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Introduction to Political Science
Journal of Economic Theory 103, 1–10 (2002) doi:10.1006/jeth.2001.2879, available online at http://www.idealibrary.com on
Introduction to Political Science David Austen-Smith Departments of Political Science and Economics, Northwestern University, Evanston, Illinois 60208
Jeffrey S. Banks Division of Humanities and Social Sciences, California Institute of Technology, Pasadena, California 91125
and Aldo Rustichini 1 Department of Economics, Boston University, Boston, Massachusetts 02215 Received June 7, 2001; final version received June 7, 2001
This introduction to the JET symposium on political science briefly reviews the main results of the papers in this issue and tries to put them in the context of the current research in political theory. Journal of Economic Literature Classification Numbers: D70, D71, D72, D78. © 2002 Elsevier Science (USA)
1. INTRODUCTION The field of political theory has an ancient pedigree and continues to be a central component of contemporary political science; the nine papers of this symposium, however, do not fall under its rubric. Instead, they are exemplars of what has come to be called positive or formal political theory, broadly understood as being ‘‘concerned with understanding political phenomena through analytical models which, it is hoped, yield insight into why political outcomes look the way they do and not some other way’’
1 The nine papers comprising the Symposium were selected prior to Jeffrey Banks’ passing in late December 2000. Although he cannot be held responsible for anything written in this Introduction, we wish to emphasize the enormous contribution Jeff made to the Symposium itself and to political science more widely. He is sorely missed and we are proud to dedicate the Symposium to him.
⁄
1 0022-0531/02 $35.00 © 2002 Elsevier Science (USA) All rights reserved.
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[2]. The seminal publications initiating positive political theory as a field are Anthony Downs’ An Economic Theory of Democracy [7] and William Riker’s A Theory of Political Coalitions [22]. Both books were distinguished for political science by their unequivocal adoption of a rational choice–theoretic methodology and distinguished for rational choice theory by their unequivocal focus on political decision making. Forty or so years on, the use of rational choice theory in political science is widespread, if not widely accepted, and growing, albeit unevenly across the various empirical subfields of the discipline. This is all well and good (at least, for those of us who find the approach compelling), but it raises the question of why a journal devoted to economic theory should publish a symposium devoted to political science. One immediate answer is that, in large part, economics concerns the allocation of scarce resources and the market is not the only mechanism for doing this. But perhaps the more important rationale for the symposium is the renaissance of political economy among economists. What is understood by the term political economy depends greatly on who is doing the understanding. Within much of political science, for instance, it refers either to essentially Marxist analyses of the state or, more generally, to any effort to understand how the macroeconomy affects the polity. As currently used in economics, however, the domain of the topic is nicely characterized by Banks and Hanushek [5], who write: Political economy... is the study of rational decisions in the context of political and economic institutions. Its central tenent is that a comprehensive understanding of economic phenomena requires knowledge of the political institutions, actors, and incentives present in the decision-making process. Conversely, these same political variables are best studied with the rational actor orientiation of economics and with a continual eye toward the economic consequences of political choices (p. 1).
Not surprisingly, although treating both dimensions of political economy with equal subtlety is clearly desirable it is hard to do: labeling an abstract set of alternatives as a set of tax rates does not make a voting model political economy any more than does renaming the representative agent in a neoclassical growth model as the median voter. And while it is the political scientists who understandably lean in the direction of oversimplifying the economics of a problem, it is the economists who are likewise more inclined to oversimplify the politics. One objective of the symposium, therefore, is to flag some recent work in political science with the intention, more-or-less implicitly, of encouraging those with a comparative advantage in game theory and rational choice (i.e., economists) to consider the politics of collective decision making more deeply than, say, forcing a median voter result under all circumstances. Indeed, if there is one thing that formal political theory has learned over
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the past 40 years or so it is that, unlike in purely economic markets, going to large numbers of agents in a collective decision fails to pin down the appropriate model of individual behavior 2 and adding opportunities for trade can only make things more complicated. 3 But such considerations are among the challenges for political economy and political science rather than prohibitions against furthering our understanding of collective decision making. The papers of the symposium fall broadly into three categories— Legislative bargaining and coalition formation, two-candidate competition, and multicandidate competition—and in various ways they all tackle aspects of the challenges posed by political science for a rigorous political economy. The papers by Eraslan [9], Eraslan and Merlo [10], and Jackson and Moselle [12] consider issues in legislative bargaining and coalition formation. The seminal contribution to legislative bargaining theory is due to Baron and Ferejohn [6]. In this paper, Baron and Ferejohn extend the Rubinstein alternating-offers model to a legislative setting: at the start of any period, Nature chooses one of a set of legislators to propose a division of the dollar among all legislators; if a majority votes in favor of the proposal then that division is implemented and the game ends; if a majority votes against the proposal then the game moves into the next session and repeats. All n \ 3 legislators are presumed to discount the future, to have linear preferences that depend exclusively on their own allocations and there are infinite possible sessions. There are many Nash equilibria to this game and attention is essentially confined to stationary (history independent) subgame perfect equilibria, which exist for their model and have the following now-familiar properties: when an individual is selected, he or she proposes a division of the dollar that gives some (n − 1)/2 others their discounted continuation values and the proposer the (larger) residual; the remaining legislators receive nothing. Therefore the very first proposal is accepted and ex post only a bare majority get a positive payoff. The Baron–Ferejohn model has become a staple for sidestepping core-existence problems in collective decision making and the subsequent literature is replete with applications; by imposing sufficient structure on the legislative process, the analyst ensures a prediction that can say something
2 Suppose we let the number of agents go to infinity. Whereas the behavior of a single consumer is appropriately modeled in the limit as an independent price-taker, that of a single voter is left unspecified as the probability of being pivotal goes rapidly to zero: it only takes one person to tip the balance in a majority decision whether there are three people or three billion and three but the likelihood of any given individual being that pivot is vanishingly small. 3 For example, under standard assumptions on preferences, moving from a one- to a twodimensional issue space leads to a catastrophic breakdown in the majority rule core.
