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date: 17 February 2020

Economic Methods in Positive Political Theory

Abstract and Keywords

This article focuses on economic methods in political science, specifically on positive political theory. It provides a sketch of the two canonical approaches to developing a positive political theory: collective preference theory and game theory. It is argued that these two techniques are distinguished by their trade-offs, despite having some clear formal differences. The article also considers other specific techniques within the game-theoretic approach, which are designed to accommodate two important analytical characteristics that are distinctive to political science.

Keywords: economic methods, positive political theory, canonical approaches, collective preference theory, game theory, game-theoretic approach, analytical characteristics

1 Introduction

Economics and political science share a common ancestry in “political economy” and both are concerned with the decisions of people facing constraints, at the individual level and in the aggregate. But while rational choice theory in some form or other has been a cornerstone of economic reasoning for over a century, with the mathematical development of this theory beginning in the middle of the nineteenth century, its introduction to political science is relatively recent and far from generally accepted within the discipline.1 Three books proved seminal with respect to the application (p. 811) of economic methods in political science, the first of which is Kenneth Arrow’s Social Choice and Individual Values (1951, 1963).2

Although mathematical models of voting can be found at least as far back as the thirteenth century (McLean 1990) and despite Paul Samuelson’s claim, in the foreword to the second edition of the book, that the subject of Arrow’s contribution was “mathematical politics,” an appreciation within political science (as opposed to economics) of the significance of both Arrow’s possibility theorem itself or, more importantly for this chapter, the axiomatic method with which it is established was slow in coming. An exception was William Riker, who quickly understood the depth of Arrow’s insight and the significance of an axiomatic theory of preference aggregation, both for normative democratic theory and for the positive analysis of agenda-setting and voting.

The remaining two of the three seminal books are Anthony Downs’s An Economic Theory of Democracy (1957) and William Riker’s The Theory of Political Coalitions (1962). These books were distinguished for political science by their use of rational choice theory and distinguished for rational choice theory by their explicit concern with politics.

Downs’s 1957 volume covers a wide set of issues but is perhaps most noted for his development of the spatial model of electoral competition and for the decision-theoretic argument suggesting rational individuals are unlikely to vote. The spatial model builds on an economic model of retail location due to Hotelling (1929) and Smithies (1941). Approximately a decade after the publication of Downs’s book, Davis and Hinich (1966, 1967) and Davis, Hinich, and Ordeshook (1970) described the multidimensional version of the (political) spatial model, the mathematics of which has given rise to a remarkable series of results, exposing the deep structure of a variety of preference aggregation rules, most notably, simple plurality rule (e.g. Plott 1967; McKelvey 1979; Schofield 1983; McKelvey and Schofield 1987; Saari 1997). Similarly, Downs’s decision-theoretic approach to turnout elicited a variety of innovations as authors sought variations on the theme to provide a better account of participation in large elections (e.g. Riker and Ordeshook 1968; Ferejohn and Fiorina 1975). But although treating voters as taking decisions independent of any consideration of others’ behaviour (as the Downsian decision-theoretic approach surely does) yields some insight, the character of most if not all political behavior is intrinsically strategic, for which the appropriate model is game theoretic.3

Riker understood the importance not only of Arrow’s theorem and a mathematical theory of preference aggregation, he also recognized that game theory, the quintessential theory of strategic interaction between rational agents, was the natural tool with which to analyze political behaviour. In his 1962 book, Riker exploited a cooperative game-theoretic model, due to von Neumann and Morgenstern (1944), to develop an (p. 812) understanding of coalition structure and provide the first thoroughgoing effort to apply game theory to understand politics. Cooperative game theory is distinguished essentially by the presumption that if gains from cooperation or collusion were available to a group of agents, then those gains would surely be realized. As such, it is closely tied to the Arrovian approach to preference aggregation and much of the early work stimulated by Riker’s contribution reflected concerns similar to those addressed in the possibility theorem. An important concept here is that of the core of a cooperative game.

Loosely speaking, if an alternative x is in the core, then any coalition of individuals who agree on a distinct alternative y that they all strictly prefer, cannot be in a position to replace x with y. For example, the majority rule core contains only alternatives that cannot be defeated under majority voting. Similarly, if we imagine that a group uses a supramajority rule requiring at least 2/3 of the group to approve any change, then x is in the core if there is no alternative y such that at least 2/3 of the group strictly prefer y to x. So if a group involves nine individuals, five of whom strictly prefer x to y and the remaining four strictly prefer y to x, then the majority rule core (when the choice is between x and y) is x alone whereas both x and y are in the 2/3 rule core.

