The Spatial Model of Politics, Study notes of Topology

Download The Spatial Model of Politics and more Study notes Topology in PDF only on Docsity! The Spatial Model of Politics Norman Schoeld November 21, 2007 v 9.2.1 Realignment and Federalism . . . . . . . . . . . . . . . . . . . . . . . .229 9.3 Coalitions of Enemies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232 9.3.1 The New Deal Coalition . . . . . . . . . . . . . . . . . . . . . . . . . . . .232 9.3.2 The Creation of the Republican Coalition . . . . . . . . . . . .233 9.3.3 Social Conservatives Ascendant in the G.O. P. . . . . . . . .236 9.3.4 Stem Cell Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .238 9.3.5 Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .240 9.4 The Changing Political Equilibrium . . . . . . . . . . . . . . . . . . . . . . .241 9.4.1 Party Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244 9.4.2 Party Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245 9.4.3 Party Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247 9.5 The Future of Republican Populism . . . . . . . . . . . . . . . . . . . . . . .250 9.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252 9.7 Appendix: Republican Senator Votes . . . . . . . . . . . . . . . . . . . . . .257 10 Final Remarks 259 10.1 The Madisonian Scheme of Government . . . . . . . . . . . . . . . . . .259 10.2 Preferences and Judgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265 vi Contents Tables 6.1 Duration (in months) of government, 1945-1987 120 6.2 Frequency of coalition types, by country, 1945-1987 121 6.3 Duration of European coalitions, 1945-1987 122 6.4 Knesset seats 128 6.5 Seats and votes in the Netherlands 137 6.6 Estimated vote shares and valences in the Netherlands 137 6.7 Seats in the Dutch Parliament, 2003 and 2006 140 6.8 Elections in Denmark, 1957 and 1964 144 6.9 Party and faction strengths in the Dáil Eireann, 1987 150 6.10 Recent elections in Europe 160 7.1 Vote shares and seats in the Knesset 180 7.2 Turkish election results 1999 185 7.3 Turkish election results 2002 185 7.4 Turkish election results 2007 190 7.5 Multinomial Logit Analysis of the 1999 Election in Turkey 199 7.6 Multinomial Logit Analysis of the 2002 Election in Turkey 200 7.7 Log Bayes factors for model comparisons in 1999 201 7.8 Log Bayes factors for model comparisons in 2002 201 9.1 Votes of Republican senators on immigration and stem cell research 257 Figures 3.1 A voting complex 57 4.1 Convex and non-convex preference 85 4.2 Non-convex social preference 87 4.3 Non-convexity of the critical preference cones 89 4.4 Condition for local cyclicity at a point 92 5.1 Euclidean preferences with the q- rule given by (n; q) = (4; 3) 108 5.2 Euclidean preferences with the q- rule given by (n; q) = (6; 4) 109 5.3 Euclidean preferences with the q- rule given by (n; q) = (5; 3) 110 5.4 The heart, the yolk and the uncovered set 113 5.5 The heart with a uniform electorate on the triangle 114 5.6 The heart with a uniform electorate on the pentagon 114 5.7 Experimental results of Fiorina and Plott (1978) 116 5.8 Experimental results of McKelvey and Ordeshook (1978) 116 vii 5.9 Experimental results of Laing and Olmstead (1978) 117 5.10 Experimental results of Laing and Olmstead (1978) 117 5.11 Experimental results of Eavey (1991) 118 5.12 Experimental results of Eavey (1991) 118 6.1 The core in the Knesset in 1992 129 6.2 The heart in the Knesset in 1988 129 6.3 Party positions in the Knesset in 1996 131 6.4 The conguration of the Knesset after the election of 2003 133 6.5 The conguration of the Knesset after the election of 2006 133 6.6 Party positions in the Netherlands in 1977 136 6.7 The Dutch Parliament in 2006 139 6.8 Finland in 2003 142 6.9 Denmark in 2001 145 6.10 Sweden in 2002 145 6.11 Norway in 2001 146 6.12 The heart in Belgium in 1999 148 6.13 The heart in Belgium in 2003 149 6.14 Ireland in 1987 151 6.15 Ireland in 2007 152 6.16 Iceland in 2003 153 6.17 Austria in 2006 154 6.18 Germany in 2002 155 6.19 The core in Italy in 1987 156 6.20 Italy in 2001 157 7.1 The Gumbel distribution 170 7.2 A local Nash equilibrium in the Knesset in 1996 182 7.3 Party positions and voter distribution in Turkey in 1999 186 7.4 The heart in Turkey in 1999 186 7.5 Party positions and voter distribution in Turkey in 2002 187 7.6 Party positions in the Netherlands 192 7.7 Party positions in the United Kingdom 194 7.8 Balance loci for parties in Britain 194 8.1 Activists in Argentina 213 8.2 The voter distribution in Argentina in 1989 219 9.1 Activists in the United States 225 10.1 Estimated positions of posssible candidates for the U.S. presidency 271 x Foreword their fruits harvested by separate communities of scholars with only the occasional cross-over (as in William Riker and Peter Ordeshook, An In- troduction to Positive Political Theory (Prentice-Hall, 1972)). One of the leading innovators, drawing from (and making important contribu- tions to) all of these traditions, is Norman Schoeld. The present volume is something of a grand synthesis. Its substantive focus is elections with its electoral deals, party activists, and voters on the one hand, and parlia- ments with their coalition-building and government-forming maneuver- ings on the other. In terms of tools, the arguments found in this volume draw heavily on social choice theory, the spatial model, and cooperative game theory. Indeed, the rst ve chapters constitute just about all one needs to know of social choice theory. But the core (pun intended) of this ne volume is found in four very rich applied chapters, constituting a profound synthesis of elections and parliaments – of voters and activists choosing political agents, agents in turn choosing governments, and governments governing. Along the way we learn about voting rules, electoral systems, the ecology of government coalitions, precipitating events, and quite a lot about the empirical con- dition of electorates, parliaments, and legislatures in the advanced indus- trial democracies of the West. (The intensive treatment of an incredibly complex coalitional situation found in Israel over the last two decades is highly instructive.) I want to single out two especially imaginative treatments found in the later chapters of this volume – imaginative both theoretically and em- pirically. First, Schoeld provides one of the most elaborated theoreti- cally grounded typologies of coalitional arrangements in parliamentary democracies (in a literature rich in typologies based mainly, even exclu- sively, on empirical patterns alone) which, in turn, provides insights into the frequent absence of centripetal forces in multiparty regimes. Sec- ond, Schoeld elaborates an analysis of electoral activists that goes far- ther, and is founded on a more rm theoretical basis, than anything that presently exists in the literature. Taking a highly original turn, Schoeld applies the logic of Duverger to interest groups, suggesting how the elec- toral rule (plurality vs. proportional representation) will affect activist coalition building. Application to the building of electoral coalitions in Argentina and the United States is quite provocative. Each of these will be of separate interest to research communities. To- xi gether, however, they provide the underpinnings for a net assessment of the effects of the centripetal pull of the voting electorate and the centrifu- gal impact of activists. The reader may have to burn a lot of intellectual energy to get to these points, but getting there not only is half the fun but also makes strikingly evident that Schoeld's large theoretical buildup is not merely an occasion for play in the theory sandbox. Norman Schoeld has the soul of a mathematician and the heart of a political scientist. He has, over a long career but especially in the present volume, combined these two impulses elegantly. In one sense this book is a nished product. In another it is but the beginning of a conversation. Kenneth A. Shepsle Harvard University September 2007 xv from Madison's dual theory of the Republic. On the one hand, Madison saw the President as a a natural way to prevent mutability or disorder in the legislature, while the ability of Congress to veto presidential risk- taking could prevent autocracy. At the same time, the election of the president would, in the extended republic, enhance the “probability of a t choice.” Madison's argument is interpreted in the light of the model of elections presented in the previous chapters. A number of chapters of this book use some gures and tables from previous work. Cambridge University Press kindly gave permission to use material from Schoeld (2006a), Schoeld and Sened (2006) and Miller and Schoeld (2008). I am grateful to Blackwell for permission to use material from Schoeld (1993) and Schoeld and Miller (2007), to Sage for permission to use material from Schoeld (1995), to Springer to use material from Schoeld (1996) and Schoeld (2006b) and to North Holland to use material from Schoeld and Cataife (2007). I received very helpful comments on the versions of the last four chapters of the book, presented at various conferences and seminars: the World Public Choice Meeting, Amsterdam, the ISNIE conference, Boulder, the conference on European Governance, Emory University, conferences on political economy in Cancún and Guanajuato, Mexico, the Meeting of the Society for the Advancement of Economic Theory, Vigo, Spain, the Conference on Modernization of the State and the Econ- omy, Moscow, and other conferences at the University of Virginia, Char- lottesville, at SUNY, Binghamton, and at the University of Hamburg. Versions of some of this work were presented at the Department of Eco- nomics, Concordia University, Montreal, at the Center for Mathematical and Statistical Modeling, Wilfred Laurier University, Waterloo, at the Center for Mathematical Modeling, University of California at Irvine, at the Higher School of Economics, St. Petersburg, at George Mason Uni- versity, and at the California Institute of Technology. The original versions of most of the chapters were typed by Cherie Moore, and many of the diagrams were drawn by Ugur Ozdemir. I am grateful to Cheryl Eavey, Joseph Godfrey, Eric Linhart, Evan Schnidman and Suumu Shikano for permission to make use of their work. I thank my coauthors, Guido Cataife, Gary Miller and Ugur Ozdemir for their collaboration. Ken Benoit and Michael Laver graciously gave permission for me to use their estimates of party position for many of the European xvi Preface polities (Benoit and Laver, 2006). I appreciate the support of the NSF (under grants SES 0241732 and 0715929), and of Washington University. The Weidenbaum Center at Washington University provided support during a visit at the International Center for Economic Research in Turin. I thank Enrico Colombatto and Alessandra Calosso for the hospitality I enjoyed at ICER. A year spent at Humboldt University, Berlin, under the auspices of the Fulbright Founda- tion, as distinguished professor of American Studies during 2002–2003, gave me the opportunity to formulate an earlier version of the formal electoral model. Finally, my thanks to Terry Clague and Robert Lang- ham, the editors at Routledge, for their willingness to wait for a number of years while the manuscript was in preparation. Norman Schoeld Washington University Saint Louis, Missouri 4 November 2007 Chapter 1 Introduction 1.1 Representative Democracy A fundamental question that may be asked about a political, economic or social system is whether it is responsive to the wishes or opinions of the members of the society and, if so, whether it can aggregate the conicting notions of these individuals in a way which is somehow ra- tional. More particularly, is it the case, for the kind of conguration of preferences that one might expect, that the underlying decision process gives rise to a set of outcomes which is natural and stable, and more im- portantly, “small” with respect to the set of all possible outcomes? If so, then it may be possible to develop a theoretical or “causal” account of the relationship between the nature of the decision process, along with the pattern of preferences, and the behavior of the social and po- litical system. For example, microeconomic theory is concerned with the analysis of a method of preference aggregation through the market. Under certain conditions this results in a particular distribution of prices for commodities and labor, and thus income. The motivation for this en- deavor is to match the ability of some disciplines in natural science to develop causal models, tying initial conditions of the physical system to a small set of predicted outcomes. The theory of democracy is to a large extent based on the assumption that the initial conditions of the politi- cal system are causally related to the essential properties