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about what might happen in majoritarian committees. 4 But despite the flurry of applications for the model, important theoretical questions regarding general existence and uniqueness of (stationary) equilibria were left unaddressed. And given an important aspect of the appeal of the model was its relative agnosticism about details of collective deliberations, answers to such questions are important. Banks and Duggan [3] provide a very general existence theorem for sequential bargaining in committees comprised of risk-averse agents that might use a variety of decision-making rules and which admit a status quo. And Eraslan’s contribution to this symposium proves a valuable uniqueness result: specifically, she shows that in the case of linear preferences, but permitting variations in discount rates and the likelihoods of being chosen to propose in any period, the vector of stationary subgame perfect equilibrium payoffs is unique. Thus any multiplicity in the equilibrium set is (ex ante) payoff irrelevant. Of course, this result does not solve all empirically relevant issues since observable behavior along different equilibrium paths will typically vary but it is a significant step. The other two papers exploiting the sequential bargaining framework push the model in more substantive directions. Eraslan and Merlo replace the assumption of a fixed dollar for distribution with that of a stochastically varying dollar. This assumption makes some sense when considering real time bargaining over government coalition formation in parliamentary democracies. Given the proliferation of parties in typical parliamentary systems with proportional representation, the appropriate interpretation of an electoral vote distribution for the prospects of any party are often obscure; in turn, such ambiguity can make the relative value of joining a coalition early rather than late unclear, in which case the stochastic payoff assumption has content. Eraslan and Merlo show, first, that uncertainty can induce delay; this is perhaps not surprising in itself but they also show that the resulting equilibrium (even if unique) can be inefficient with the inefficiency coming from an agreement being reached too soon in the sequence. Thus, it is not so much delay itself that induces inefficiency as it is insufficient delay. Second, they show that Eraslan’s uniqueness result for stationary subgame perfect equilibrium payoffs breaks down: with a stochastic cake, there are not only multiple equilibria but also multiple equilibrium payoff’s. Consequently, unlike the implication of the uniqueness result for the deterministic setup, parties are not ex ante indifferent over the set of equilibrium governments and selection issues matter beyond the identification of observable behaviors. 4 On the other hand, absent the stationarity assumption the Baron–Ferejohn model exhibits a folk theorem so it can be argued that, while core-emptiness implies no prediction at all, the Baron–Ferejohn model predicts pretty much anything can happen.
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Jackson and Moselle explore a different variation within the sequential bargaining framework. They distinguish a purely distributive (divide-thedollar) issue and a purely policy (ideological) issue and examine bargaining over both. Individuals selected by Nature to be the proposer in any session are free to propose decisions on either or both of the dimensions and their main results are that, in equilibrium, proposals involve both issues simultaneously and that there are always multiple ideological positions with positive probability of being implemented. Moreover, the median legislator on the ideological issue may not be included in a winning coalition. The intuition behind these findings lies in the implicit trading across the issue dimensions, a feature that is perhaps most easily seen in the comparative statics: the more important is the distributive relative to the ideological issue the more diverse the support of equilibrium policy positions; conversely, as this relative importance goes to zero the support of equilibrium policy positions shrinks in the limit to the median’s ideological position. In turn, Jackson and Moselle’s results provide a perspective on the rationale for party formation, in the sense of predetermined coalitions. Moving from questions of legislative to electoral behavior, it is useful to distinguish the remaining contributions to the symposium by the assumption on the number of electoral competitors. Four papers—Banks et al. [4], Laslier and Picard [16], Aragones and Palfrey [1], and Prat [21]— assume two-candidate competition and focus on particular aspects of the policy competition, and two papers—Dutta et al. [8] and Myerson [18] —focus on the complications induced by the existence of more than two candidates, largely without concern regarding any final policy outcome. Of the four papers assuming two-candidate competition, that of Banks et al. and of Laslier and Picard consider abstract solution concepts for multidimensional spatial voting models. The basic game underlying such models involves a large set of voters, each with a strictly quasi-concave utility function defined over a given set of alternatives X ı R d, and two candidates competing for a single elected office under majority rule; the candidate who wins office (equivalently, a positive plurality) has a normalized payoff of one and the loser receives zero. The timing of the game is that candidates simultaneously choose policies (x, y) in X following which voters simultaneously cast one vote for x or for y. Unfortunately, as mentioned earlier, a central difficulty when there is more than a single issue (i.e., d > 1) is the absence of a core and, since the core coincides with the set of undominated pure strategy Nash equilibrium outcomes to the basic spatial voting game, this is more than a problem with adopting a solution concept from cooperative game theory. In response to the problems posed by core-emptiness in the spatial voting game, analysts have either proposed some alternative solution concept (for example, Kramer’s minmax set [13]) or sought equilibria in mixed
