assumed to be single-peaked (monotonic).

While single-peaked preferences (preference orderings that can be represented by a line) are helpful in producing social consensus in the absence of unanimity, they are also an important aspect of the spatial representations of politics. If we take a group of individuals (voters in an election, legislators on a committee) who are considering a policy along one dimension (say, candidates in an ideological dimension or the amount of tax dollars to budget for defense spending), and their utility function is single-peaked, then the outcome of this process has some interesting characteristics. Specifically, the voter or committee member located at the center of the group on the relevant dimension determines the outcome. Black’s median voter theorem tells us that the ideal point of the median voter has an empty win set. A win set W(x) is the set of all points that beat (are collectively preferred to) x under a decision rule. If the ideal point of the median voter has an empty win set, then the median voter’s preference commands a majority against all other possible points on the policy or candidate ideology dimension (Black 1958).

This argument for a centripetal force in politics is an important finding. The commanding stature of the median voter was first posited by Harold Hotelling in predicting the geographic congregation of firms at one location, such as hot dog vendors on a street (Hotelling 1929). Downs adopted Hotelling’s proximity model in his now famous median voter theorem in elections. The theorem states that the median voter in a single dimension cannot be defeated in a pair-wise vote with full turnout and sincere voting (I vote according to my true preference ordering rather than trying to ‘game’ the vote by strategically voting for a less-preferred alternative). Given the median voter theorem (MVT), Downs predicted that party (or candidate) platforms would converge to the median voter’s policy preference. It is called a proximity model because Downs assumed that voters used the spatial distance between themselves and candidates to determine who they should vote for. Rational voters in this model vote for the candidate or party with a platform closest to their most preferred policy. Parties converge because that’s where the votes are (Downs 1957). Convergence has been observed in real political contexts, but there are a variety of complications that can prevent convergence (Frohlich et al. 1978). Empirically speaking, there is evidence from a plethora of elections here and abroad where parties and candidates failed to converge to a single policy point or even a vector of policy points in a continuum of policies. Divergence appears to be the norm rather than the exception (Morton 1993).

Multiple dimensions are a complication for the median voter theorem (Riker 1980). Political competition can occur along multiple dimensions. After all, the cost of a policy is only one consideration when it comes to deciding how to authoritatively allocate resources. Fairness, effectiveness, efficiency, and other considerations can come in to play. Ideology is one way to rate candidates, but what about affect (likeability), trust, and performance considerations? Non-policy attributes, or valence, may influence election outcomes (Groseclose 2001). Scholars have extended the spatial model of voting developed by Downs into multiple dimensions using a Euclidian model of utility functions (Hinich and Pollard 1981; Enelow and Hinich 1984). Plott’s theorem suggests that a median voter result is possible in multiple dimensions, but that the conditions for it are attenuated. It is dependent upon radial symmetry among the alternatives (Plott 1967). McKelvey’s chaos theorem asserts that there is no majority rule empty-winset point in a multidimensional spatial setting other than Plott’s special case. In other words, we can start at an arbitrary point in the space and a majority can move us to any other point in the space. With no Condorcet winner, policy cycles endlessly (McKelvey 1976, 1979). However, is this chaos ephemeral? Consider that policy cycling isn’t frequent in legislatures. As Gordon Tullock famously queried, why all the stability (Tullock 1981)? It remains a point of contention, though Shepsle suggests that institutions impose policy stability through restrictive rules (Shepsle 1979). Institutions may impose stability, but this merely changes the choice context (rules instead of policy). Of course, legislatures involve sophisticated players who may vote strategically and thus account for some of the apparent regularity.

Aside from policy dimensions, what other factors could lead to candidate or party divergence? Theories of candidate divergence suggest alternative specifications such as non-normal voter distributions, directional logic, permitting third party entry, valence, and turnout variance as a basis for moving away from the stylized median voter model. Can the distribution of voters produce platform divergence on their own? Where multiple modes exist, candidate divergence may be rational. The implications for polarization in the median voter model were anticipated by Downs (Downs 1957). He argued that the location of equilibria in an election would be dependent on the shape of the distribution of citizen preferences (Downs 1957). Downs, on this at least, was wrong. The pure MVT with complete turnout and sincere voting predicts that the median voter is, in fact, a Condorcet winner: no position defeats the ideal point of the median voter in a pair-wise vote irrespective of distributional qualities (Black 1958).