of the system. That is to say it is assumed that the interaction of cross-cutting interest groups in a democracy leads to an “equilibrium” outcome that is nat- ural in the sense of balancing the divergent interests of the members of 1 4 Chapter 1. Introduction quently the “core” party can, if it so chooses, form a minority govern- ment, one without a majority of the seats in the legislature. This property of the core provides an explanation for what has appeared to be a puz- zle. The data set collected by Laver and Schoeld (1990) dealing with coalition governments in 12 European countries in the period 1945–1987 shows that about one-third of the governments were minority. About one- third were minimal winning, with just enough seats for a majority, and the remaining third were surplus, with parties included in the coalition unnecessary for the majority. In the absence of a core, the spatial the- ory suggests that bargaining between the parties will focus on a domain in the policy space known as the “heart.” In the simplest case where it is assumed that parties have “Euclidean” preferences determined by policy distance, the “heart” will be a domain bounded by the compromise sets of various minimal winning coalitions. These minimal winning coalitions are natural candidates for coalition government. Indeed, in some cases a bounding minimal winning coalition may costlessly include a surplus party. This notion of the heart suggests that in the absence of a core, one or other of these minimal winning or surplus coalitions will form. Chapter 6 illustrates the difference between a core and the heart by considering recent elections in Israel in the period 1988 to 2006 and in the Netherlands in 1977, 1981 and 2006. In Israel, the core party was Labor, under Rabin in 1992, and a new party, “Kadima,” founded by Ariel Sharon in 2005, but under the leadership of Ehud Olmert. After the elections of 1988 and 2003 the bargaining domain of the heart was bounded by various coalitions, involving the larger parties, Labor and Likud, and smaller parties like Shas. These examples raise another theoretical problem: if party leaders are aware that by adopting a centrist position they can create minority, dom- inant government, then why are parties located so far from the electoral center? Chapter 6 illustrates the great variety of political congurations in Europe: bipolar political systems, such as the Netherlands and Fin- land; left unipolar systems such as Denmark, Sweden and Norway; center unipolar systems such as Belgium, Luxembourg and Ireland; right unipo- lar such as Iceland. Italy is unique in that it had a dominant center party, the Christian Democrats until 1994, after which the political system was totally transformed by the elimination of the core. Models of elections also suggest that the electoral center will be an at- 1.1 Representative Democracy 5 tractor for political parties, since parties will calculate that they will gain most votes at the center.4 Chapter 7 presents an electoral model where this centripetal tendency will only occur under specic conditions. The model is based on the idea of valence, derived from voters' judgements about characteristics of the candidates, or party leaders. These valences or judgements are rst assumed to be independent of the policy choice of the party. The theory shows that parties will converge to the electoral cen- ter only if the valence differences between the parties are small, relative to the other parameters of the model. The empirical analysis considers elections in Israel in 1996, in Turkey in 1999 and 2002, in the Netherlands in 1977–1981 and in Britain in 1997. The results show that the estimated parameters of the model did not satisfy the necessary condition for convergence in Israel. The theory thus gives an explanation for the dispersion of political parties in Israel and Turkey along a principal electoral axis. However, the condition sufcient for convergence of the parties was satised in the British election of 1997, and in the Dutch elections of 1977–1981. Because there was no evidence of convergence in these elec- tions, the conict between theory and evidence suggests that the stochas- tic electoral model be modied to provide a better explanation of party policy choice. The chapter goes on to consider a more general valence model based on activist support for the parties 5. This activist valence model presupposes that party activists donate time and other resources to their party. Such resources allow a party to present itself more effectively to the electorate, thus increasing its valence. The main theorem of this chapter indicates how parties might balance the centrifugal tendency as- sociated with activist support, and the centripetal tendency generated by the attraction of the electoral center. One aspect of this theory is that it implies that party leaders will act as though they have policy preferences, since they must accommodate the demands of political activists to maintain support for future elections. A further feature is that party positions will be sensitive to the nature 4An extensive literature has developed in an attempt to explain why parties do not converge to the electoral center. See, for example, Adams (1999a,b, 2001); Adams and Merrill (1999a, 2005); Adams, Merrill and Grofman (2005); Merrill and Grofman (1999); Merrill and Adams (2001); Macdonald and Rabinowitz (1998). 5See Aldrich (1983a,b, 1995); Aldrich and McGinnis (1989). 6 Chapter 1. Introduction of electoral judgements and to the willingness of activists to support the party. As these shift with time, then so will the positions of the parties. The theory thus gives an explanation of one of the features that comes from the discussion in Chapter 6: the general conguration of parties in each of the countries shifts slowly with time. In particular, under pro- portional representation, there is no strong impulse for parties to cohere into blocks. As a consequence, activist groups may come into existence relatively easily, and induce the creation of parties, leading to political fragmentation. Chapters 8 and 9 apply this activist electoral model to examine elec- tions under plurality rule. Chapter 8 considers presidential elections in Argentina in 1989 and 1995. In 1989, a populist leader on the left, Car- los Menem, was able to use a new dimension of policy (dened in terms of the nancial structure of the economy) to gain new middle-class ac- tivist supporters, and win the election of 1995. Chapter 9 considers recent elections in the United States, and argues that there has been a slow re- alignment of the principal dimensions of political competition. Since the presidential contest between Johnson and Goldwater in 1964, the party positions have rotated (in a clockwise direction) in a space created by economic and social axes. In recent elections, the increasing importance of the social dimension, characterized by attitudes associated with civil and personal rights, have made policy making for political candidates very confusing. Aspects of policy making, such as stem cell research and immigration, are discussed at length to give some background to the nature of current politics in the United States. It is worth summarizing the results from the formal model and the empirical analyses presented in this volume. 1. The results on the formal spatial model, presented in Chapters 2 to 5, indicate that the occurence of a core, or unbeaten alternative, is very unlikely in a direct democracy using majority rule, when the dimen- sion of the policy is at least two. However, a social choice concept known as the heart, a generalization of the core, will exist, and con- verges to the core when the core is non-empty. A legislative body, made up of democratically elected representatives, can be modeled in social choice terms. Because party strengths will be disparate, a large, centrally located party may be located at a core position. Such a party, in a situation with no majority party, may be able to form a 1.2 The Theory of Social Choice 9 tional” preference relation pi. The society is represented by a prole of preference relations, p = (pi; : : : ; pn); one for each individual. Let the set of possible alternatives beW = fx; y; : : :g. If person i prefers x to y then write (x; y) 2 pi, or more commonly xpiy. The social mechanism or preference function, , translates any prole p into a preference rela- tion (p). The point of the theory is to examine conditions on  which are sufcient to ensure that whatever “rationality properties” are held by the individual preferences, then these same properties are held by (p). Arrow's Impossibility Theorem (1951) essentially showed that if the ra- tionality property under consideration is that preference be a weak order then  must be dictatorial. To see what this means, let Ri be the weak preference for i induced from pi. That is to say xRiy if and only if it is not the case that ypix: Then pi is called a weak order if and only if Ri is transitive, i.e., if xRiy and yRiz for some x; y; z inW , then xRiz. Arrow's theorem effectively demonstrated that if  (p) is a weak order whenever every individual has a weak order preference then there must be some dictatorial individual i, say, who is characterized by the ability to enforce every social choice. It was noted some time afterwards that the result was not true if the conditions of the theorem were weakened. For example, the requirement that  (p) be a weak order means that “social indifference” must be tran- sitive. If it is only required that strict social preference be transitive, then there can indeed be a non-dictatorial social preference mechanism with this weaker rationality property (Sen, 1970). To see this, suppose  is dened by the strong Pareto rule: x(p)y if and only if there is no in- dividual who prefers y to x but there is some individual who prefers x to y. It is evident that  is non-dictatorial. Moreover if each pi is transi- tive then so is (p). However, (p) cannot be a weak order. To illustrate this, suppose that the society consists of two individuals f1; 2g who have preferences 1 2 x y z x y z This means xp1zp1y etc. Since f1; 2g disagree on the choice between x and y and also on the choice between y and z both x; y and y; z must be 10 Chapter 1. Introduction socially indifferent. But then if (p) is to be a weak order, it must be the case that x and z are indifferent. However, f1; 2g agree that x is superior to z, and by the denition of the strong Pareto rule, x must be chosen over z. This of course contradicts transitivity of social indifference. A second criticism due to Fishburn (1970) was that the theorem was not valid in the case that the society was innite. Indeed since democ- racy often involves the aggregation of preferences of many millions of voters the conclusion could be drawn that the theorem was more or less irrelevant. However, three papers by Gibbard (1969), Hanssen (1976) and Kir- man and Sondermann (1972) showed that the result on the existence of a dictator was quite robust. The rst three sections of Chapter 2 essentially parallel the proof by Kirman and Sondermann. The key notion here is that of a decisive coalition: a coalitionM is decisive for a social choice function, ; if and only if xpiy for all i belonging toM for the prole p implies x(p)y. Let D represent the set of decisive coalitions dened by : Suppose now that there is some coalition, perhaps the whole society N , which is decisive. If  preserves transitivity (i.e., (p) is transitive) then the intersection of any two decisive coalitions must itself be deci- sive. The intersection of all decisive coalitions must then be decisive: this smallest decisive coalition is called an oligarchy. The oligarchy may indeed consist of more than one individual. If it comprises the whole so- ciety then the rule is none other than the Pareto rule. However, in this case every individual has a veto. A standard objection to such a rule is that the set of chosen alternatives may be very large, so that the rule is effectively indeterminate. Suppose the further requirement is imposed that (p) al- ways be a weak order. In this case it can be shown that for any coalition, M; either M itself or its complement NnM must be decisive. Take any decisive coalition A, and consider a proper subset B say of A. If B is not decisive then NnB is, and so A\ (NnB)= AnB is decisive. In other words every decisive coalition contains a strictly smaller decisive coali- tion. Clearly, if the society is nite then some individual is the smallest decisive coalition, and consequently is a dictator. Even in the case when N is innite, there will be a smallest “invisible” dictator. It turns out, therefore, that reasonable and relatively weak rationality properties on  impose certain restrictions on the class D of decisive coalitions. These restrictions on D do not seem to be similar to the characteristics that po- 1.2 The Theory of Social Choice 11 litical systems display. As a consequence these rst attempts by Sen and Fishburn and others to avoid the Arrow Impossibility Theorem appear to have little force. A second avenue of escape is to weaken the requirement that (p) always be transitive. For example a more appropriate mechanism might be to make a choice fromW of all those unbeaten alternatives. Then an alternative x is chosen if and only if there is no other alternative y such that y(p)x. The set of unbeaten alternatives is also called the core for (p); and is dened by Core(; p) = fx 2 W : y(p)x for no y 2 Wg: In the case thatW is nite the existence of a core is essentially equivalent to the requirement that (p) be acyclic (Sen, 1970). Here a preference, p, is called acyclic if and only if whenever there is a chain of preferences x0px1px2p