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strategies. For the basic multidimensional spatial voting game, however, the existence of mixed strategy Nash equilibria has not been generally established. 5 On the other hand, McKelvey [17] has argued that the support of any equilibrium mixed strategies for the game, when they exist, must lie in the uncovered set. Although there are some subtle variations, an alternative x in a set X ı R d is set to cover an alternative y in X if x is strictly (majority) preferred to y and, for every alternative z such that y is preferred to z, x is (at least) weakly preferred to z. The uncovered set is the set of alternatives that are not covered and, evidently, this set is precisely the core when the latter exists. 6 Banks et al. study the same issues in a much more general setting, establish McKelvey’s claim as a special case, and provide existence and partial characterization results for the general case. Their results therefore provide a deeper understanding of the uncovered set itself and, as a consequence, of the connection between the noncooperative notion of mixed strategy Nash equilibrium and the cooperative notion of the core for voting games. Unfortunately, there remain some problems for which the uncovered set offers little guidance, the most important of which is the pure distributional problem motivating Baron and Ferejohn’s [6] sequential decision-making model mentioned earlier. In this case, not only does the core fail to exist but (with linear and selfish preferences) the uncovered set includes the entire relative interior of the alternative space. And it is this last observation that motivates Laslier and Picard to ask if there might be tighter bounds on the support of equilibrium mixed strategies here: their answer is yes. Laslier and Picard consider the canonical model for studying purely distributional politics: there are n individuals each of whom has preferences linear in his or her allocation of a given fully divisible dollar; two identical candidates, or parties, offer distributions of the dollar (i.e., probability distributions on the simplex) in an effort to win a plurality of the votes and so secure elected office. The authors specify a proper subset of the simplex with the property that no individual’s share of the dollar exceeds 2/n, prove existence, and demonstrate that the support of any mixed strategy equilibrium must be contained in this set. In particular, any distribution in which each individual’s share is uniformly distributed over [0, 2/n] is an optimal strategy. But other equilibria exist and the authors characterize some of these at least partially, concluding that majority rule elections cannot lead to marked distributional inequalities. This is a striking result, both from a technical perspective and when juxtaposed 5 Kramer [14] proves an existence result for a model with a continuum of voters and vote-maximizing candidates. 6 In part because of this feature and in part because of the importance of the set in the theory of majority tournaments [15], the uncovered set has been proposed as a solution concept in its own right.
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against alternative treatments of the issue such as Myerson [20] where the message is majority rule promotes inequality. The contributions by Aragones and Palfrey and by Prat are more applied. Core existence is not a problem in the one-dimensional spatial model and so pure strategy undominated Nash equilibria typically exist for such models of two-candidate competition. Although exploiting a twocandidate, single-issue model, Aragones and Palfrey perturb the setup slightly by adding a fixed exogenous difference between the candidates that is a valence issue for the voters; that is, an issue that is independent of policy positions and over which all voters’ preferences are the same. Although there now exist one or two formal treatments of this issue, Aragones and Palfrey are the first to explore the structure of mixed strategy equilibria to the game—this is important as the existence of pure strategy equilibria is at best fragile and contingent on the inclusion of various ad hoc modeling assumptions. (The reason why pure strategy equilibria fail to exist here is easy to see: if candidate a has a given valence advantage over candidate b, however small, then a is sure to win if she adopts a policy platform sufficiently close to that of b; consequently, b always seeks to distance himself from a.) In their model, Aragones and Palfrey assume there is some uncertainty about the exact location of the median voter and assume candidates simultaneously choose strategies to maximize their respective probabilities of winning. Inter alia, they obtain existence and some characterization results under various assumptions on the distribution of the median’s ideal point. In particular, if the valence difference is sufficiently small then the support of the mixed strategy equilibria converges to the expected median’s ideal point as the size of the electorate goes to infinity. Prat too develops a model with given valence differences between candidates but here the goal is to explore the role of money in electoral competition. Unlike in Aragones and Palfrey, it is common knowledge that voters with quadratic preferences are distributed uniformly over a hypercube in R d so a (policy) core exists. On the other hand, voters do not know surely the valence characteristic (evaluated identically across voters) of the incumbent or the challenger. These data, however, are known to the incumbent and each of the 2d single-issue interest groups, one pair of opposing groups per policy dimension. Prat assumes the incumbent has selected a policy position in his or her first period of office and seeks money from the groups at the start of the second and final period; having attracted funds (observed by the voters) the incumbent responds by implementing a policy implicitly determined in the equilibrium to the common agency game between lobbies and incumbent (see, for example, [11]); voters then vote between the incumbent and a challenger. 7 With this 7
Prat assumes for convenience that only the incumbent receives money from the groups.