The concept of the core is intuitively appealing as a predictor of what might happen. Given the actions available to individuals under the rules governing any social interaction, and assuming that coalitions can freely form and coordinate on mutually advantageous courses of action, the core describes those outcomes that cannot be overturned: even if a coalition does not like a particular core outcome, the very fact that the outcome is in the core means that the coalition is powerless to overturn it. On the other hand, when the core is empty (that is, fails to contain any alternatives) then its use as a solution concept for a cooperative game-theoretic model is suspect. For example, suppose a group of three persons has to use majority rule to decide how to share a dollar and suppose every individual cares exclusively about their own share. Then every possible outcome (that is, division of the dollar) can be upset by a majority coalition. To see this, suppose a fair division of $(1/3) to each individual is proposed; then individuals (say, A and B) can propose and vote to share the dollar evenly between themselves and give nothing to individual C; but then A and C can propose and vote to give $(2/3) to A and $(1/3) to C. Because A and C care only about their own shares, this proposal upsets the proposal favoring A and B. But by the same token, a division that shares the dollar equally between C and B, giving nothing to A, upsets the outcome that gives B nothing; and so on. In this example, the core offers no guidance about what to expect as a final outcome. Furthermore, it does not follow that the core being empty implies instability or continued change. Rather, core emptiness means only that every possible outcome can in principle be overturned; as such, the model offers no prediction at all. It is an unfortunate fact, therefore, that, save in constrained environments, the core of any cooperative game-theoretic model of political behavior is typically empty, attenuating the predictive or explanatory content of the model.

Discovering the extent to which the core failed to exist was disappointing and induced at least some pessimism about the general value of formal economic reasoning as a tool for political science. Things changed with the development of (p. 813) techniques within economics and game theory that greatly extended the scope and power of non-cooperative game theory, in which there is no presumption that the existence of gains from cooperation are realized. And at least at the time of writing this chapter, it is non-cooperative game theory that dominates contemporary positive political theory.

This chapter concerns economic methods in political science. It is confined exclusively to positive (formal) political theory, paying no attention to econometric methods for empirical political science. Furthermore, I adopt the perspective that a central task for positive political theory is to understand the relationship between the preferences of individuals comprising a polity and the collective choices from a set of possible alternatives over which the individuals’ preferences are defined.4 The next section sketches the two canonical approaches to developing a positive political theory, collective preference theory and game theory. I briefly argue that despite some clear formal differences, these two techniques are essentially distinguished by the trade-off each makes with respect to a minimal democracy constraint and a demand that well-defined predictions are generally guaranteed. Moreover, the attempt to develop collective preference theory as an explanatory framework for political science reveals two important analytical characteristics, distinctive to political science rather than economics. The subsequent section, therefore, considers some more specific techniques within the game-theoretic approach designed to accommodate these characteristics. A third section concludes.

2 Two Approaches From Economics

Economics is rooted in the choices of individuals, albeit with a broad notion of what counts as an “individual” when useful, as in the theory of markets where firms are often treated as individuals. And the basic economic model of individual choice is decision theoretic: in its simplest variant, individuals are assumed to have preferences over a set of feasible alternatives that are complete (every pair of alternatives can be ranked) and transitive (for any three alternatives, say x, y, z, if x is preferred to y and y is preferred to z, then x must be preferred to z) and to choose an alternative (e.g. purchasing bundles of groceries, cars, education, …) that maximizes their preferences, or payoffs, over this set. The predictions of the model, therefore, are given (p. 814) by studying how the set of maximal elements varies with changes in the feasible set. Now it is certainly true that individuals make political decisions but those decisions of interest to political science are not primarily individual consumption or investment decisions; rather, they are decisions to vote, to participate in collective action, to adopt a platform on which to run for elected office, and so forth. In contrast to canonical decision-making in economics, therefore, what an individual chooses in politics is not always what an individual obtains (e.g. voting for some electoral candidate does not ensure that the candidate is elected). Thus, the link between an individual’s decisions in politics and the consequent payoffs to the individual is attenuated relative to that for economic decisions: the basic political model of individual choice is game theoretic.