Other scholars have taken issue with the proximity calculus where voters choose the candidate or party that is “closest” to them. Rabinowitz argues that platform space cannot be represented in terms of an ordered and continuous set of policy alternatives. Rather than a policy continuum, the directional theory of voting suggests policy alternatives are dichotomous and thus candidates are judged by their intensity and policy direction (Rabinowitz 1989). Furthermore, the directional model predicts platform divergence from the median voter. A divergent candidate may be preferred because he is perceived as more credible on the favored policy direction and thus more likely to move policy towards the voter’s preferred position. Rabinowitz and MacDonald argue that this is a more realistic and empirically confirmed model, though other scholars have disputed this claim and maintain that the proximity model is superior (Rabinowitz 1989; MacDonald et al. 1998; Rabinowitz et al. 1982; Westholm 1997).
Finally, the prospect of entry by a third party may cause parties to diverge from the median in order to discourage a third party challenge on their extremes in a polarized electorate (Fiorina 1999; Palfrey 1984). Hinich and Munger develop a theory of ideology which permits party divergence (Hinich and Munger 1994). Incorporating previous modifications to the MVT, such as incomplete information and uncertainty in voter policy locations and candidate locations, they argue that the creation and maintenance of an ideology by parties is a necessary component of political competition (Hinich and Munger 1994). In a political environment where Republicans have become much more consistently and strongly conservative (likewise for Democrats and liberals), vote-seeking parties rationally diverge in order to create a credible ideology which they can ‘sell’ to their constituents. Establishing an ideological flag at one of the ‘poles’ in a bimodal distribution would account for platform divergence. The median voter theorem has received a lot of attention in political science. While its pure form has been falsified on turnout and numerous theoretical modifications have been required to account for party divergence from the median, the centripetal logic that underlies this theorem remains an important factor in electoral and legislative settings.

Reflections on Formal and Spatial Theory

The major paradigms of formal and spatial theory in social choice, voting, institutions, and political behavior have spawned decades of empirical and theoretical research as well as countless additional, alternative, and contrary models of political decision-making. Many of the early formal models that we discuss here have either been falsified or significantly modified to account for empirical deficiencies. Riker’s theory of minimum winning coalitions doesn’t well describe many legislative contexts and the behavior of legislators often violates his theoretical expectations. Downs’ turnout prediction has been falsified and his theory on party convergence at the median has been serially violated in the real world of politics. Downs’ median voter theorem is problematic in multiple dimensions and platform divergence is an apparent empirical norm. The cycling and instability of social choice identified by McKelvey is not a consistent characteristic of government institutions, causing some political scientists to puzzle over the apparent stability in these institutions and their incorporated choice mechanisms.

These empirical failures and the ad hoc modifications aimed at rescuing them have led some scholars to suggest that political behavior is inherently irrational or at minimum a poverty of realistic and empirically supported rational choice models rendering them to be of little use or relevance (Green and Shapiro 1994). This is a mistake. It ignores important empirical validations of formal models, over-emphasizes point predictions in relation to comparative statics, and sets up a straw man in the form of ‘the rational choice theory’ when there is not one but rather a multitude of formal, spatial, and rational choice theories. The failure of one or more formal models does not prove that formal theory has little utility in empirical investigations of politics. Those failures spur puzzle-solving, the development of better and alternative models, and the exploration of new and innovative empirical tests of model predictions.

We can see this in the variety of models, evidence, and arguments that we examined that were in direct response to the Downs-Hotelling proximity model on platform convergence. The Riker-Ordeshook model of turnout that we consider at the beginning of the essay modified the traditional Downsian turnout model by incorporating a new variable: a psychic benefit from participation. We learned quite a bit from the ‘failure’ of the Downsian turnout and proximity models. We now know that instrumental calculation (the cost of voting combined with the probability of affecting the outcome) is insufficient to spur a voter to participate. Rather, the experiential benefit characterized as a psychic ‘civic duty’ benefit by Riker and Ordeshook is the decisive consideration. Other modelers have incorporated alienation, abstention, and other modifications to account for positive turnout in elections. Thus an apparent formal model failure has actually resulted in numerous and significant contributions to our understanding of voting behavior. The research in formal and spatial modeling is ongoing and continues to be a necessary tool for understanding political institutions and behavior (Grofman 1993).



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1 More recent innovations have moved away from the atomistic individual assumption in adopting institution and context-specific models.

2 see Shepsley and Bonchek p. 251-259 for a detailed solution of the Riker-Ordeshook turnout model in a two candidate election.


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