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setting, Prat explores the informational value of contributions to voters, the relationship between money and policy choice, and considers some welfare implications of limiting campaign contributions. As a by-product, his results provide a rationale for the observation that campaign spending has a low effect on electoral outcomes. All in all, this is a nice contribution to an as yet small but growing literature seeking to integrate both informational and quid-pro-quo models of campaign contributions. The final two papers of the symposium, by Myerson and by Dutta et al., tackle very important and very difficult problems in analyzing voting over multiple candidates. A central problem with the strategic analysis of large elections is how best to model voter behavior. Historically, the preferred assumption has been to assume voters are sincere and fail to use best response strategies. The sincere voting assumption is often defended on grounds of analytical simplicity and an empirical conjecture that, since the probability of making a difference is negligible, the effort of doing anything else is not worth it. While the former reason has merit, the latter is worrisome: it does not take many strategic voters to tip a large election and, while individuals may find it tricky to figure out how best to vote, political parties often employ full-time analysts to provide suggestions. Furthermore, the empirical observation that individuals avoid casting a wasted vote is hard to reconcile with the sincere voting assumption. From a gametheoretic perspective, the obvious approach is to assume voters condition their vote on the event that they are pivotal—even if the electorate is large, it is finite and so this event has positive probability. But doing this rapidly leads to considerable analytical complexity when there are more than two candidates; if there are three candidates, for example, then there are multiple pivot events to consider and best response behavior for one eventuality (say, a and b tied for first place) may be at odds for another (say, a and c tied for first place). In a creative move to develop a suitable framework for handling this kind of complexity, Myerson developed the Poisson model for large voting games: in these games the number of voters is uncertain and Myerson [19] develops general results that make the analysis of equilibria tractable. It is this model that he brings to bear on the three-candidate electoral voting problem in this symposium. Assuming three given candidates, or alternatives, for a single electoral office, Myerson compares the sets of equilibria for a family of scoring rules: ‘‘In an (A, B)-scoring rule, each voter must choose a vote vector that is a permutation of either (1, B, 0) or (1, A, 0). That is, the voter must give a maximum of 1 point to one candidate, a minimum of 0 points to some other candidate, and A or B points to the remaining candidate’’ [18]. By suitably choosing A and B, these rules include Borda, plurality, negative plurality, approval, and many more. Through a set of carefully chosen examples, Myerson shows that different rules induce some very different
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properties of the equilibrium set. These examples illustrate the power of the Poisson game approach to modeling large electorates but, at the same time, also suggest that the approach is not always going to make the analytical work more tractable; dealing with large elections is, it seems, an inherently complicated problem in general. The final paper by Dutta, Jackson, and LeBreton returns us to the legislative setting. These authors explore the properties of the successive elimination committee voting procedure when the agenda consists of potential candidates strategically choosing whether to enter the election. Committees involve small numbers of voters and, with sequential elimination, a well-defined procedure that makes the analysis of rational voting tractable (albeit hardly trivial). The properties of the successive elimination and similar committee procedures for a fixed set of alternatives are well known, at least for the complete information case. But very little is understood about endogenous agendas here. Dutta et al.’s study, therefore, is a welcome start on a large class of empirically important issues. Their principal finding in the paper is that the set of equilibrium outcomes with strategic candidacy can, in direct contrast to the fixed agenda case, include Pareto dominated alternatives. The extent to which this result extends to other decision-making schemes is an important issue for future research.
REFERENCES 1. E. Aragones and T. Palfrey, Mixed equilibrium in a Downsian model with a favored candidate, J. Econ. Theory 103 (2002), 131–161. 2. D. Austen-Smith and J. S. Banks, ‘‘Positive Political Theory I: Collective Preference,’’ University of Michigan Press, Ann Arbor, 1999. 3. J. S. Banks and J. Duggan, A bargaining model of collective choice, Amer. Polit. Sci. Rev. 94 (2000), 73–88. 4. J. S. Banks, J. Duggan, and M. Le Breton, Bounds for mixed strategy equilibria and the spatial model of elections, J. Econ. Theory 103 (2002), 88–105. 5. J. S. Banks and R. Hanushek, ‘‘Modern Political Economy,’’ Cambridge University Press, Cambridge, UK, 1995. 6. D. Baron and J. Ferejohn, Bargaining in legislatures, Amer. Polit. Sci. Rev. 83 (1989), 1181–1206. 7. A. Downs, ‘‘An Economic Theory of Democracy,’’ Harper & Rowe, Cambridge, UK, 1957. 8. B. Dutta, M. Jackson, and M. Le Breton, Voting by successive elimination and strategic candidacy, J. Econ. Theory 103 (2002), 190–218. 9. H. Eraslan, Uniqueness of stationary equilibrium payoffs in the Baron–Ferejohn model, J. Econ. Theory 103 (2002), 11–30. 10. H. Eraslan and A. Merlo, Majority rule in a stochastic model of bargaining, J. Econ. Theory 103 (2002), 31–48. 11. G. Grossman and E. Helpmann, Electoral competition and special interest politics, Rev. Econ. Stud. 63 (1996), 265–286.
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12. M. Jackson and B. Moselle, Coalition and party formation in a legislative voting game, J. Econ. Theory 103 (2002), 49–87. 13. G. Kramer, A dynamical model of political equilibrium, J. Econ. Theory 16 (1977), 310–334. 14. G. Kramer, Existence of electoral equilibrium, in ‘‘Game Theory and Political Science’’ (P. Ordeshook, Ed.), New York Univ. Press, New York, 1978. 15. J.-F. Laslier, ‘‘Tournament Solutions and Majority Voting,’’ Springer-Verlag, New York, 1997. 16. J.-F. Laslier and N. Picard, Distributive politics and electoral competition, J. Econ. Theory 103 (2002), 106–130. 17. R. McKelvey, Covering, dominance and institution-free properties of social choice, Amer. J. Polit. Sci. 30 (1986), 283–314. 18. R. Myerson, Comparison of scoring rules in Poisson voting games, J. Econ. Theory 103 (2002), 219–251. 19. R. Myerson, Large Poisson games, J. Econ. Theory 94 (2000), 7–45. 20. R. Myerson, Incentives to cultivate favored minorities under alternative electoral systems, Amer. Polit. Sci. Rev. 87 (1993), 856–869. 21. A. Prat, Campaign spending with office-seeking politicians, rational voters, and multiple lobbies, J. Econ. Theory 103 (2002), 162–189. 22. W. H. Riker, ‘‘The Theory of Political Coalitions,’’ Yale Univ. Press, New Haven, 1962.