The preceding observations suggest two approaches to understanding how individual preferences connect to political, or collective, choices and both are pursued: a direct approach through extending the individual decision-theoretic model to the collectivity as a whole, an approach essentially begun with Arrow (1951, 1963) and Black (1958); and an indirect approach through exploring the consequences of mutually consistent sets of strategic decisions by instrumentally rational agents, an approach with roots in von Neumann and Morgenstern (1944) and Nash (1951).

Under the direct (collective preference) approach to social choice, individuals’ preferences are directly aggregated into a “social preference” which, as in individual decision theory, is then maximized to yield a set of best (relative to the maximand) alternatives, the collective choices. But although individual preferences surely influence individual decisions such as voting, there is no guarantee that individuals’ preferences are revealed by their decisions (for example, an individual may have strict preferences over candidates for electoral office, yet choose to vote strategically or to abstain). Under the indirect (game-theoretic) approach to social choice, therefore, it is individuals’ actions that are aggregated to arrive at collective choices. Faced with a particular decision problem, individuals rarely have to declare their preferences directly but instead have to take some action. For example, in a multicandidate election under plurality rule, individuals must choose the candidate for whom to vote and may abstain; the collective choice from the election is then decided by counting the recorded votes and not by direct observation of all individuals’ preferences over the entire list of candidates. It is useful to be a little more precise.

A preference profile is a list of preferences, one for each individual in the society, over a set of alternatives for that society. An abstract collective choice rule is a rule that assigns collective choices to each and every profile; that is, for any list of preferences, a collective choice rule identifies the set of outcomes chosen by society. Similarly, a preference aggregation rule is a rule that aggregates individuals’ preferences into a single, complete, “social preference” relation over the set of alternatives; that is, for any profile, the preference aggregation rule collects individual preferences into a social preference relation over alternatives. It is important to note that while the theory (following economics) presumes individual preferences are complete and transitive, the only requirement at this point of a social preference relation is that it is complete. For any profile and preference aggregation rule, we can identify those (p. 815) alternatives (if any) that are ranked best by the social preference relation derived from the profile by the rule. With a slight abuse of the language, this set is known as the core of the preference aggregation rule at the particular profile of concern.5 Taken together, therefore, a preference aggregation rule and its associated core for all possible preference profiles is an instance of an abstract collective choice rule. Thus, the extension of the classical economic decision-theoretic model of individual choice to the problem of collective decision-making, the direct approach mentioned above, can be described as the analysis of the abstract collective choice rules defined by the core of various preference aggregation rules.

The analytical challenge confronted by the direct approach is to find conditions under which preference aggregation relations exist and yield well-defined, that is, non-empty, cores. This approach has focused on two complementary issues: delineating classes of preference aggregation rule that are consistent with various sets of desiderata (for instance, Arrow’s possibility theorem (1951, 1963) and May’s theorem (May 1952) characterizing majority rule) and describing the properties of particular preference aggregation rules in various environments6 (for instance, Plott’s characterization of majority cores (Plott 1967) in the spatial model and the chaos theorems of McKelvey 1976 1979, and Schofield 1978, 1983). Contributions to the first issue rely heavily on axiomatic methods whereas contributions to the second have, for the most part, exploited the spatial voting model in which the feasible set of alternatives is some subset of (typically) k-dimensional Euclidean space and individuals’ preferences can be described by continuous quasi-concave (loosely, single peaked in every direction) utility functions.

From the perspective of developing a decision-theoretic approach to prediction and explanation at the collective level, the results from collective preference theory are a little disappointing. There exist aggregation rules that justify treating collective choice in a straightforward decision-theoretic way only if the environment is very simple, having a minimal number of alternatives from which to choose or satisfying severe restrictions on the sorts of preference profiles that can exist (for instance, profiles of single-peaked preferences over a fixed ordering of the alternatives), or if the preferences of all but a very few are ignored in the aggregation (as in dictatorships). Moreover, in the context of the spatial model, most of the aggregation procedures observed in the world are, at least in principle, subject to chronic instability unless politics concerns only a single issue.7 Nevertheless, a great deal has been learned from the collective preference approach to political decision-making about the properties and implications of preference aggregation and voting rules, (p. 816) and about the normative and descriptive trade-offs inherent in choosing one rule over another.8