pxr then it is not the case that xrpx0. However, acyclicity of  also imposes a restriction on D. Dene the collegium (D) for the family D of decisive coalitions of  to be the intersection (possibly empty) of all the decisive coalitions. If the collegium is empty then it is always possible to construct a “rational” prole p such that (p) is cyclic (Brown, 1973). Therefore, a necessary condition for  to be acyclic is that  exhibit a non-empty collegium. We say  is collegial in this case. Obviously, if the collegium is large then the rule is indeterminate, whereas if the collegium is small the rule is almost dictatorial. A third possibility is that the preferences of the members of the society are restricted in some way, so that a natural social choice function, such as majority rule, will be “well behaved.” For example, suppose that the set of alternatives is a closed subset of a single dimensional “left–right” continuum. Suppose further that each individual i has convex preference on W , with a most preferred point (or bliss point) xi, say.6 Then a well- known result by Black (1958) asserts that the core for majority rule is 6Convexity of the preference p just means that for any y the set {x : xpy} is convex. A natural preference to use is Euclidean preference dened by xpiy if and only if jjxxijj < jjyxijj, for some bliss point, xi, inW , and norm jjjj onW . Clearly Euclidean preference is convex. 14 Chapter 1. Introduction Conversely, if q   r 1 r  n then such a prole could certainly be constructed. Another way of ex- pressing this is that a q-rule  is acyclic for all acyclic proles if and only if jW j < n n q : Note that we assume that q < n. Nakamura (1979) later proved that this result could be generalized to the case of an arbitrary social preference function. The result depends on the notion of a Nakamura number v() for . Given a non-collegial family D of coalitions, a memberM of D is minimal decisive if and only ifM belongs to D, but for no member i ofM doesMnfig belong to D. If D0 is a subfamily of D consisting of minimal decisive coalitions, and moreover D0 has an empty collegium then call D0 a Nakamura subfamily of D. Now consider the collection of all Nakamura subfamilies of D. Since N is nite these subfamilies can be ranked by their cardinality. Dene v(D) to be the cardinality of the smallest Nakamura subfamily, and call v(D) the Nakamura number of D. Any Nakamura subfamily D0, with cardinality jD0j = v(D), is called aminimal non-collegial subfamily. When  is a social preference function with decisive familyD dene the Nakamura number v() of  to be equal to v(D). More formally v() = minfjD0j : D0  D and (D0) = g: In the case that  is collegial then dene v() = v(D) =1 (innity): Nakamura showed that for any voting rule, ; ifW is nite, with jW j < v(); then (p) must be acyclic whenever p is an acyclic prole. On the other hand, if  is a social preference function and jW j  v() then it is always possible to construct an acyclic prole on W such that (p) is cyclic. Thus the cardinality restriction onW which is necessary and suf- cient for  to be acyclic is that jW j < v(). To relate this to Ferejohn– Grether's result for a q-rule, dene v(n; q) to be the largest integer such that v(n; q) < q n q : 1.2 The Theory of Social Choice 15 It is an easy matter to show that when q is a q-rule then v(q) = 2 + v(n; q): The Ferejohn–Grether restriction jW j < n nq may also be written jW j < 1 + q n q which is the same as jW j < v(q): Thus Nakamura's result is a generalization of the earlier result on q-rules. The interest in this analysis is that Greenberg (1979) showed that a core would exist for a q-rule as long as preferences were convex and the choice space, W , was of restricted dimension. More precisely suppose that W is a compact,7 convex subset of Euclidean space of dimension w, and suppose each individual preference is continuous8 and convex. If q > ( w w+1 )n then the core of (p) must be non-empty, and if q  ( w w+1 )n then a convex prole can be constructed such that the core is empty. From a result by Walker (1977) the second result also implies, for the constructed prole p; that (p) is cyclic. Rewriting Greenberg's inequality it can be seen that the necessary and sufcient dimensionality condition (given convexity and compactness) for the existence of a core and the non-existence of cycles for a q-rule, q, is that dim(W )  v(n; q) where dim(W ) = w is the dimension ofW . Since v(q) = 2 + v(n; q): where v(q) is the Nakamura number of the q-rule, this suggests that for an arbitrary non-collegial voting rule  there is a stability dimension, namely v() = v() 2, such that dim(W )  v() is a necessary and sufcient condition for the existence of a core and the non-existence of 7Compactness just means the set is closed and bounded. 8The continuity of the preference, p; that is required is that for each x 2 W; the set p1{y 2W : xpy} is open in the topology onW: 16 Chapter 1. Introduction cycles. Chapters 3, 4 and 5 of this volume prove this result and present a number of further applications. An important procedure in this proof is the construction of a represen- tation  for an arbitrary social preference function. Let D = fM1; : : : ;Mvg be a minimal non-collegial subfamily for . Note that D has empty collegium and cardinality v() = v. Then  can be represented by a (v 1) dimensional simplex
in Rv1. Moreover, each of the v faces of this simplex can be identied with one of the v coalitions in D. Each proper subfamily Dt = f::;Mt1;Mt+1; ::g has a non-empty collegium, (Dt), and each of these can be identied with one of the vertices of
. To each i 2 (Dt) we can assign a preference pi; for i = f1; : : : ; vg on a set x = fx1;x2; : : : ; xvg giving a permutation prole (D1) (D2) : : : (Dv) x1 x2 xv x2 x3 x1 : : : : : : : : : xv x1 : : : xv1 : From this construction it follows that x1(p)x2


(p)xv(p)x1: Thus wheneverW has cardinality at least v, then it is possible to construct a prole p such that (p) has a permutation cycle of this kind. This representation theorem is used in Chapter 4 to prove Nakamura's result and to extend Greenberg's Theorem to the case of an arbitrary rule. The principal technique underlying Greenberg's Theorem is an impor- tant result due to Fan (1961). Suppose thatW is a compact convex subset of Rw, and suppose P is a correspondence from W into itself which is convex and continuous.9 Then there exists an “equilibrium” point x inW such that P (x) is empty. In the case under question if each individual preference, pi, is continuous, then so is the preference correspondence P associated with (p). Moreover, ifW is a subset of Euclidean space with 9Again, continuity of the preference correspondence, P; means that for each x 2 W; the set P1(x) = fy 2W : x 2 P (y)g is open in the topology onW: 1.2 The Theory of Social Choice 19 A smooth prole for the society N is a differentiable function u = (u1; : : : ; un) : W ! Rn: We assume in the following analysis thatW is compact, and let U(W )N be the space of all such proles endowed with the Whitney C1-topology (Golubitsky and Guillemin, 1973; Hirsch, 1976). Essentially two proles u1 and u2 are close in this topology if all values and the rst derivatives are close. Restricting attention to smooth utility proles whose associated pref- erences are convex gives the space Ucon(W )N . We say that the core Core(; u) for a rule  is structurally stable (inUcon(W )N ) ifCore(; u) is non-empty and there exists a neighborhood V of u in Ucon(W )N such that Core(; u0) is non-empty for all u0 in V . To illustrate, if Core(; u) is non-empty but not structurally unstable then an arbitrary small pertur- bation of u; to a different but still convex smooth preference prole, u0, is sufcient to destroy the core by rendering Core(; u0) empty. By the previous result if dim(W )  v() then Core(; u) is non- empty for every smooth, convex prole, and thus this dimension con- straint is sufcient for Core(; u) to be structurally stable. It had earlier been shown by Rubinstein (1979) that the set of contin- uous proles such that the majority rule core is non-empty is in fact a nowhere dense set in a particular topology on proles, independently of the dimension. However, the perturbation involved deformations induced by creating non-convexities in the preferred sets. Thus the construction did not deal with the question of structural stability in the topological space Ucon(W )N . Chapter 5 continues with the result by McKelvey and Schoeld (1987) and Saari (1997) which indicates that, for any q-rule, q; there is an in- stability dimension, w(q). If dim(W )  w(q) andW has no boundary then the q-core will be empty for a dense set of proles in Ucon(W )N . This immediately implies that the core cannot be structurally stable, so any sufciently small perturbation in Ucon(W )N will destroy the core. The same result holds if W has a non-empty boundary but dim(W )  w(q) + 1. Theorem 5.1.1 shows that if a point belongs to the core of a voting game, dened by a set, D, of decisive coalitions, then the direc- tion gradients must satisfy certain generalized symmetry conditions on the utility gradients of the voters at that point. This theorem is an exten- 20 Chapter 1. Introduction sion of an earlier result by Plott (1967) for majority rule. The easiest case to examine is where the core, Core(; u); is charac- terized by the property that exactly one individual has a bliss point at the core. We denote this by BCore(; u). The Thom Transversality Theo- rem can then be used to show that BCore(q; u) is generically empty (in the space Ucon(W )N ), whenever the dimension exceeds 2q n+1. This suggests that the instability dimension satises w(q) = 2q n+ 1. Saari (1997) extended this result in two directions, by showing that if dim(W )  2q n then BCore(q; u) could be structurally stable. Moreover, he was able to compute the instability dimension for the case of a non-bliss core, when no individual has a bliss point at the core. For example, with majority rule the instability dimension is two or three depending on whether n is odd or even. For n odd, neither bliss nor non-bliss cores can be structurally stable in two or more dimensions, since the Plott (1967) symmetry conditions cannot be generically satis- ed. On the other hand, when (n; q) = (4; 3); the Nakamura number is four, and hence a core will exist in two dimensions. Indeed, both bliss cores and non-bliss cores can occur in a structurally stable fashion. How- ever, in three dimensions the cycle set is contained in, but lls the Pareto set. For all majority rules with n  6, and even, a structurally stable bliss-core can occur in two dimensions. However, when n even, in three dimensions the core cannot be structurally stable and the cycle set need not be constrained to the Pareto set (in contradiction to Tullock's hypoth- esis). For general weighted voting games, dened by a non-collegial family, D, the core symmetry condition can be satised in a structurally stable fashion. This provides a technique for examining when a core exists in the legislatures discussed in Chapter 6. Since the core may be empty, the notion of the heart is presented as an alternative solution idea. The heart can be interpreted in terms of a local uncovering relation, and can be shown to be non-empty under fairly weak conditions. This idea is illustrated by considering various voting rules in low dimensions. The last section of Chapter 5 presents the experimental results obtained by Fiorina and Plott (1978), McKelvey, Ordeshook and Winer (1978), Laing and Olmstead (1978) and Eavey (1996) to indicate the nature of the heart in two dimensions. Chapter 2 Social Choice 2.1 Preference Relations Social choice is concerned with a fundamental question in political or economic theory: is there some process or rule for decision making which can give consistent social choices from individual preferences? In this framework denoted by W is a universal set of