Introduction to Political Science David Austen-Smith Departments of Political Science and Economics, Northwestern University, Evanston, Illinois 60208
Jeffrey S. Banks Division of Humanities and Social Sciences, California Institute of Technology, Pasadena, California 91125
and Aldo Rustichini 1 Department of Economics, Boston University, Boston, Massachusetts 02215 Received June 7, 2001; final version received June 7, 2001
This introduction to the JET symposium on political science briefly reviews the main results of the papers in this issue and tries to put them in the context of the current research in political theory. Journal of Economic Literature Classification Numbers: D70, D71, D72, D78. © 2002 Elsevier Science (USA)
1. INTRODUCTION The field of political theory has an ancient pedigree and continues to be a central component of contemporary political science; the nine papers of this symposium, however, do not fall under its rubric. Instead, they are exemplars of what has come to be called positive or formal political theory, broadly understood as being ‘‘concerned with understanding political phenomena through analytical models which, it is hoped, yield insight into why political outcomes look the way they do and not some other way’’
1 The nine papers comprising the Symposium were selected prior to Jeffrey Banks’ passing in late December 2000. Although he cannot be held responsible for anything written in this Introduction, we wish to emphasize the enormous contribution Jeff made to the Symposium itself and to political science more widely. He is sorely missed and we are proud to dedicate the Symposium to him.
⁄
1 0022-0531/02 $35.00 © 2002 Elsevier Science (USA) All rights reserved.
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[2]. The seminal publications initiating positive political theory as a field are Anthony Downs’ An Economic Theory of Democracy [7] and William Riker’s A Theory of Political Coalitions [22]. Both books were distinguished for political science by their unequivocal adoption of a rational choice–theoretic methodology and distinguished for rational choice theory by their unequivocal focus on political decision making. Forty or so years on, the use of rational choice theory in political science is widespread, if not widely accepted, and growing, albeit unevenly across the various empirical subfields of the discipline. This is all well and good (at least, for those of us who find the approach compelling), but it raises the question of why a journal devoted to economic theory should publish a symposium devoted to political science. One immediate answer is that, in large part, economics concerns the allocation of scarce resources and the market is not the only mechanism for doing this. But perhaps the more important rationale for the symposium is the renaissance of political economy among economists. What is understood by the term political economy depends greatly on who is doing the understanding. Within much of political science, for instance, it refers either to essentially Marxist analyses of the state or, more generally, to any effort to understand how the macroeconomy affects the polity. As currently used in economics, however, the domain of the topic is nicely characterized by Banks and Hanushek [5], who write: Political economy... is the study of rational decisions in the context of political and economic institutions. Its central tenent is that a comprehensive understanding of economic phenomena requires knowledge of the political institutions, actors, and incentives present in the decision-making process. Conversely, these same political variables are best studied with the rational actor orientiation of economics and with a continual eye toward the economic consequences of political choices (p. 1).
Not surprisingly, although treating both dimensions of political economy with equal subtlety is clearly desirable it is hard to do: labeling an abstract set of alternatives as a set of tax rates does not make a voting model political economy any more than does renaming the representative agent in a neoclassical growth model as the median voter. And while it is the political scientists who understandably lean in the direction of oversimplifying the economics of a problem, it is the economists who are likewise more inclined to oversimplify the politics. One objective of the symposium, therefore, is to flag some recent work in political science with the intention, more-or-less implicitly, of encouraging those with a comparative advantage in game theory and rational choice (i.e., economists) to consider the politics of collective decision making more deeply than, say, forcing a median voter result under all circumstances. Indeed, if there is one thing that formal political theory has learned over
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the past 40 years or so it is that, unlike in purely economic markets, going to large numbers of agents in a collective decision fails to pin down the appropriate model of individual behavior 2 and adding opportunities for trade can only make things more complicated. 3 But such considerations are among the challenges for political economy and political science rather than prohibitions against furthering our understanding of collective decision making. The papers of the symposium fall broadly into three categories— Legislative bargaining and coalition formation, two-candidate competition, and multicandidate competition—and in various ways they all tackle aspects of the challenges posed by political science for a rigorous political economy. The papers by Eraslan [9], Eraslan and Merlo [10], and Jackson and Moselle [12] consider issues in legislative bargaining and coalition formation. The seminal contribution to legislative bargaining theory is due to Baron and Ferejohn [6]. In this paper, Baron and Ferejohn extend the Rubinstein alternating-offers model to a legislative setting: at the start of any period, Nature chooses one of a set of legislators to propose a division of the dollar among all legislators; if a majority votes in favor of the proposal then that division is implemented and the game ends; if a majority votes against the proposal then the game moves into the next session and repeats. All n \ 3 legislators are presumed to discount the future, to have linear preferences that depend exclusively on their own allocations and there are infinite possible sessions. There are many Nash equilibria to this game and attention is essentially confined to stationary (history independent) subgame perfect equilibria, which exist for their model and have the following now-familiar properties: when an individual is selected, he or she proposes a division of the dollar that gives some (n − 1)/2 others their discounted continuation values and the proposer the (larger) residual; the remaining legislators receive nothing. Therefore the very first proposal is accepted and ex post only a bare majority get a positive payoff. The Baron–Ferejohn model has become a staple for sidestepping core-existence problems in collective decision making and the subsequent literature is replete with applications; by imposing sufficient structure on the legislative process, the analyst ensures a prediction that can say something
2 Suppose we let the number of agents go to infinity. Whereas the behavior of a single consumer is appropriately modeled in the limit as an independent price-taker, that of a single voter is left unspecified as the probability of being pivotal goes rapidly to zero: it only takes one person to tip the balance in a majority decision whether there are three people or three billion and three but the likelihood of any given individual being that pivot is vanishingly small. 3 For example, under standard assumptions on preferences, moving from a one- to a twodimensional issue space leads to a catastrophic breakdown in the majority rule core.