Unlike the direct collective preference approach, the indirect approach to collective choice through the aggregation of the strategic decisions of instrumentally rational individuals begins by specifying the collection of possible decision, or strategy, profiles that could arise. A strategy for an individual specifies what the individual would do in every possible contingency that could arise in the given setting. A strategy profile is then a list of individual strategies, one for each member of the polity. An outcome function is a rule that identifies a unique alternative in the set of possible social alternatives with every feasible strategy profile. A specification of all possible strategy profiles along with an outcome function is called a mechanism. An abstract theory of how individuals make their respective decisions under any mechanism is a rule that associates a strategy profile with every preference profile; that is, for any given preference profile, an abstract decision theory assigns a set of possible strategy profiles consistent with the theory when individuals’ preferences are described by the given list of preferences.9 For any preference profile, mechanism, and decision theory, we can identify those alternatives that could arise as outcomes from strategy profiles consistent with the decision theory at that preference profile. This is the set of equilibrium outcomes under the mechanism at the preference profile. Then the indirect approach to preference aggregation can be described as the analysis of the collective choice rules defined by the sets of equilibrium outcomes of various mechanisms and theories of individual decision-making.

There is no effort under the indirect approach to treat collective decision-making as in any way analogous to individual decision-making. Instead, individuals make choices (vote, contribute to collective action, and so forth) taking account of the choices of others and the likely consequences of various combinations of the individuals’ decisions. Although such choices are expected to reflect individual preferences, there is no presumption that they do so in any immediately transparent or literal fashion. The approach therefore requires both a theory of how individuals make their decisions (the abstract theory of decision-making considered above) and a description of how the resulting decisions are mapped into collective choices (a specification of the outcome function). Putting these two components together with preferences then yields a model of collective choice through the aggregation of individual decisions.

Unlike the collective preference approach, there is little difficulty with developing coherent predictive models of collective choice within the game-theoretic framework. That is, while the social choice mapping derived from a preference aggregation rule rarely yields maximal elements, the mapping derived from a mechanism and theory of individual behavior is typically well defined. This fact, coupled with the flexibility of the approach with respect to modeling institutional details, uncertainty, and (p. 817) incomplete information, has led to game theory dominating contemporary formal theory. Indeed, it has been argued that the adoption of (in particular) non-cooperative game-theoretic techniques represents a fundamental shift in methodology from those of collective preference theory (e.g. Baron 1994; Diermeier 1997). Yet, at least from a formal perspective, the difference between the collective preference and the game-theoretic approaches is not so stark.

Both approaches to collective choice, the direct and the indirect, yield social choice rules, taking preference profiles into collective choices. Thus any result concerning such rules must apply equally to both. In particular, it is true that a social choice rule is generally guaranteed to yield a non-empty core only if it violates a “minimal democracy” property, where “minimal democracy” means that if all but at most one individuals strictly prefer some alternative x to another y, then y should not be ranked strictly better than x under the choice rule (Austen-Smith and Banks 1998). Whence it follows that the indirect approach ensures existence of well-defined solutions by violating minimal democracy, whereas the direct approach insists on minimal democracy at the expense of ensuring non-empty cores in any but the simplest settings. On this account, the direct and the indirect approaches to understanding collective decision-making are complementary rather than competitive. Which sort of model is most appropriate depends on the problem at hand and, in some important cases, their respective predictions are intimately related (Austen-Smith and Banks 1998, 2004). Moreover, the collective preference approach has revealed two general analytical characteristics of collective decision-making peculiar to political science relative to economics, characteristics that have stimulated important methodological and substantive innovations in the game-theoretic approach. It is to these characteristics and the innovations they have induced that I now turn.