alternatives. Members of W will be written x; y etc. The society is denoted by N , and the individuals in the society are called 1; : : : ; i; : : : ; j; : : : ; n. The values of an individual i are represented by a preference relation pi on the setW . Thus xpiy is taken to mean that individual i prefers alternative x to alternative y. It is also assumed that each pi is strict, in the way to be described below. The rest of this section considers the abstract properties of a preference relation p onW: Denition 2.1.1. A strict preference relation p onW is (i) Irreexive: for no x 2 W does xpx; (ii) Asymmetric: for any x; y;2 W ;xpy ) not(ypx). The strict preference relations are regarded as fundamental primitives in the discussion. No attempt is made to determine how individuals ar- rive at their preferences, nor is the problem considered how preferences might change with time. A preference relation p may be represented by a utility function. Denition 2.1.2. A preference relation p is representable by a utility function 21 24 Chapter 2. Social Choice order, and the class of these is written T (W ). Finally the class of acyclic strict preference relations onW is written A(W ). If p 2 O(W ) then it follows from the denition that R(p) is transitive. Indeed I(p) will also be transitive. Lemma 2.1.1. If p 2 O(W ) then I(p) is transitive. Proof. Suppose xI(p)y; xI(p)z but not(xI(p)z). Because of not(xI(p)z) suppose xpz. By asymmetry of p, not(zpx) so xR(p)z. By symmetry of I(p); yI(p)x and zI(p)y; and thus yR(p)x and zR(p)y: But xR(p)z and zR(p)y and yR(p)x contradicts the transitivity ofR(p). Hence not(xpz). In the same way not(zpx); and so xI(p)z; with the result that I(p) must be transitive. Lemma 2.1.2. If p 2 O(W ) then xR(p)y and ypz ) xpz: Proof. Suppose xR(p)y, ypz and not(xpz). But not(xpz), zR(p)x: By transitivity of R(p), zR(p)y. By denition not(ypz) which contradicts ypz by asymmetry. Lemma 2.1.3. O(W )  T (W )  A(W ). Proof. (i) Suppose p 2 O(W ) but xpy, ypz yet not(xpz), for some x, y, z. By p asymmetry, not(ypx) and not(zpy): Since p 2 O(W ), not(xpz) and not(zpy) ) not(xpy). But not(ypx). So xI(p)y. But this violates xpy. By contradiction, p 2 T (W ). (ii) Suppose xjpxj+1 for j = 1; : : : ; r 1. If p 2 T (W ), then x1pxr. By asymmetry, not(xrpx1), so p is acyclic. 2.2 Social Preference Functions Let the society be N = f1; : : : ; i; : : : ; ng: A prole for N on W is an 2.2 Social Preference Functions 25 assignment to each individual i in N of a strict preference relation pi on W . Such an n-tuple (p1; : : : ; pn) will be written p. A subset M  N is called a coalition. The restriction of p toM will be written p=M = (::pi:: : i 2M): If p is a prole for N on W , write xpNy iff xpiy for all xpiy for all i 2 N . In the same way for M a coalition in N write xpMy whenever xpiy for all i 2M . Write B(W )N for the class of proles on N . When there is no possi- bility of misunderstanding we shall simply write BN for B(W )N . On occasion the analysis concerns proles each of whose component individual preferences are assumed to belong to some subset F (W ) of B(W ); for example F (W ) might be taken to be O(W ), T (W ) or A(W ). In this case write F (W )N , or FN , for the class of such proles. Let X be the class of all subsets of W . A member V 2 X will be called a feasible set. Suppose that p 2 B(W )N is a prole forN onW . For some x; y 2 W write “pi(x; y)” for the preference expressed by i on the alternatives x; y under the prole p. Thus “pi(x; y)” will give either xpiy or xI(pi)y or ypix. If f; g 2 B (W )N are two proles onW , and V 2 X , use f=M = g=M on V to mean that for any x; y 2 V , any i 2M; fi(x; y) = gi(x; y). In more abbreviated form write fV=M = gV=M . Implicitly this implies con- sideration of a restriction operator V M : B(W )N ! B(V )M : f ! fV=M ; where B(V )M means naturally enough the set of proles forM on V . A social preference function is a method of aggregating preference information, and only preference information, on a feasible set in order to construct a social preference relation. Denition 2.2.1. A method of preference aggregation (MPA), , assigns to any feasible set V , and prole p for N onW a strict social preference relation (V; p) 2 B(V ). Such a method is written as  : X BN ! B. 26 Chapter 2. Social Choice As before write (V; p)(x; y) for x; y 2 V to mean “the social preference relation declared by (V; p) between x and y.” If f; g 2 BN , write (V; f) = (V; g) whenever (V; f)(x; y) = (V; g)(x; y) for any x; y 2 V . Denition 2.2.2. A method of preference aggregation  : X  BN ! B is said to satisfy the weak axiom of independence of infeasible alterna- tives (II) iff f V = g V ) (V; f) = (V; g): Such a method is called a social preference function (SF). Note that an SF, , is functionally dependent on the feasible set V . Thus there need be no specic relationship between (V1; f) and (V2; f) for V2  V1 say. However, suppose (V1; f) is the preference relation induced by  from f on V1. Let V2  V1, and let (V1; f)=V2 be the preference relation induced by (V1; f) on V2 from the denition [(V1; f)=V2](x; y)] = [(V1; f)(x; y)] whenever x; y 2 V2. A binary preference function is one which is consistent with this re- striction operator. Denition 2.2.3. A social preference function  is said to satisfy the strong axiom of independence of infeasible alternatives (II) iff for f 2 B(V1) N , g 2 B(V2)N , and f V = g V for V = V1 \ V2 non-empty, then (V1; f)=V = (V2; g)=V: For (V1; f) to be meaningful when  is an SF, we only require that f be a prole dened on V1. This indicates that II is an extension property. For suppose f; g are dened on V1; V2 respectively, and agree on V . Then it is possible to nd a prole p dened on V1[V2, which agrees with f on V1 and with g on V2. Furthermore if  is an SF which satises II, then (V1 [ V2; p)=V1 = (V1; f) (V1 [ V2; p)=V2 = (V2; g) (V1 [ V2; p)=V1 \ V2 = (V1; f)=V = V2; g)=V: 2.3 Arrowian Impossibility Theorems 29 ui(x) = (v ri) where jV j = v and ri is the rank that x has in i's prefer- ence schedule. Although this gives a well-dened SF, (V; p), it nonethe- less results in a certain inconsistency, since (V1; p1) and (V2; p2) may not agree on the intersection V1 \ V2, even though p1 and p2 do. Although a BF avoids this difculty, other inconsistencies are intro- duced by the strong independence axiom. 2.3 Arrowian Impossibility Theorems This section considers the question of the existence of a binary social preference function,  : FN ! F , where F is some subset of B. In this notation  : FN ! F means the following: Let V be any feasible set inW , and F (V )N the set of proles, dened on V , each of whose component preferences belong to F . The domain of  is the union of F (V )N across all V inW . That is for each f 2 F (V )N ; we write (f) for the binary social preference on V; and require that (f) 2 F (V ). Denition 2.3.1. A BF  : BN ! B satises (i) The weak Pareto property (P) iff for any p 2 BN , any x; y 2 W , xpNy ) x(p)y: (ii) Non-dictatorship (ND) iff there is no i 2 N such that for all x; y in W , xpiy ) x(p)y: A BF  which satises (P) and (ND) and maps ON ! O is called a binary welfare function (BWF). Arrow's Impossibility Theorem 2.3.1. For N nite, there is no BWF. This theorem was originally obtained by Arrow (1951). To prove it we introduce the notion of a decisive coalition. Denition 2.3.2. LetM be a coalition, and  a BF. (i) DeneM to be decisive under  for x against y iff for all p 2 BN xpMy ) x(p)y: 30 Chapter 2. Social Choice (ii) Dene M to be decisive under  iff for all x; y 2 W;M is decisive for x against y. (iii) LetD(x; y) be the family of decisive coalitions under  for x against y, and D be the family of decisive coalitions under . To prove the theorem we rst introduce the idea of an ultralter. Denition 2.3.3. A family of coalitions can satisfy the following proper- ties. (F1) monotonicity: A  B and A 2 D) B 2 D; (F2) identity: N 2 D and  2 D (where  is the empty set); (F3) closed intersection: A;B 2 D) A \B 2 D; (F4) negation: for any A  N , either A 2 D or NnA 2 D. A familyD of subsets ofN which satises (F1), (F2) and (F3) is called a lter. A lter D1 is said to be ner than a lter D2 if each member of D2 belongs to D1. D1 is strictly ner than D2 iff D1 is ner than D2 and there exists A 2 D1 with A 2 D2. A lter which has no strictly ner lter is called an ultralter. A lter is called free or xed depending on whether the intersection of all its members is empty or non-empty. In the case that N is nite then by (F2) and (F3) any lter, and thus any ultralter, is xed. Lemma 2.3.2. (Kirman and Sondermann, 1972). If  : ON ! O is a BF and satises the weak Pareto property (P), then the family of decisive coalitions, D; satises (F1), F(2), F(3) and F(4). We shall prove this lemma below. Arrow's theorem follows from Lemma 2.3.2 since D will be an ultralter which denes a unique dicta- tor. This can be shown by the following three lemmas. Lemma 2.3.3. Let D be a family of subsets of N , which satises (F1), (F2), (F3) and (F4). Then if A 2 D, there is some proper subset B of A which belongs to D. Proof. Let B be a proper subset of A with B =2 D. By (F4), NnB 2 D. But then by (F3), A \ (NnB) = AnB 2 D. 2.3 Arrowian Impossibility Theorems 31 Hence if B  A, either B 2 D or AnB 2 D. Lemma 2.3.4. If D satises (F1), (F2), (F3) and (F4) then it is an ultra- lter. Proof. Suppose D1 is a lter which is strictly ner than D. Then there is some A;B 2 D1, with A 2 D but B =2 D. By the previous lemma, either AnB or A \ B must belong to D. Suppose AnB 2 D. Then AnB 2 D1. But since D1 is a lter (AnB) \ B =  must belong to D, which contradicts (F2). Hence A\B belongs to D. But by (F1), B 2 D. Hence D is an ultralter. Lemma 2.3.5. If N is nite and D is an ultralter with D = fAjg then \Aj = fig; where fig is decisive and consists of a single member of N . Proof. Consider any Aj 2 D, and let i 2 Aj . By (F4) either fig 2 D or Ajfig 2 D. If fig 2 D; then Ajfig 2 D. Repeat the process a nite number of times to obtain a singleton fig, say, belonging to D. Proof of Theorem 2.3.1. For N nite, by the previous four lemmas, the family of -decisive coalitions forms an ultralter. The intersection of all decisive coalitions is a single individual i, say. Since this intersection is nite, fig 2 D. Thus i is a dictator. Consequently any BF  : ON ! O which satises (P) must be dictatorial. Hence there is no BWF. Note that when N is innite there can exist a BWF  (Fishburn, 1970). However, its family of decisive coalitions still forms an ultralter. See Schmitz (1977) and Armstrong (1980) for further discussion on the exis- tence of a BWF when N is an innite society. The rest of this section will prove Lemma 2.3.2. The following den- itions are required. Denition 2.3.4. Let  be a BF,M a coalition, p a prole, x; y 2 W . (i) M is almost decisive for x against y with respect to p iff xpMy; ypNMx and x(p)y. 34 Chapter 2. Social Choice fashion: zp1xp1y xp2yp2z yp3zp2x yp4xp4z: Since A = V1 [ V2 2 D; we obtain x(p)y; B = V1 [ V3 2 D; we obtain z(p)x: By transitivity, z(p)y. Now zpV1y, ypNV1z and z(p)y. By Lemma 2.3.6, V1 2 D0(z; y) = D. Thus A \B 2 D. This lemma demonstrates that if  : TN ! T is a BF which satises (P) then D is a lter. However, if p 2 ON then p 2 TN , and if (p) 2 O then (p) 2 T , by Lemma 2.1.3. Hence to complete the proof of Lemma 2.3.2 only the following lemma needs to be shown. Lemma 2.3.9. If  : ON ! O is a BF and satises (P) then D satises (F4). Proof. SupposeM 2 D. We seek to show thatNnM 2 D. If for any f; there exist x; y 2 W such that yfMx and y(f)x, thenM would belong to D(x; y) and so be decisive. Thus for any f; there exist x; y 2 W; with yfMx and not (y(f)x) i.e., xR((f))y: Now consider g 2 ON ; with g = f on fx; yg and xgNMz; ygNMz and ygMz: By II; xR((g))y: By (P), y(g)z: Since (g) 2 O it is negatively transitive, and by Lemma 2.1.2, x(g)z: Thus NnM 2 D(x; z) and so NnM 2 D: 2.4 Power and Rationality Arrow's theorem showed that there is no binary social preference func- tion which maps weak orders to weak orders and satises the Pareto and non-dictatorship requirements when N is nite. Although there may ex- ist a BWF when N is innite, nonetheless “power” is concentrated in the sense that there is an “invisible dictator.” It can be argued that the require- 2.4 Power and Rationality 35 ment of negative transitivity is too strong, since this property requires that indifference be transitive. Individual indifference may well display intransitivities, because of just perceptible differences, and so may social indifference. To illustrate the problem with transitivity of indifference, consider the binary social preference function, called the weak Pareto rule written n and dened by: xn(p)y iff xpNy: In this case fNg = Dn . This rule is a BF, satises (P) by denition, and is non-dictatorial. However, suppose the preferences are zpMxpMy ypNMzpNMx for some proper subgroupM in N . Since there is not unanimous agree- ment, this implies xI(n)yI(n)z. If negative transitivity is required, then it must be the case that xI(n)z: Yet zpNx; so z(p)x: Such an ex- ample suggests that the Impossibility Theorem is due to the excessive rationality requirement. For this reason Sen (1970) suggested weakening the rationality requirement. Denition 2.4.1. A BF  : ON ! T which satises (P) and (ND) is called a binary decision function (BDF). Lemma 2.4.1. There exists a BDF. To show this, say a BF  satises the strong Pareto property (P) iff, for any p 2 BN ; ypix for no i 2 N; and xpjy for some j 2 N ) x(p)y: Note that the strong Pareto property (P) implies the weak Pareto property (P). Now dene a BF n, called the strong Pareto rule, by: xn(p)y iff ypix for no i 2 N and xpjy for some j 2 N: n may be called the extension of n; since it is clear that xn (p) y ) xn(p)y: Obviously n satises (P) and thus (P). However, just as n violates transitive indifference, so does n: On the other hand n satises transi- tive strict preference. 