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about what might happen in majoritarian committees. 4 But despite the flurry of applications for the model, important theoretical questions regarding general existence and uniqueness of (stationary) equilibria were left unaddressed. And given an important aspect of the appeal of the model was its relative agnosticism about details of collective deliberations, answers to such questions are important. Banks and Duggan [3] provide a very general existence theorem for sequential bargaining in committees comprised of risk-averse agents that might use a variety of decision-making rules and which admit a status quo. And Eraslan’s contribution to this symposium proves a valuable uniqueness result: specifically, she shows that in the case of linear preferences, but permitting variations in discount rates and the likelihoods of being chosen to propose in any period, the vector of stationary subgame perfect equilibrium payoffs is unique. Thus any multiplicity in the equilibrium set is (ex ante) payoff irrelevant. Of course, this result does not solve all empirically relevant issues since observable behavior along different equilibrium paths will typically vary but it is a significant step. The other two papers exploiting the sequential bargaining framework push the model in more substantive directions. Eraslan and Merlo replace the assumption of a fixed dollar for distribution with that of a stochastically varying dollar. This assumption makes some sense when considering real time bargaining over government coalition formation in parliamentary democracies. Given the proliferation of parties in typical parliamentary systems with proportional representation, the appropriate interpretation of an electoral vote distribution for the prospects of any party are often obscure; in turn, such ambiguity can make the relative value of joining a coalition early rather than late unclear, in which case the stochastic payoff assumption has content. Eraslan and Merlo show, first, that uncertainty can induce delay; this is perhaps not surprising in itself but they also show that the resulting equilibrium (even if unique) can be inefficient with the inefficiency coming from an agreement being reached too soon in the sequence. Thus, it is not so much delay itself that induces inefficiency as it is insufficient delay. Second, they show that Eraslan’s uniqueness result for stationary subgame perfect equilibrium payoffs breaks down: with a stochastic cake, there are not only multiple equilibria but also multiple equilibrium payoff’s. Consequently, unlike the implication of the uniqueness result for the deterministic setup, parties are not ex ante indifferent over the set of equilibrium governments and selection issues matter beyond the identification of observable behaviors. 4 On the other hand, absent the stationarity assumption the Baron–Ferejohn model exhibits a folk theorem so it can be argued that, while core-emptiness implies no prediction at all, the Baron–Ferejohn model predicts pretty much anything can happen.
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Jackson and Moselle explore a different variation within the sequential bargaining framework. They distinguish a purely distributive (divide-thedollar) issue and a purely policy (ideological) issue and examine bargaining over both. Individuals selected by Nature to be the proposer in any session are free to propose decisions on either or both of the dimensions and their main results are that, in equilibrium, proposals involve both issues simultaneously and that there are always multiple ideological positions with positive probability of being implemented. Moreover, the median legislator on the ideological issue may not be included in a winning coalition. The intuition behind these findings lies in the implicit trading across the issue dimensions, a feature that is perhaps most easily seen in the comparative statics: the more important is the distributive relative to the ideological issue the more diverse the support of equilibrium policy positions; conversely, as this relative importance goes to zero the support of equilibrium policy positions shrinks in the limit to the median’s ideological position. In turn, Jackson and Moselle’s results provide a perspective on the rationale for party formation, in the sense of predetermined coalitions. Moving from questions of legislative to electoral behavior, it is useful to distinguish the remaining contributions to the symposium by the assumption on the number of electoral competitors. Four papers—Banks et al. [4], Laslier and Picard [16], Aragones and Palfrey [1], and Prat [21]— assume two-candidate competition and focus on particular aspects of the policy competition, and two papers—Dutta et al. [8] and Myerson [18] —focus on the complications induced by the existence of more than two candidates, largely without concern regarding any final policy outcome. Of the four papers assuming two-candidate competition, that of Banks et al. and of Laslier and Picard consider abstract solution concepts for multidimensional spatial voting models. The basic game underlying such models involves a large set of voters, each with a strictly quasi-concave utility function defined over a given set of alternatives X ı R d, and two candidates competing for a single elected office under majority rule; the candidate who wins office (equivalently, a positive plurality) has a normalized payoff of one and the loser receives zero. The timing of the game is that candidates simultaneously choose policies (x, y) in X following which voters simultaneously cast one vote for x or for y. Unfortunately, as mentioned earlier, a central difficulty when there is more than a single issue (i.e., d > 1) is the absence of a core and, since the core coincides with the set of undominated pure strategy Nash equilibrium outcomes to the basic spatial voting game, this is more than a problem with adopting a solution concept from cooperative game theory. In response to the problems posed by core-emptiness in the spatial voting game, analysts have either proposed some alternative solution concept (for example, Kramer’s minmax set [13]) or sought equilibria in mixed