3 Two Analytical Characteristics

The first characteristic exposed by collective preference theory involves the role of opportunities for trade. In economics, it is typically the case that, for any given society, the greater are the opportunities for trade the more likely it is that welfare-improving trade takes place. The analogue to increasing opportunities for trade in politics is increasing the dimensionality of the policy space in the spatial model or the number of alternatives in the finite-alternative model. As the number of alternatives or issue dimensions on which the preferences of a given population can differ grows, so too does the number of opportunities for winning coalitions to agree on a change from any policy; with one dimension, for example, coalitions must agree either to “move policy to the left” or to “move policy to the right” but, with two dimensions, there are uncountable directions in which to change policy, and preferences can be distributed over the plane, permitting more coalitions to form against any given (p. 818) policy. But it is precisely in such complex settings that preference aggregation rules are most poorly behaved: with one dimension the median voter theorem ensures a well-defined collective choice under majority rule (Black 1958) but, with two dimensions, the existence of such a choice is an extremely rare event and virtually any pair of alternatives can be connected by a finite sequence of majority-preferred steps (McKelvey 1979). Thus increasing “opportunities for trade” in the political setting exacerbate the problems of reaching a collective choice rather than ameliorate them.

The second characteristic concerns large populations. In economics it is the market that aggregates individual decisions into a collective outcome. As the number of individuals grows, the influence of any single agent becomes negligible and, in the limit, instrumentally rational individuals act as price-takers; moreover, large populations tend to smooth over non-convexities and irregularities at the individual level, justifying an approximation that all members of the population act as canonic economic theory presumes. These nice properties do not hold in political settings. Individual decisions are aggregated through voting and although the likelihood that any individual is pivotal vanishes as the electorate grows, for any finite society that likelihood is not zero: under majority rule in the classical Downsian spatial model, the median voter is pivotal whether there are three voters or three billion and three. So not only can the collective choice depend critically on a single person’s decision, it is unjustified to treat each agent analogously to a “price-taker” and non-convexities and irregularities can matter a great deal depending on precisely where they are located in the population. The “correct” model of decision-making here is therefore to presume individuals condition their choices on being pivotal and act as if their vote or contribution or whatever tips the balance in favour of one or other collective choice: either they are not in fact pivotal in which case their decision is irrelevant, or they are pivotal in which case their decision determines the collective choice.10

The conclusion that rational individuals condition their vote decisions on the event that they are pivotal is a strategic, game-theoretic, perspective and it is within this framework that efforts to tackle the problems raised by each characteristic have been undertaken.

3.1 Institutions and Explanation

A virtue of the collective preference methodology (and, to a large extent, the cooperative game-theoretic methodology exploited by Riker 1962 and others) is that it is essentially “institution free,” focusing exclusively on how domains of preference profiles are mapped into collective choices without attention to how the profiles might be recorded, from where the alternatives might arise, and so on. The idea underlying the axiomatic method of collective preference theory is to abstract from empirical and detailed institutional complications and study whole classes of possible institution-satisfying (p. 819) particular properties. A limitation of this method for an explanatory theory, however, is the typical emptiness of the core in complex settings, that is, those with many issues or alternatives over which to choose. And although non-cooperative game theory typically requires an exhaustive description of the relevant institutional details in any application, it is rarely hampered by questions of the existence of solutions. This observation prompted a shift in emphasis away from a collective preference methodology tailored to avoid concerns with the details of any application, to a non-cooperative game-theoretic approach that embraces such details as intrinsic to the analysis.

Two illustrations of the role of institutional detail in finessing problems of existence in complex political environments are provided by the use of particular sorts of agenda in committee decision-making from finite sets of alternatives (see Miller 1995 for an overview) and the citizen-candidate approach to electoral competition in the spatial model, whereby the candidates contesting an election are themselves voters who strategically choose whether or not to run for office at some cost (Osborne and Slivinsky 1996; Besley and Coate 1997).

In the classical preference profile to illustrate the instability of majority rule (the Condorcet paradox), three committee members have strict preferences over three alternatives such that each alternative is best in one person’s ordering, middle ranked in a second person’s ordering, and worst in a third person’s ordering. There is no majority core in this example, with every alternative being beaten by one of the others under majority preference. However, committee decisions are often governed by rules, such as the amendment agenda. Under the amendment agenda, one alternative is first voted against another and the majority winner of the vote (not preference) is then put against the residual alternative in a final majority vote to determine the outcome. It is well known that the unique subgame perfect Nash equilibrium (an instance of a theory of individual decision-making in the earlier language) prescribes that individuals vote with their immediate (sincere) preferences at the final division to yield two conditional outcomes, one for each possible winner at the first division, and then vote sincerely at the first division with respect to these two conditional outcomes. Assuming majority preference is always strict, this backwards induction procedure invariably produces a unique prediction which in general depends on the ordering of the alternatives as well as the distribution of individual preferences per se.11 Moreover, the set of possible outcomes from amendment agendas (on any given finite set of alternatives and any finite committee) as a function of the preference profiles inducing a strict majority preference relation is now completely characterized (Banks 1985).