36 Chapter 2. Social Choice Lemma 2.4.2. (Sen, 1970). n is a BDF. Proof. Suppose xn(p)y and yn(p)z: Now xn(p)y , xR(pi)y for all i 2 N and xpjy for some j 2 N: Similarly for fy; zg. By transitivity of R(pi); we obtain xR(pi)z for all i 2 N: By Lemma 2.1.2, xpjz for some j 2 N: Hence xn(p)z. While this seems to refute the relevance of the impossibility theorem, note that the only decisive coalition for n is fNg. Indeed the strong Pareto rule is somewhat indeterminate, since any individual can effec- tively veto a decision. Any attempt to make the rule more “determinate” runs into the following problem. Denition 2.4.2. An oligarchy  for a BF  is a minimally decisive coalition which belongs to every decisive coalition. Lemma 2.4.3. (Gibbard, 1969). If N is nite, then any BF  : ON ! T which satises P has an oligarchy. Proof. Restrict  to  : TN ! T . By Lemma 2.3.8, since  satises (P), its decisive coalitions form a lter. Let  = \Aj , where the intersection runs over all Aj 2 D: Since N is nite, this intersection is nite, and so  2 D: Obviously  fig 2 D for any i 2 : Consequently  is a minimally decisive coalition or oligarchy. The following lemma shows that members of an oligarchy can block social decisions that they oppose. Lemma 2.4.4. (Schwartz, 1986). If  : ON ! T and p 2 ON ; and  is the oligarchy for ; then dene x(p) = fi 2  : xpiyg y(p) = fj 2  : ypjxg: Then (i) x(p) 6=  and  = x(p) [ y(p)) not (y(p)x) (ii) x(p) 6= ; y(p) 6=  and  = x [ y ) xI((p))y: 2.5 Choice Functions 39 Denition 2.5.2. A choice function C : X BN ! X satises the weak axiom of revealed preference (WARP) iff wherever V  V 0, and p is dened on V 0, with V \ C(V 0; p) 6= , then V \ C(V 0; p) = C(V; pV ); where pV is the restriction of p to V . Note the analogue with (II). If we write C(V 0; p)=V for V \ C(V 0; p) when this is non-empty, then WARP requires that C(V; pV ) = C(V 0; p)=V: Denition 2.5.3. (i) A choice function C : X  BN ! X is said to be rationalized by an SF  : X BN ! B iff for any V 2 X and any p 2 BN ; C(V; p) = fx : y(V; p)x for no y 2 V g: (ii) A choice function C : X  BN ! X is said to be rationalized by a BF  : BN ! B iff for any p 2 BN ; and any x; y 2 W;x 6= y, C(fx; yg; p) = fxg , x(p)y: (iii) A choice function C : X  BN ! X is said to satisfy the binary choice axiom (BICH) iff there is a BF  : BN ! B such that for any V 2 X; any p 2 BN ; C(V; p) = fx 2 V : y(p)x for no y 2 V g: Say C satises BICH w.r.t.  in this case. (iv) Given a choice function C : X  BN ! X dene the induced BF C : B N ! B by C(fx; yg; p) = fxg , xC(p)y: (v) Given a BF  : BN ! B dene the choice procedure C : X BN ! X by C(V; p) = fx 2 V : y(p)x for no y 2 V g: 40 Chapter 2. Social Choice Note that C(V; p) may be empty for some V; p: Lemma 2.5.1. If C satises BICH w.r.t.  then  rationalizes C. Proof. (i) C(fx; yg; p) = fxg )not(y(p)x: If yI((p))x then not (x(p)y); so y 2 C(fx; yg; p): Hence C(fx; yg; p) = fxg ) x(p)y: (ii) x(p)y ) not (y(p)x): Hence C(fx; yg; p) = fxg: Another way of putting this lemma is that if C = C is a choice function then  = C : In the following we delete reference to p when there is no ambiguity, and simply regard C as a mapping from X to itself. Example 2.5.1. (i) Suppose C is dened on the pair sets ofW = fx; y; zg by C(fx; yg) = fxg; C(fy; zg) = fyg and C(fx; zg) = fzg: If C satises BICH w.r.t. ; then it is necessary that xyzx; so C(fx; y; zg) = : Hence C cannot satisfy BICH. (ii) Suppose C(fx; yg) = fxg C(fy; zg) = fy; zg: If C(fx; zg) = fx; zg; then xyI()z and xI()z; so C(fx; y; zg) = fx; zg: While C satises BICH w.r.t. ;  does not give a weak order, although  may give a strict partial order. The Spatial Model of Politics Norman Schoeld November 21, 2007 v 9.2.1 Realignment and Federalism . . . . . . . . . . . . . . . . . . . . . . . .229 9.3 Coalitions of Enemies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232 9.3.1 The New Deal Coalition . . . . . . . . . . . . . . . . . . . . . . . . . . . .232 9.3.2 The Creation of the Republican Coalition . . . . . . . . . . . .233 9.3.3 Social Conservatives Ascendant in the G.O. P. . . . . . . . .236 9.3.4 Stem Cell Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .238 9.3.5 Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .240 9.4 The Changing Political Equilibrium . . . . . . . . . . . . . . . . . . . . . . .241 9.4.1 Party Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244 9.4.2 Party Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245 9.4.3 Party Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247 9.5 The Future of Republican Populism . . . . . . . . . . . . . . . . . . . . . . .250 9.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252 9.7 Appendix: Republican Senator Votes . . . . . . . . . . . . . . . . . . . . . .257 10 Final Remarks 259 10.1 The Madisonian Scheme of Government . . . . . . . . . . . . . . . . . .259 10.2 Preferences and Judgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .265 vi Contents Tables 6.1 Duration (in months) of government, 1945-1987 120 6.2 Frequency of coalition types, by country, 1945-1987 121 6.3 Duration of European coalitions, 1945-1987 122 6.4 Knesset seats 128 6.5 Seats and votes in the Netherlands 137 6.6 Estimated vote shares and valences in the Netherlands 137 6.7 Seats in the Dutch Parliament, 2003 and 2006 140 6.8 Elections in Denmark, 1957 and 1964 144 6.9 Party and faction strengths in the Dáil Eireann, 1987 150 6.10 Recent elections in Europe 160 7.1 Vote shares and seats in the Knesset 180 7.2 Turkish election results 1999 185 7.3 Turkish election results 2002 185 7.4 Turkish election results 2007 190 7.5 Multinomial Logit Analysis of the 1999 Election in Turkey 199 7.6 Multinomial Logit Analysis of the 2002 Election in Turkey 200 7.7 Log Bayes factors for model comparisons in 1999 201 7.8 Log Bayes factors for model comparisons in 2002 201 9.1 Votes of Republican senators on immigration and stem cell research 257 Figures 3.1 A voting complex 57 4.1 Convex and non-convex preference 85 4.2 Non-convex social preference 87 4.3 Non-convexity of the critical preference cones 89 4.4 Condition for local cyclicity at a point 92 5.1 Euclidean preferences with the q- rule given by (n; q) = (4; 3) 108 5.2 Euclidean preferences with the q- rule given by (n; q) = (6; 4) 109 5.3 Euclidean preferences with the q- rule given by (n; q) = (5; 3) 110 5.4 The heart, the yolk and the uncovered set 113 5.5 The heart with a uniform electorate on the triangle 114 5.6 The heart with a uniform electorate on the pentagon 114 5.7 Experimental results of Fiorina and Plott (1978) 116 5.8 Experimental results of McKelvey and Ordeshook (1978) 116 vii 5.9 Experimental results of Laing and Olmstead (1978) 117 5.10 Experimental results of Laing and Olmstead (1978) 117 5.11 Experimental results of Eavey (1991) 118 5.12 Experimental results of Eavey (1991) 118 6.1 The core in the Knesset in 1992 129 6.2 The heart in the Knesset in 1988 129 6.3 Party positions in the Knesset in 1996 131 6.4 The conguration of the Knesset after the election of 2003 133 6.5 The conguration of the Knesset after the election of 2006 133 6.6 Party positions in the Netherlands in 1977 136 6.7 The Dutch Parliament in 2006 139 6.8 Finland in 2003 142 6.9 Denmark in 2001 145 6.10 Sweden in 2002 145 6.11 Norway in 2001 146 6.12 The heart in Belgium in 1999 148 6.13 The heart in Belgium in 2003 149 6.14 Ireland in 1987 151 6.15 Ireland in 2007 152 6.16 Iceland in 2003 153 6.17 Austria in 2006 154 6.18 Germany in 2002 155 6.19 The core in Italy in 1987 156 6.20 Italy in 2001 157 7.1 The Gumbel distribution 170 7.2 A local Nash equilibrium in the Knesset in 1996 182 7.3 Party positions and voter distribution in Turkey in 1999 186 7.4 The heart in Turkey in 1999 186 7.5 Party positions and voter distribution in Turkey in 2002 187 7.6 Party positions in the Netherlands 192 7.7 Party positions in the United Kingdom 194 7.8 Balance loci for parties in Britain 194 8.1 Activists in Argentina 213 8.2 The voter distribution in Argentina in 1989 219 9.1 Activists in the United States 225 10.1 Estimated positions of posssible candidates for the U.S. presidency 271 x Foreword their fruits harvested by separate communities of scholars with only the occasional cross-over (as in William Riker and Peter Ordeshook, An In- troduction to Positive Political Theory (Prentice-Hall, 1972)). One of the leading innovators, drawing from (and making important contribu- tions to) all of these traditions, is Norman Schoeld. The present volume is something of a grand synthesis. Its substantive focus is elections with its electoral deals, party activists, and voters on the one hand, and parlia- ments with their coalition-building and government-forming maneuver- ings on the other. In terms of tools, the arguments found in this volume draw heavily on social choice theory, the spatial model, and cooperative game theory. Indeed, the rst ve chapters constitute just about all one needs to know of social choice theory. But the core (pun intended) of this ne volume is found in four very rich applied chapters, constituting a profound synthesis of elections and parliaments – of voters and activists choosing political agents, agents in turn choosing governments, and governments governing. Along the way we learn about voting rules, electoral systems, the ecology of government coalitions, precipitating events, and quite a lot about the empirical con- dition of electorates, parliaments, and legislatures in the advanced indus- trial democracies of the West. (The intensive treatment of an incredibly complex coalitional situation found in Israel over the last two decades is highly instructive.) I want to single out two especially imaginative treatments found in the later chapters of this volume – imaginative both theoretically and em- pirically. First, Schoeld provides one of the most elaborated theoreti- cally grounded typologies of coalitional arrangements in parliamentary democracies (in a literature rich in typologies based mainly, even exclu- sively, on empirical patterns alone) which, in turn, provides insights into the frequent absence of centripetal forces in multiparty regimes. Sec- ond, Schoeld elaborates an analysis of electoral activists that goes far- ther, and is founded on a more rm theoretical basis, than anything that presently exists in the literature. Taking a highly original turn, Schoeld applies the logic of Duverger to interest groups, suggesting how the elec- toral rule (plurality vs. proportional representation) will affect activist coalition building. Application to the building of electoral coalitions in Argentina and the United States is quite provocative. Each of these will be of separate interest to research communities. To- xi gether, however, they provide the underpinnings for a net assessment of the effects of the centripetal pull of the voting electorate and the centrifu- gal impact of activists. The reader may have to burn a lot of intellectual energy to get to these points, but getting there not only is half the fun but also makes strikingly evident that Schoeld's large theoretical buildup is not merely an occasion for play in the theory sandbox. Norman Schoeld has the soul of a mathematician and the heart of a political scientist. He has, over a long career but especially in the present volume, combined these two impulses elegantly. In one sense this book is a nished product. In another it is but the beginning of a conversation. Kenneth A. Shepsle Harvard University September 2007 xv from Madison's dual theory of the Republic. On the one hand, Madison saw the President as a a natural way to prevent mutability or disorder in the legislature, while the ability of Congress to veto presidential risk- taking could prevent autocracy. At the same time, the election of the president would, in the