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strategies. For the basic multidimensional spatial voting game, however, the existence of mixed strategy Nash equilibria has not been generally established. 5 On the other hand, McKelvey [17] has argued that the support of any equilibrium mixed strategies for the game, when they exist, must lie in the uncovered set. Although there are some subtle variations, an alternative x in a set X ı R d is set to cover an alternative y in X if x is strictly (majority) preferred to y and, for every alternative z such that y is preferred to z, x is (at least) weakly preferred to z. The uncovered set is the set of alternatives that are not covered and, evidently, this set is precisely the core when the latter exists. 6 Banks et al. study the same issues in a much more general setting, establish McKelvey’s claim as a special case, and provide existence and partial characterization results for the general case. Their results therefore provide a deeper understanding of the uncovered set itself and, as a consequence, of the connection between the noncooperative notion of mixed strategy Nash equilibrium and the cooperative notion of the core for voting games. Unfortunately, there remain some problems for which the uncovered set offers little guidance, the most important of which is the pure distributional problem motivating Baron and Ferejohn’s [6] sequential decision-making model mentioned earlier. In this case, not only does the core fail to exist but (with linear and selfish preferences) the uncovered set includes the entire relative interior of the alternative space. And it is this last observation that motivates Laslier and Picard to ask if there might be tighter bounds on the support of equilibrium mixed strategies here: their answer is yes. Laslier and Picard consider the canonical model for studying purely distributional politics: there are n individuals each of whom has preferences linear in his or her allocation of a given fully divisible dollar; two identical candidates, or parties, offer distributions of the dollar (i.e., probability distributions on the simplex) in an effort to win a plurality of the votes and so secure elected office. The authors specify a proper subset of the simplex with the property that no individual’s share of the dollar exceeds 2/n, prove existence, and demonstrate that the support of any mixed strategy equilibrium must be contained in this set. In particular, any distribution in which each individual’s share is uniformly distributed over [0, 2/n] is an optimal strategy. But other equilibria exist and the authors characterize some of these at least partially, concluding that majority rule elections cannot lead to marked distributional inequalities. This is a striking result, both from a technical perspective and when juxtaposed 5 Kramer [14] proves an existence result for a model with a continuum of voters and vote-maximizing candidates. 6 In part because of this feature and in part because of the importance of the set in the theory of majority tournaments [15], the uncovered set has been proposed as a solution concept in its own right.
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against alternative treatments of the issue such as Myerson [20] where the message is majority rule promotes inequality. The contributions by Aragones and Palfrey and by Prat are more applied. Core existence is not a problem in the one-dimensional spatial model and so pure strategy undominated Nash equilibria typically exist for such models of two-candidate competition. Although exploiting a twocandidate, single-issue model, Aragones and Palfrey perturb the setup slightly by adding a fixed exogenous difference between the candidates that is a valence issue for the voters; that is, an issue that is independent of policy positions and over which all voters’ preferences are the same. Although there now exist one or two formal treatments of this issue, Aragones and Palfrey are the first to explore the structure of mixed strategy equilibria to the game—this is important as the existence of pure strategy equilibria is at best fragile and contingent on the inclusion of various ad hoc modeling assumptions. (The reason why pure strategy equilibria fail to exist here is easy to see: if candidate a has a given valence advantage over candidate b, however small, then a is sure to win if she adopts a policy platform sufficiently close to that of b; consequently, b always seeks to distance himself from a.) In their model, Aragones and Palfrey assume there is some uncertainty about the exact location of the median voter and assume candidates simultaneously choose strategies to maximize their respective probabilities of winning. Inter alia, they obtain existence and some characterization results under various assumptions on the distribution of the median’s ideal point. In particular, if the valence difference is sufficiently small then the support of the mixed strategy equilibria converges to the expected median’s ideal point as the size of the electorate goes to infinity. Prat too develops a model with given valence differences between candidates but here the goal is to explore the role of money in electoral competition. Unlike in Aragones and Palfrey, it is common knowledge that voters with quadratic preferences are distributed uniformly over a hypercube in R d so a (policy) core exists. On the other hand, voters do not know surely the valence characteristic (evaluated identically across voters) of the incumbent or the challenger. These data, however, are known to the incumbent and each of the 2d single-issue interest groups, one pair of opposing groups per policy dimension. Prat assumes the incumbent has selected a policy position in his or her first period of office and seeks money from the groups at the start of the second and final period; having attracted funds (observed by the voters) the incumbent responds by implementing a policy implicitly determined in the equilibrium to the common agency game between lobbies and incumbent (see, for example, [11]); voters then vote between the incumbent and a challenger. 7 With this 7
Prat assumes for convenience that only the incumbent receives money from the groups.