Similar to the difficulty with many alternatives illustrated by the Condorcet paradox, the majority rule core in the multidimensional spatial model is typically empty, and core emptiness means the model offers no positive predictions beyond the claim that for every policy, there exists an alternative policy and a majority that strictly (p. 820) prefers that alternative. But policies are offered by candidates and candidates are themselves members of the electorate. It is natural, therefore, to treat the set of potential candidates as being exactly the set of citizens. Furthermore, since citizens are endowed with policy preferences, other things equal and conditional on being elected, a successful candidate has no incentive to implement any platform other than his or her most preferred policy. In turn, rational voters recognize that whatever a candidate drawn from the electorate might promise in the campaign, should the candidate be elected then that person’s ideal policy is the final outcome. When the set of potential candidates coincides with the set of voters, therefore, there is no essential difference between the problem of electoral platform selection and the problem of candidate entry: explaining the distribution of electoral policy platforms in the citizen-candidate model is equivalent to explaining the distribution of citizens who choose to run for electoral office. And assuming that it is costly to run for office, individuals weigh the expected gains (which depend, inter alia, on who else is running) from entering an election against this cost when deciding whether to run for office. It follows that alternatives are costly to place on the agenda in the citizen-candidate model and it is not hard to see, then, that these institutional details introduce sufficient stickiness to ensure the existence of equilibria. Moreover, by varying parameters such as the cost of entry, the electoral rule of concern, and so forth, various comparative predictions concerning policy outcomes and electoral system are available.12

3.2 Information and Large Populations

Beyond questions of core existence, the application of the collective preference model to environments in which individuals face considerable uncertainty, either about the implications of any collective decision (imperfect information) or about the preferences of others (incomplete information), is awkward. Non-cooperative game theory, however, can readily accommodate uncertainty and informational variations.13 And whereas uncertainty is unnecessary for developing a coherent theory of economic behavior among large populations (in particular, the theory of perfect competition), it is uncertainty that provides a hook on which to develop a coherent theory of political behavior among large populations.

As remarked above, a peculiarity of political decision-making relative to economic decision-making is that consideration of large populations greatly complicates rather than simplifies the analysis of individual decisions. In markets, each consumer becomes negligible with respect to influencing price as the number of consumers (p. 821) grows and, therefore, is properly conceived as taking prices as given; in electorates, however, while it remains true that the likelihood that any single vote tips the outcome becomes vanishingly small as the number of voters grows, it is not true (at least for finite electorates) that the behavior of any given voter should be conditioned on the almost sure event that the voter’s decision is consequentially irrelevant. This is not usually a problem for classical collective preference theory which, as the name suggests, focuses on aggregating given preference profiles, not vote profiles. Nor is it any problem for Nash equilibrium theory insofar as there are a huge number of equilibrium patterns of voting in any large election (other than with unanimity rule), most of which look empirically silly. But empirical voting patterns are not arbitrary. And once account is taken of the fact that preferring one candidate to another in an election does not imply voting for that candidate (individuals can abstain or vote strategically), there is clearly a severe methodological problem with respect to analysing equilibrium behavior in large electorates.

One line of attack has been by brute force, using combinatorial techniques to compute the probability that a particular vote is pivotal, conditional on the specified (undominated) votes of others (Ledyard 1984; Cox 1994; Palfrey 1989). But this is cumbersome and places considerable demands on exactly what it is that individuals know about the behavior of others. In particular, individuals are assumed to know the exact size of the population. Myerson (1998, 2000, 2002) relaxes this assumption and develops a novel theory of Poisson games to analyse strategic behavior in large populations.

Rather than assume the size of the electorate is known, suppose that the actual number of potential voters is a random variable distributed according to a Poisson distribution with mean n, where n is large. Then the probability that there is any particular number of voters in the society is easily calculated. As a statistical model underlying the true size of any electorate, the Poisson distribution uniquely exhibits a very useful technical property, environmental equivalence: under the Poisson distribution, any individual in the realized electorate believes that the number of other individuals in the electorate is also a random variable distributed according to a Poisson distribution with the same mean. And because the number and identity of realized individuals in the electorate is a random variable, it is enough to identify voters by type rather than their names, where an individual’s type describes all of the strategically relevant characteristics of the individual (for example the individual’s preferences over the candidates seeking office in any election). If the list of possible individual types is fixed and known, then the distribution of each type in a realized population of any size is itself given by a Poisson distribution.