extended republic, enhance the “probability of a t choice.” Madison's argument is interpreted in the light of the model of elections presented in the previous chapters. A number of chapters of this book use some gures and tables from previous work. Cambridge University Press kindly gave permission to use material from Schoeld (2006a), Schoeld and Sened (2006) and Miller and Schoeld (2008). I am grateful to Blackwell for permission to use material from Schoeld (1993) and Schoeld and Miller (2007), to Sage for permission to use material from Schoeld (1995), to Springer to use material from Schoeld (1996) and Schoeld (2006b) and to North Holland to use material from Schoeld and Cataife (2007). I received very helpful comments on the versions of the last four chapters of the book, presented at various conferences and seminars: the World Public Choice Meeting, Amsterdam, the ISNIE conference, Boulder, the conference on European Governance, Emory University, conferences on political economy in Cancún and Guanajuato, Mexico, the Meeting of the Society for the Advancement of Economic Theory, Vigo, Spain, the Conference on Modernization of the State and the Econ- omy, Moscow, and other conferences at the University of Virginia, Char- lottesville, at SUNY, Binghamton, and at the University of Hamburg. Versions of some of this work were presented at the Department of Eco- nomics, Concordia University, Montreal, at the Center for Mathematical and Statistical Modeling, Wilfred Laurier University, Waterloo, at the Center for Mathematical Modeling, University of California at Irvine, at the Higher School of Economics, St. Petersburg, at George Mason Uni- versity, and at the California Institute of Technology. The original versions of most of the chapters were typed by Cherie Moore, and many of the diagrams were drawn by Ugur Ozdemir. I am grateful to Cheryl Eavey, Joseph Godfrey, Eric Linhart, Evan Schnidman and Suumu Shikano for permission to make use of their work. I thank my coauthors, Guido Cataife, Gary Miller and Ugur Ozdemir for their collaboration. Ken Benoit and Michael Laver graciously gave permission for me to use their estimates of party position for many of the European xvi Preface polities (Benoit and Laver, 2006). I appreciate the support of the NSF (under grants SES 0241732 and 0715929), and of Washington University. The Weidenbaum Center at Washington University provided support during a visit at the International Center for Economic Research in Turin. I thank Enrico Colombatto and Alessandra Calosso for the hospitality I enjoyed at ICER. A year spent at Humboldt University, Berlin, under the auspices of the Fulbright Founda- tion, as distinguished professor of American Studies during 2002–2003, gave me the opportunity to formulate an earlier version of the formal electoral model. Finally, my thanks to Terry Clague and Robert Lang- ham, the editors at Routledge, for their willingness to wait for a number of years while the manuscript was in preparation. Norman Schoeld Washington University Saint Louis, Missouri 4 November 2007 Chapter 1 Introduction 1.1 Representative Democracy A fundamental question that may be asked about a political, economic or social system is whether it is responsive to the wishes or opinions of the members of the society and, if so, whether it can aggregate the conicting notions of these individuals in a way which is somehow ra- tional. More particularly, is it the case, for the kind of conguration of preferences that one might expect, that the underlying decision process gives rise to a set of outcomes which is natural and stable, and more im- portantly, “small” with respect to the set of all possible outcomes? If so, then it may be possible to develop a theoretical or “causal” account of the relationship between the nature of the decision process, along with the pattern of preferences, and the behavior of the social and po- litical system. For example, microeconomic theory is concerned with the analysis of a method of preference aggregation through the market. Under certain conditions this results in a particular distribution of prices for commodities and labor, and thus income. The motivation for this en- deavor is to match the ability of some disciplines in natural science to develop causal models, tying initial conditions of the physical system to a small set of predicted outcomes. The theory of democracy is to a large extent based on the assumption that the initial conditions of the politi- cal system are causally related to the essential properties of the system. That is to say it is assumed that the interaction of cross-cutting interest groups in a democracy leads to an “equilibrium” outcome that is nat- ural in the sense of balancing the divergent interests of the members of 1 4 Chapter 1. Introduction quently the “core” party can, if it so chooses, form a minority govern- ment, one without a majority of the seats in the legislature. This property of the core provides an explanation for what has appeared to be a puz- zle. The data set collected by Laver and Schoeld (1990) dealing with coalition governments in 12 European countries in the period 1945–1987 shows that about one-third of the governments were minority. About one- third were minimal winning, with just enough seats for a majority, and the remaining third were surplus, with parties included in the coalition unnecessary for the majority. In the absence of a core, the spatial the- ory suggests that bargaining between the parties will focus on a domain in the policy space known as the “heart.” In the simplest case where it is assumed that parties have “Euclidean” preferences determined by policy distance, the “heart” will be a domain bounded by the compromise sets of various minimal winning coalitions. These minimal winning coalitions are natural candidates for coalition government. Indeed, in some cases a bounding minimal winning coalition may costlessly include a surplus party. This notion of the heart suggests that in the absence of a core, one or other of these minimal winning or surplus coalitions will form. Chapter 6 illustrates the difference between a core and the heart by considering recent elections in Israel in the period 1988 to 2006 and in the Netherlands in 1977, 1981 and 2006. In Israel, the core party was Labor, under Rabin in 1992, and a new party, “Kadima,” founded by Ariel Sharon in 2005, but under the leadership of Ehud Olmert. After the elections of 1988 and 2003 the bargaining domain of the heart was bounded by various coalitions, involving the larger parties, Labor and Likud, and smaller parties like Shas. These examples raise another theoretical problem: if party leaders are aware that by adopting a centrist position they can create minority, dom- inant government, then why are parties located so far from the electoral center? Chapter 6 illustrates the great variety of political congurations in Europe: bipolar political systems, such as the Netherlands and Fin- land; left unipolar systems such as Denmark, Sweden and Norway; center unipolar systems such as Belgium, Luxembourg and Ireland; right unipo- lar such as Iceland. Italy is unique in that it had a dominant center party, the Christian Democrats until 1994, after which the political system was totally transformed by the elimination of the core. Models of elections also suggest that the electoral center will be an at- 1.1 Representative Democracy 5 tractor for political parties, since parties will calculate that they will gain most votes at the center.4 Chapter 7 presents an electoral model where this centripetal tendency will only occur under specic conditions. The model is based on the idea of valence, derived from voters' judgements about characteristics of the candidates, or party leaders. These valences or judgements are rst assumed to be independent of the policy choice of the party. The theory shows that parties will converge to the electoral cen- ter only if the valence differences between the parties are small, relative to the other parameters of the model. The empirical analysis considers elections in Israel in 1996, in Turkey in 1999 and 2002, in the Netherlands in 1977–1981 and in Britain in 1997. The results show that the estimated parameters of the model did not satisfy the necessary condition for convergence in Israel. The theory thus gives an explanation for the dispersion of political parties in Israel and Turkey along a principal electoral axis. However, the condition sufcient for convergence of the parties was satised in the British election of 1997, and in the Dutch elections of 1977–1981. Because there was no evidence of convergence in these elec- tions, the conict between theory and evidence suggests that the stochas- tic electoral model be modied to provide a better explanation of party policy choice. The chapter goes on to consider a more general valence model based on activist support for the parties 5. This activist valence model presupposes that party activists donate time and other resources to their party. Such resources allow a party to present itself more effectively to the electorate, thus increasing its valence. The main theorem of this chapter indicates how parties might balance the centrifugal tendency as- sociated with activist support, and the centripetal tendency generated by the attraction of the electoral center. One aspect of this theory is that it implies that party leaders will act as though they have policy preferences, since they must accommodate the demands of political activists to maintain support for future elections. A further feature is that party positions will be sensitive to the nature 4An extensive literature has developed in an attempt to explain why parties do not converge to the electoral center. See, for example, Adams (1999a,b, 2001); Adams and Merrill (1999a, 2005); Adams, Merrill and Grofman (2005); Merrill and Grofman (1999); Merrill and Adams (2001); Macdonald and Rabinowitz (1998). 5See Aldrich (1983a,b, 1995); Aldrich and McGinnis (1989). 6 Chapter 1. Introduction of electoral judgements and to the willingness of activists to support the party. As these shift with time, then so will the positions of the parties. The theory thus gives an explanation of one of the features that comes from the discussion in Chapter 6: the general conguration of parties in each of the countries shifts slowly with time. In particular, under pro- portional representation, there is no strong impulse for parties to cohere into blocks. As a consequence, activist groups may come into existence relatively easily, and induce the creation of parties, leading to political fragmentation. Chapters 8 and 9 apply this activist electoral model to examine elec- tions under plurality rule. Chapter 8 considers presidential elections in Argentina in 1989 and 1995. In 1989, a populist leader on the left, Car- los Menem, was able to use a new dimension of policy (dened in terms of the nancial structure of the economy) to gain new middle-class ac- tivist supporters, and win the election of 1995. Chapter 9 considers recent elections in the United States, and argues that there has been a slow re- alignment of the principal dimensions of political competition. Since the presidential contest between Johnson and Goldwater in 1964, the party positions have rotated (in a clockwise direction) in a space created by economic and social axes. In recent elections, the increasing importance of the social dimension, characterized by attitudes associated with civil and personal rights, have made policy making for political candidates very confusing. Aspects of policy making, such as stem cell research and immigration, are discussed at length to give some background to the nature of current politics in the United States. It is worth summarizing the results from the formal model and the empirical analyses presented in this volume. 