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setting, Prat explores the informational value of contributions to voters, the relationship between money and policy choice, and considers some welfare implications of limiting campaign contributions. As a by-product, his results provide a rationale for the observation that campaign spending has a low effect on electoral outcomes. All in all, this is a nice contribution to an as yet small but growing literature seeking to integrate both informational and quid-pro-quo models of campaign contributions. The final two papers of the symposium, by Myerson and by Dutta et al., tackle very important and very difficult problems in analyzing voting over multiple candidates. A central problem with the strategic analysis of large elections is how best to model voter behavior. Historically, the preferred assumption has been to assume voters are sincere and fail to use best response strategies. The sincere voting assumption is often defended on grounds of analytical simplicity and an empirical conjecture that, since the probability of making a difference is negligible, the effort of doing anything else is not worth it. While the former reason has merit, the latter is worrisome: it does not take many strategic voters to tip a large election and, while individuals may find it tricky to figure out how best to vote, political parties often employ full-time analysts to provide suggestions. Furthermore, the empirical observation that individuals avoid casting a wasted vote is hard to reconcile with the sincere voting assumption. From a gametheoretic perspective, the obvious approach is to assume voters condition their vote on the event that they are pivotal—even if the electorate is large, it is finite and so this event has positive probability. But doing this rapidly leads to considerable analytical complexity when there are more than two candidates; if there are three candidates, for example, then there are multiple pivot events to consider and best response behavior for one eventuality (say, a and b tied for first place) may be at odds for another (say, a and c tied for first place). In a creative move to develop a suitable framework for handling this kind of complexity, Myerson developed the Poisson model for large voting games: in these games the number of voters is uncertain and Myerson [19] develops general results that make the analysis of equilibria tractable. It is this model that he brings to bear on the three-candidate electoral voting problem in this symposium. Assuming three given candidates, or alternatives, for a single electoral office, Myerson compares the sets of equilibria for a family of scoring rules: ‘‘In an (A, B)-scoring rule, each voter must choose a vote vector that is a permutation of either (1, B, 0) or (1, A, 0). That is, the voter must give a maximum of 1 point to one candidate, a minimum of 0 points to some other candidate, and A or B points to the remaining candidate’’ [18]. By suitably choosing A and B, these rules include Borda, plurality, negative plurality, approval, and many more. Through a set of carefully chosen examples, Myerson shows that different rules induce some very different
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properties of the equilibrium set. These examples illustrate the power of the Poisson game approach to modeling large electorates but, at the same time, also suggest that the approach is not always going to make the analytical work more tractable; dealing with large elections is, it seems, an inherently complicated problem in general. The final paper by Dutta, Jackson, and LeBreton returns us to the legislative setting. These authors explore the properties of the successive elimination committee voting procedure when the agenda consists of potential candidates strategically choosing whether to enter the election. Committees involve small numbers of voters and, with sequential elimination, a well-defined procedure that makes the analysis of rational voting tractable (albeit hardly trivial). The properties of the successive elimination and similar committee procedures for a fixed set of alternatives are well known, at least for the complete information case. But very little is understood about endogenous agendas here. Dutta et al.’s study, therefore, is a welcome start on a large class of empirically important issues. Their principal finding in the paper is that the set of equilibrium outcomes with strategic candidacy can, in direct contrast to the fixed agenda case, include Pareto dominated alternatives. The extent to which this result extends to other decision-making schemes is an important issue for future research.
REFERENCES 1. E. Aragones and T. Palfrey, Mixed equilibrium in a Downsian model with a favored candidate, J. Econ. Theory 103 (2002), 131–161. 2. D. Austen-Smith and J. S. Banks, ‘‘Positive Political Theory I: Collective Preference,’’ University of Michigan Press, Ann Arbor, 1999. 3. J. S. Banks and J. Duggan, A bargaining model of collective choice, Amer. Polit. Sci. Rev. 94 (2000), 73–88. 4. J. S. Banks, J. Duggan, and M. Le Breton, Bounds for mixed strategy equilibria and the spatial model of elections, J. Econ. Theory 103 (2002), 88–105. 5. J. S. Banks and R. Hanushek, ‘‘Modern Political Economy,’’ Cambridge University Press, Cambridge, UK, 1995. 6. D. Baron and J. Ferejohn, Bargaining in legislatures, Amer. Polit. Sci. Rev. 83 (1989), 1181–1206. 7. A. Downs, ‘‘An Economic Theory of Democracy,’’ Harper & Rowe, Cambridge, UK, 1957. 8. B. Dutta, M. Jackson, and M. Le Breton, Voting by successive elimination and strategic candidacy, J. Econ. Theory 103 (2002), 190–218. 9. H. Eraslan, Uniqueness of stationary equilibrium payoffs in the Baron–Ferejohn model, J. Econ. Theory 103 (2002), 11–30. 10. H. Eraslan and A. Merlo, Majority rule in a stochastic model of bargaining, J. Econ. Theory 103 (2002), 31–48. 11. G. Grossman and E. Helpmann, Electoral competition and special interest politics, Rev. Econ. Stud. 63 (1996), 265–286.
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12. M. Jackson and B. Moselle, Coalition and party formation in a legislative voting game, J. Econ. Theory 103 (2002), 49–87. 13. G. Kramer, A dynamical model of political equilibrium, J. Econ. Theory 16 (1977), 310–334. 14. G. Kramer, Existence of electoral equilibrium, in ‘‘Game Theory and Political Science’’ (P. Ordeshook, Ed.), New York Univ. Press, New York, 1978. 15. J.-F. Laslier, ‘‘Tournament Solutions and Majority Voting,’’ Springer-Verlag, New York, 1997. 16. J.-F. Laslier and N. Picard, Distributive politics and electoral competition, J. Econ. Theory 103 (2002), 106–130. 17. R. McKelvey, Covering, dominance and institution-free properties of social choice, Amer. J. Polit. Sci. 30 (1986), 283–314. 18. R. Myerson, Comparison of scoring rules in Poisson voting games, J. Econ. Theory 103 (2002), 219–251. 19. R. Myerson, Large Poisson games, J. Econ. Theory 94 (2000), 7–45. 20. R. Myerson, Incentives to cultivate favored minorities under alternative electoral systems, Amer. Polit. Sci. Rev. 87 (1993), 856–869. 21. A. Prat, Campaign spending with office-seeking politicians, rational voters, and multiple lobbies, J. Econ. Theory 103 (2002), 162–189. 22. W. H. Riker, ‘‘The Theory of Political Coalitions,’’ Yale Univ. Press, New Haven, 1962.