The preceding implications of modeling population size as an unobserved draw from a Poisson distribution allow a relatively tractable and appealing strategic theory of elections. Because only types are relevant, individual strategies are appropriately defined as depending only on voter type rather than on voter identity. Thus individuals know only their own types, the distribution of possible types in the population, that the population size is a random draw from a Poisson distribution, and that all individuals of the same type behave in the same way. Call such a strategic model a (p. 822) Poisson game. An equilibrium to a Poisson game is then a specification of strategies, one for each type, such that, for any individual of any type, the individual is taking a best decision taking as given the strategies of all other types, conditional on his or her beliefs regarding the numbers of individuals of each type in the electorate. Such equilibria exist and have well-defined limits with strictly positive turnout as the mean population size increases. Myerson proposes using these limiting equilibria as the basis of predictions about political behavior in large populations. And to illustrate the relative elegance of the method over the usual combinatoric approach, in Myerson (2000) he provides a version of a theorem on turnout and candidate platform convergence due to Ledyard (1984) and, in Myerson (1998), he establishes a Condorcet jury theorem (see also Myerson 2002 for a comparative analysis of three-candidate elections under scoring rules using a Poisson game framework).14

In economics, markets also serve to aggregate information through relative prices. There are no relative prices explicit in elections, yet it is not only implausible to presume voters know the true size of the electorate, it is also implausible that they know the full implications of electing one candidate rather than another. Because the likelihood that any single vote is pivotal in a large election is negligible, the incentives for any one voter in a large electorate to invest in becoming better informed regarding the candidates for election are likewise negligible. Thus Downs (1957) argued that voters in large populations would be “rationally ignorant.” But just as is the case with his theory of participation, Downs’s argument is decision-theoretic and does not necessarily apply once the strategic character of political behavior is made explicit. In particular, an instrumentally rational voter conditions her vote on the event that she is pivotal; and in the presence of asymmetric information throughout the electorate, conditioning on the event of being pivotal can yield a great deal of information about what others know. To see this, consider an example in which two candidates are competing for a majority of votes in a three-person electorate. Suppose each voter receives a noisy private signal correlated with which of the two candidates would be best and (for simplicity) suppose further that all voters share identical full-information preferences. Now if the first two voters are voting sincerely relative to their signals and the third voter is pivotal, it must be the case that the first two voters have received conflicting information about the candidates, in which case the third voter can base her vote on all of the available information distributed through the electorate, even though that distribution was not publicly known.

Exactly what are the information aggregation properties of various electoral schemes is currently subject to much research. In some settings, the logic sketched above can yield quite perverse results; for example, Ordeshook and Palfrey (1988) provide an example in which an almost sure Condorcet winner (that is, an alternative against which no alternative is preferred by a strict majority) is surely defeated in an amendment agenda with incomplete information. And in other settings, it turns (p. 823) out that elections are remarkably efficient at aggregating information; Feddersen and Pesendorfer (1996) prove a striking full-information equivalence theorem for two-candidate elections with costly voting under any majority or super-majority rule shy of unanimity, namely despite the fact that a significant proportion of the electorate might abstain, the limiting outcome as the population grows is almost surely the outcome that would arise if all individuals voted under complete information.15

4 Conclusion

To all intents and purposes, the methods of contemporary positive political theory coincide with the methods of contemporary economic theory. The most widespread framework for models of campaign contributions at present derives from the common agency problem introduced by Bernheim and Whinston (1986); Rubinstein’s model of alternating offer bargaining (Rubinstein 1982) has been developed and extended to ground a general theory of legislative decision-making and coalition formation; Spence’s theory of costly signaling games (Spence 1974) and Crawford and Sobel’s (1982) extension of this theory to costless (cheap-talk) signaling provide the tools for a theory of legislative committees, delegation, informational lobbying, debate, and so on. More recently, the growth of interest in behavioral economics, experimental research, and so forth is beginning to appear in the political science literature. Rather than sketch these and other applications of economic methods to political science, this chapter attempts to articulate a broader (likely idiosyncratic) view of positive political theory since the importation of formal rational choice theory to politics. After all, political decision-making has at least as much of a claim to being subject to rational choice as economic decision-making; political agents make purposive decisions to promote their interests subject to constraints. It would be odd, then, to discover that the methods of economics are of no value to the study of politics.