1. The results on the formal spatial model, presented in Chapters 2 to 5, indicate that the occurence of a core, or unbeaten alternative, is very unlikely in a direct democracy using majority rule, when the dimen- sion of the policy is at least two. However, a social choice concept known as the heart, a generalization of the core, will exist, and con- verges to the core when the core is non-empty. A legislative body, made up of democratically elected representatives, can be modeled in social choice terms. Because party strengths will be disparate, a large, centrally located party may be located at a core position. Such a party, in a situation with no majority party, may be able to form a 1.2 The Theory of Social Choice 9 tional” preference relation pi. The society is represented by a prole of preference relations, p = (pi; : : : ; pn); one for each individual. Let the set of possible alternatives beW = fx; y; : : :g. If person i prefers x to y then write (x; y) 2 pi, or more commonly xpiy. The social mechanism or preference function, , translates any prole p into a preference rela- tion (p). The point of the theory is to examine conditions on  which are sufcient to ensure that whatever “rationality properties” are held by the individual preferences, then these same properties are held by (p). Arrow's Impossibility Theorem (1951) essentially showed that if the ra- tionality property under consideration is that preference be a weak order then  must be dictatorial. To see what this means, let Ri be the weak preference for i induced from pi. That is to say xRiy if and only if it is not the case that ypix: Then pi is called a weak order if and only if Ri is transitive, i.e., if xRiy and yRiz for some x; y; z inW , then xRiz. Arrow's theorem effectively demonstrated that if  (p) is a weak order whenever every individual has a weak order preference then there must be some dictatorial individual i, say, who is characterized by the ability to enforce every social choice. It was noted some time afterwards that the result was not true if the conditions of the theorem were weakened. For example, the requirement that  (p) be a weak order means that “social indifference” must be tran- sitive. If it is only required that strict social preference be transitive, then there can indeed be a non-dictatorial social preference mechanism with this weaker rationality property (Sen, 1970). To see this, suppose  is dened by the strong Pareto rule: x(p)y if and only if there is no in- dividual who prefers y to x but there is some individual who prefers x to y. It is evident that  is non-dictatorial. Moreover if each pi is transi- tive then so is (p). However, (p) cannot be a weak order. To illustrate this, suppose that the society consists of two individuals f1; 2g who have preferences 1 2 x y z x y z This means xp1zp1y etc. Since f1; 2g disagree on the choice between x and y and also on the choice between y and z both x; y and y; z must be 10 Chapter 1. Introduction socially indifferent. But then if (p) is to be a weak order, it must be the case that x and z are indifferent. However, f1; 2g agree that x is superior to z, and by the denition of the strong Pareto rule, x must be chosen over z. This of course contradicts transitivity of social indifference. A second criticism due to Fishburn (1970) was that the theorem was not valid in the case that the society was innite. Indeed since democ- racy often involves the aggregation of preferences of many millions of voters the conclusion could be drawn that the theorem was more or less irrelevant. However, three papers by Gibbard (1969), Hanssen (1976) and Kir- man and Sondermann (1972) showed that the result on the existence of a dictator was quite robust. The rst three sections of Chapter 2 essentially parallel the proof by Kirman and Sondermann. The key notion here is that of a decisive coalition: a coalitionM is decisive for a social choice function, ; if and only if xpiy for all i belonging toM for the prole p implies x(p)y. Let D represent the set of decisive coalitions dened by : Suppose now that there is some coalition, perhaps the whole society N , which is decisive. If  preserves transitivity (i.e., (p) is transitive) then the intersection of any two decisive coalitions must itself be deci- sive. The intersection of all decisive coalitions must then be decisive: this smallest decisive coalition is called an oligarchy. The oligarchy may indeed consist of more than one individual. If it comprises the whole so- ciety then the rule is none other than the Pareto rule. However, in this case every individual has a veto. A standard objection to such a rule is that the set of chosen alternatives may be very large, so that the rule is effectively indeterminate. Suppose the further requirement is imposed that (p) al- ways be a weak order. In this case it can be shown that for any coalition, M; either M itself or its complement NnM must be decisive. Take any decisive coalition A, and consider a proper subset B say of A. If B is not decisive then NnB is, and so A\ (NnB)= AnB is decisive. In other words every decisive coalition contains a strictly smaller decisive coali- tion. Clearly, if the society is nite then some individual is the smallest decisive coalition, and consequently is a dictator. Even in the case when N is innite, there will be a smallest “invisible” dictator. It turns out, therefore, that reasonable and relatively weak rationality properties on  impose certain restrictions on the class D of decisive coalitions. These restrictions on D do not seem to be similar to the characteristics that po- 1.2 The Theory of Social Choice 11 litical systems display. As a consequence these rst attempts by Sen and Fishburn and others to avoid the Arrow Impossibility Theorem appear to have little force. A second avenue of escape is to weaken the requirement that (p) always be transitive. For example a more appropriate mechanism might be to make a choice fromW of all those unbeaten alternatives. Then an alternative x is chosen if and only if there is no other alternative y such that y(p)x. The set of unbeaten alternatives is also called the core for (p); and is dened by Core(; p) = fx 2 W : y(p)x for no y 2 Wg: In the case thatW is nite the existence of a core is essentially equivalent to the requirement that (p) be acyclic (Sen, 1970). Here a preference, p, is called acyclic if and only if whenever there is a chain of preferences x0px1px2p


pxr then it is not the case that xrpx0. However, acyclicity of  also imposes a restriction on D. Dene the collegium (D) for the family D of decisive coalitions of  to be the intersection (possibly empty) of all the decisive coalitions. If the collegium is empty then it is always possible to construct a “rational” prole p such that (p) is cyclic (Brown, 1973). Therefore, a necessary condition for  to be acyclic is that  exhibit a non-empty collegium. We say  is collegial in this case. Obviously, if the collegium is large then the rule is indeterminate, whereas if the collegium is small the rule is almost dictatorial. A third possibility is that the preferences of the members of the society are restricted in some way, so that a natural social choice function, such as majority rule, will be “well behaved.” For example, suppose that the set of alternatives is a closed subset of a single dimensional “left–right” continuum. Suppose further that each individual i has convex preference on W , with a most preferred point (or bliss point) xi, say.6 Then a well- known result by Black (1958) asserts that the core for majority rule is 6Convexity of the preference p just means that for any y the set {x : xpy} is convex. A natural preference to use is Euclidean preference dened by xpiy if and only if jjxxijj < jjyxijj, for some bliss point, xi, inW , and norm jjjj onW . Clearly Euclidean preference is convex. 14 Chapter 1. Introduction Conversely, if q   r 1 r  n then such a prole could certainly be constructed. Another way of ex- pressing this is that a q-rule  is acyclic for all acyclic proles if and only if jW j < n n q : Note that we assume that q < n. Nakamura (1979) later proved that this result could be generalized to the case of an arbitrary social preference function. The result depends on the notion of a Nakamura number v() for . Given a non-collegial family D of coalitions, a memberM of D is minimal decisive if and only ifM belongs to D, but for no member i ofM doesMnfig belong to D. If D0 is a subfamily of D consisting of minimal decisive coalitions, and moreover D0 has an empty collegium then call D0 a Nakamura subfamily of D. Now consider the collection of all Nakamura subfamilies of D. Since N is nite these subfamilies can be ranked by their cardinality. Dene v(D) to be the cardinality of the smallest Nakamura subfamily, and call v(D) the Nakamura number of D. Any Nakamura subfamily D0, with cardinality jD0j = v(D), is called aminimal non-collegial subfamily. When  is a social preference function with decisive familyD dene the Nakamura number v() of  to be equal to v(D). More formally v() = minfjD0j : D0  D and (D0) = g: In the case that  is collegial then dene v() = v(D) =1 (innity): Nakamura showed that for any voting rule, ; ifW is nite, with jW j < v(); then (p) must be acyclic whenever p is an acyclic prole. On the other hand, if  is a social preference function and jW j  v() then it is always possible to construct an acyclic prole on W such that (p) is cyclic. Thus the cardinality restriction onW which is necessary and suf- cient for  to be acyclic is that jW j < v(). To relate this to Ferejohn– Grether's result for a q-rule, dene v(n; q) to be the largest integer such that v(n; q) < q n q : 1.2 The Theory of Social Choice 15 It is an easy matter to show that when q is a q-rule then v(q) = 2 + v(n; q): The Ferejohn–Grether restriction jW j < n nq may also be written jW j < 1 + q n q which is the same as jW j < v(q): Thus Nakamura's result is a generalization of the earlier result on q-rules. The interest in this analysis is that Greenberg (1979) showed that a core would exist for a q-rule as long as preferences were convex and the choice space, W , was of restricted dimension. More precisely suppose that W is a compact,7 convex subset of Euclidean space of dimension w, and suppose each individual preference is continuous8 and convex. If q > ( w w+1 )n then the core of (p) must be non-empty, and if q  ( w w+1 )n then a convex prole can be constructed such that the core is empty. From a result by Walker (1977) the second result also implies, for the constructed prole p; that (p) is cyclic. Rewriting Greenberg's inequality it can be seen that the necessary and sufcient dimensionality condition (given convexity and compactness) for the existence of a core and the non-existence of cycles for a q-rule, q, is that dim(W )  v(n; q) where dim(W ) = w is the dimension ofW . Since v(q) = 2 + v(n; q): where v(q) is the Nakamura number of the q-rule, this suggests that for an arbitrary non-collegial voting rule  there is a stability dimension, namely v() = v() 2, such that dim(W )  v() is a necessary and sufcient condition for the existence of a core and the non-existence of 7Compactness just means the set is closed and bounded. 8The continuity of the preference, p; that is required is that for each x 2 W; the set p1{y 2W : xpy} is open in the topology onW: 16 Chapter 1. Introduction cycles. Chapters 3, 4 and 5 of this volume prove this result and present a number of further applications. An important procedure in this proof is the construction of a represen- tation  for an arbitrary social preference function. Let D = fM1; : : : ;Mvg be a minimal non-collegial subfamily for . Note that D has empty collegium and cardinality v() = v. Then  can be represented by a (v 1) dimensional simplex
in Rv1. Moreover, each of the v faces of this simplex can be identied with one of the v coalitions in D. Each proper subfamily Dt = f::;Mt1;Mt+1; ::g has a non-empty collegium, (Dt), and each of these can be identied with one of the vertices of
. To each i 2 (Dt) we can assign a preference pi; for i = f1; : : : ; vg on a set x = fx1;x2; : : : ; xvg giving a permutation prole (D1) (D2) : : : (Dv) x1 x2 xv x2 x3 x1 : : : : : : : : : xv x1 : : : xv1 : From this construction it follows that x1(p)x2