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Notes:

(1) A suggestion (first made to me in conversation many years ago by Barry Weingast) as to why the two disciplines differ so markedly with respect to the use of mathematical modeling is that political science has no analogous concept to that of the margin in economics. And the importance of the margin in this respect lies less with its substantive content than with the amenability of its logic to elementary diagrammatic representation. Economic theorizing evolved into its contemporary mathematical form through a diagrammatic development of the logic of the margin, whereas positive political theory, almost of necessity, bypassed any such graphical development and jumped directly to applied game theory.

(2) Duncan Black’s The Theory of Committees and Elections (1958) has some claim to be included as a fourth such book. However, although Black considers similar issues to those taxing Arrow, his concern was more limited than that of Arrow and his particular contribution to political science was generally recognized only after the importance of Arrow’s work had begun to be appreciated.

(3) Palfrey and Rosenthal 1983 and Ledyard 1984 provide the earliest fully strategic models of turnout.

(4) Of course, this understanding itself reflects a largely consequentialist perspective intrinsic to economics. Insofar as there is consideration with any economic process, it is rarely with the process per se but with respect to the outcomes supported or induced by that process. This remains true for normative analysis. For example, axiomatic characterizations of procedures for dispute resolution (such as bargaining or bankruptcy) rarely exclude all references to the consequences of using such procedures: Pareto efficiency and individual rationality are common instances of such consequentialist properties. In contrast, a consequentialist perspective is less well accepted within political science at large, where (inter alia) there is widespread concern with, say, the legitimacy of procedures independent of the outcomes they might induce.

(5) The abuse arises since, strictly speaking, the core is defined with respect to a given family of coalitions. To the extent that a preference aggregation rule can be defined in terms of so-called decisive, or winning coalitions, the use of the term is standard. But not all rules can be so defined in which case the set of best elements induced by such a rule is not a core in the strict sense (see, for example, Austen-Smith and Banks 1999, ch. 3). The terminology in these instances is therefore an abuse but a useful and harmless one nevertheless.

(6) That is, various admissible classes of preference profiles and sorts of feasible sets of alternatives.

(7) See Austen-Smith and Banks 1999 for an elaboration of these claims.

(8) It is worth pointing out here, too, that the typical emptiness of the core has stimulated work on solutions concepts other than the core for collective preference theory (e.g. Schwarz 1972; Miller 1980; McKelvey 1986).

(9) Examples of such decision theories include Nash equilibrium and its refinements. See Fudenberg and Tirole 1991 or Myerson 1991.

(10) While this applies to any large finite electorate, proceeding to the limit in which each voter is infinitessimally small removes even this prescription regarding how strategically rational agents may behave, on which more below.

(11) It is worth noting in this example that the equilibrium outcome surely violates minimal democracy because, for every possible decision, there is an alternative that is strictly preferred by two of the three individuals.

(12) The contemporary literature on game-theoretic models of comparative institutions is large and growing. Examples include Austen-Smith and Banks 1988; Cox 1990; Persson, Roland, and Tabellini 1997; Myerson 1999; and Diermeier, Eraslan, and Merlo 2003.

(13) The theoretical foundations were laid by Harsanyi 1967–8. Two particularly important papers since then for political science are Spence 1973, who introduced the class of signaling games, and Crawford and Sobel 1982, who extended this class to include cheap-talk (costless) signaling.

(14) Condorcet jury theorems address the problem of choosing one of two alternatives when voters are uncertain about which is most in their interests. Typically, the theorems connect the size of the electorate (jury) to the probability that majority voting outcomes coincide with the majority choice that would be made under no uncertainty.

(15) The logic of this result is that the while, as Downs suggested, the relative number of voters voting informatively declines as the electorate grows due to the diminishing likelihood of being pivotal, the absolute number of voters voting informatively increases as the electorate grows at a faster rate, and it is the latter that dominates the information aggregation.