(p)xv(p)x1: Thus wheneverW has cardinality at least v, then it is possible to construct a prole p such that (p) has a permutation cycle of this kind. This representation theorem is used in Chapter 4 to prove Nakamura's result and to extend Greenberg's Theorem to the case of an arbitrary rule. The principal technique underlying Greenberg's Theorem is an impor- tant result due to Fan (1961). Suppose thatW is a compact convex subset of Rw, and suppose P is a correspondence from W into itself which is convex and continuous.9 Then there exists an “equilibrium” point x inW such that P (x) is empty. In the case under question if each individual preference, pi, is continuous, then so is the preference correspondence P associated with (p). Moreover, ifW is a subset of Euclidean space with 9Again, continuity of the preference correspondence, P; means that for each x 2 W; the set P1(x) = fy 2W : x 2 P (y)g is open in the topology onW: 1.2 The Theory of Social Choice 19 A smooth prole for the society N is a differentiable function u = (u1; : : : ; un) : W ! Rn: We assume in the following analysis thatW is compact, and let U(W )N be the space of all such proles endowed with the Whitney C1-topology (Golubitsky and Guillemin, 1973; Hirsch, 1976). Essentially two proles u1 and u2 are close in this topology if all values and the rst derivatives are close. Restricting attention to smooth utility proles whose associated pref- erences are convex gives the space Ucon(W )N . We say that the core Core(; u) for a rule  is structurally stable (inUcon(W )N ) ifCore(; u) is non-empty and there exists a neighborhood V of u in Ucon(W )N such that Core(; u0) is non-empty for all u0 in V . To illustrate, if Core(; u) is non-empty but not structurally unstable then an arbitrary small pertur- bation of u; to a different but still convex smooth preference prole, u0, is sufcient to destroy the core by rendering Core(; u0) empty. By the previous result if dim(W )  v() then Core(; u) is non- empty for every smooth, convex prole, and thus this dimension con- straint is sufcient for Core(; u) to be structurally stable. It had earlier been shown by Rubinstein (1979) that the set of contin- uous proles such that the majority rule core is non-empty is in fact a nowhere dense set in a particular topology on proles, independently of the dimension. However, the perturbation involved deformations induced by creating non-convexities in the preferred sets. Thus the construction did not deal with the question of structural stability in the topological space Ucon(W )N . Chapter 5 continues with the result by McKelvey and Schoeld (1987) and Saari (1997) which indicates that, for any q-rule, q; there is an in- stability dimension, w(q). If dim(W )  w(q) andW has no boundary then the q-core will be empty for a dense set of proles in Ucon(W )N . This immediately implies that the core cannot be structurally stable, so any sufciently small perturbation in Ucon(W )N will destroy the core. The same result holds if W has a non-empty boundary but dim(W )  w(q) + 1. Theorem 5.1.1 shows that if a point belongs to the core of a voting game, dened by a set, D, of decisive coalitions, then the direc- tion gradients must satisfy certain generalized symmetry conditions on the utility gradients of the voters at that point. This theorem is an exten- 20 Chapter 1. Introduction sion of an earlier result by Plott (1967) for majority rule. The easiest case to examine is where the core, Core(; u); is charac- terized by the property that exactly one individual has a bliss point at the core. We denote this by BCore(; u). The Thom Transversality Theo- rem can then be used to show that BCore(q; u) is generically empty (in the space Ucon(W )N ), whenever the dimension exceeds 2q n+1. This suggests that the instability dimension satises w(q) = 2q n+ 1. Saari (1997) extended this result in two directions, by showing that if dim(W )  2q n then BCore(q; u) could be structurally stable. Moreover, he was able to compute the instability dimension for the case of a non-bliss core, when no individual has a bliss point at the core. For example, with majority rule the instability dimension is two or three depending on whether n is odd or even. For n odd, neither bliss nor non-bliss cores can be structurally stable in two or more dimensions, since the Plott (1967) symmetry conditions cannot be generically satis- ed. On the other hand, when (n; q) = (4; 3); the Nakamura number is four, and hence a core will exist in two dimensions. Indeed, both bliss cores and non-bliss cores can occur in a structurally stable fashion. How- ever, in three dimensions the cycle set is contained in, but lls the Pareto set. For all majority rules with n  6, and even, a structurally stable bliss-core can occur in two dimensions. However, when n even, in three dimensions the core cannot be structurally stable and the cycle set need not be constrained to the Pareto set (in contradiction to Tullock's hypoth- esis). For general weighted voting games, dened by a non-collegial family, D, the core symmetry condition can be satised in a structurally stable fashion. This provides a technique for examining when a core exists in the legislatures discussed in Chapter 6. Since the core may be empty, the notion of the heart is presented as an alternative solution idea. The heart can be interpreted in terms of a local uncovering relation, and can be shown to be non-empty under fairly weak conditions. This idea is illustrated by considering various voting rules in low dimensions. The last section of Chapter 5 presents the experimental results obtained by Fiorina and Plott (1978), McKelvey, Ordeshook and Winer (1978), Laing and Olmstead (1978) and Eavey (1996) to indicate the nature of the heart in two dimensions. Chapter 2 Social Choice 2.1 Preference Relations Social choice is concerned with a fundamental question in political or economic theory: is there some process or rule for decision making which can give consistent social choices from individual preferences? In this framework denoted by W is a universal set of alternatives. Members of W will be written x; y etc. The society is denoted by N , and the individuals in the society are called 1; : : : ; i; : : : ; j; : : : ; n. The values of an individual i are represented by a preference relation pi on the setW . Thus xpiy is taken to mean that individual i prefers alternative x to alternative y. It is also assumed that each pi is strict, in the way to be described below. The rest of this section considers the abstract properties of a preference relation p onW: Denition 2.1.1. A strict preference relation p onW is (i) Irreexive: for no x 2 W does xpx; (ii) Asymmetric: for any x; y;2 W ;xpy ) not(ypx). The strict preference relations are regarded as fundamental primitives in the discussion. No attempt is made to determine how individuals ar- rive at their preferences, nor is the problem considered how preferences might change with time. A preference relation p may be represented by a utility function. Denition 2.1.2. A preference relation p is representable by a utility function 21 24 Chapter 2. Social Choice order, and the class of these is written T (W ). Finally the class of acyclic strict preference relations onW is written A(W ). If p 2 O(W ) then it follows from the denition that R(p) is transitive. Indeed I(p) will also be transitive. Lemma 2.1.1. If p 2 O(W ) then I(p) is transitive. Proof. Suppose xI(p)y; xI(p)z but not(xI(p)z). Because of not(xI(p)z) suppose xpz. By asymmetry of p, not(zpx) so xR(p)z. By symmetry of I(p); yI(p)x and zI(p)y; and thus yR(p)x and zR(p)y: But xR(p)z and zR(p)y and yR(p)x contradicts the transitivity ofR(p). Hence not(xpz). In the same way not(zpx); and so xI(p)z; with the result that I(p) must be transitive. Lemma 2.1.2. If p 2 O(W ) then xR(p)y and ypz ) xpz: Proof. Suppose xR(p)y, ypz and not(xpz). But not(xpz), zR(p)x: By transitivity of R(p), zR(p)y. By denition not(ypz) which contradicts ypz by asymmetry. Lemma 2.1.3. O(W )  T (W )  A(W ). Proof. (i) Suppose p 2 O(W ) but xpy, ypz yet not(xpz), for some x, y, z. By p asymmetry, not(ypx) and not(zpy): Since p 2 O(W ), not(xpz) and not(zpy) ) not(xpy). But not(ypx). So xI(p)y. But this violates xpy. By contradiction, p 2 T (W ). (ii) Suppose xjpxj+1 for j = 1; : : : ; r 1. If p 2 T (W ), then x1pxr. By asymmetry, not(xrpx1), so p is acyclic. 2.2 Social Preference Functions Let the society be N = f1; : : : ; i; : : : ; ng: A prole for N on W is an 2.2 Social Preference Functions 25 assignment to each individual i in N of a strict preference relation pi on W . Such an n-tuple (p1; : : : ; pn) will be written p. A subset M  N is called a coalition. The restriction of p toM will be written p=M = (::pi:: : i 2M): If p is a prole for N on W , write xpNy iff xpiy for all xpiy for all i 2 N . In the same way for M a coalition in N write xpMy whenever xpiy for all i 2M . Write B(W )N for the class of proles on N . When there is no possi- bility of misunderstanding we shall simply write BN for B(W )N . On occasion the analysis concerns proles each of whose component individual preferences are assumed to belong to some subset F (W ) of B(W ); for example F (W ) might be taken to be O(W ), T (W ) or A(W ). In this case write F (W )N , or FN , for the class of such proles. Let X be the class of all subsets of W . A member V 2 X will be called a feasible set. Suppose that p 2 B(W )N is a prole forN onW . For some x; y 2 W write “pi(x; y)” for the preference expressed by i on the alternatives x; y under the prole p. Thus “pi(x; y)” will give either xpiy or xI(pi)y or ypix. If f; g 2 B (W )N are two proles onW , and V 2 X , use f=M = g=M on V to mean that for any x; y 2 V , any i 2M; fi(x; y) = gi(x; y). In more abbreviated form write fV=M = gV=M . Implicitly this implies con- sideration of a restriction operator V M : B(W )N ! B(V )M : f ! fV=M ; where B(V )M means naturally enough the set of proles forM on V . A social preference function is a method of aggregating preference information, and only preference information, on a feasible set in order to construct a social preference relation. Denition 2.2.1. A method of preference aggregation (MPA), , assigns to any feasible set V , and prole p for N onW a strict social preference relation (V; p) 2 B(V ). Such a method is written as  : X BN ! B. 26 Chapter 2. Social Choice As before write (V; p)(x; y) for x; y 2 V to mean “the social preference relation declared by (V; p) between x and y.” If f; g 2 BN , write (V; f) = (V; g) whenever (V; f)(x; y) = (V; g)(x; y) for any x; y 2 V . Denition 2.2.2. A method of preference aggregation  : X  BN ! B is said to satisfy the weak axiom of independence of infeasible alterna- tives (II) iff f V = g V ) (V; f) = (V; g): Such a method is called a social preference function (SF). Note that an SF, , is functionally dependent on the feasible set V . Thus there need be no specic relationship between (V1; f) and (V2; f) for V2  V1 say. However, suppose (V1; f) is the preference relation induced by  from f on V1. Let V2  V1, and let (V1; f)=V2 be the preference relation induced by (V1; f) on V2 from the denition [(V1; f)=V2](x; y)] = [(V1; f)(x; y)] whenever x; y 2 V2. A binary preference function is one which is consistent with this re- striction operator. Denition 2.2.3. A social preference function  is said to satisfy the strong axiom of independence of infeasible alternatives (II) iff for f 2 B(V1) N , g 2 B(V2)N , and f V = g V for V = V1 \ V2 non-empty, then (V1; f)=V = (V2; g)=V: For (V1; f) to be meaningful when  is an SF, we only require that f be a prole dened on V1. This indicates that II is an extension property. For suppose f; g are dened on V1; V2 respectively, and agree on V . Then it is possible to nd a prole p dened on V1[V2, which agrees with f on V1 and with g on V2. Furthermore if  is an SF which satises II, then (V1 [ V2; p)=V1 = (V1; f) (V1 [ V2; p)=V2 = (V2; g) (V1 [ V2; p)=V1 \ V2 = (V1; f)=V = V2; g)=V: 2.3 Arrowian Impossibility Theorems 29 ui(x) = (v ri) where jV j = v and ri is the rank that x has in i's prefer- ence schedule. Although this gives a well-dened SF, (V; p), it nonethe- less results in a certain inconsistency, since (V1; p1) and (V2; p2) may not agree on the intersection V1 \ V2, even though p1 and p2 do. Although a BF avoids this difculty, other inconsistencies are intro- duced by the strong independence axiom. 2.3 Arrowian Impossibility Theorems This section considers the question of the existence of a binary social preference function,  : FN ! F , where F is some subset of B. In this notation  : FN ! F means the following: Let V be any feasible set inW , and F (V )N the set of proles, dened on V , each of whose component preferences belong to F . The domain of  is the union of F (V )N across all V inW . That is for each f 2 F (V )N ; we write (f) for the binary social preference on V; and require that (f) 2 F (V ). Denition 2.3.1. A BF  : BN ! B satises (i) The weak Pareto property (P) iff for any p 2 BN , any x; y 2 W , xpNy ) x(p)y: (ii) Non-dictatorship (ND) iff there is no i 2 N such that for all x; y in W , xpiy ) x(p)y: A BF  which satises (P) and (ND) and maps ON ! O is called a binary welfare function (BWF). Arrow's Impossibility Theorem 2.3.1. For N nite, there is no BWF. This theorem was originally obtained by Arrow (1951). To prove it we introduce the notion of a decisive coalition. Denition 2.3.2. LetM be a coalition, and  a BF. (i) DeneM to be decisive under  for x against y iff for all p 2 BN xpMy ) x(p)y: 30 Chapter 2. Social Choice (ii) Dene M to be decisive under  iff for all x; y 2 W;M is decisive for x against y. (iii) LetD(x; y) be the family of decisive coalitions under  for x against y, and D be the family of decisive coalitions under . To prove the theorem we rst introduce the idea of an ultralter. Denition 2.3.3. A family of coalitions can satisfy the following proper- ties. (F1) monotonicity: A  B and A 2 D) B 2 D; (F2) identity: N 2 D and  2 D (where  is the empty set); (F3) closed intersection: A;B 2 D) A \B 2 D; (F4) negation: for any A  N , either A 2 D or NnA 2 D. A familyD of subsets ofN which satises (F1), (F2) and (F3) is called a lter. A lter D1 is said to be ner than a lter D2 if each member of D2 belongs to D1. D1 is strictly ner than D2 iff D1 is ner than D2 and there exists A 2 D1 with A 2 D2. A lter which has no strictly ner lter is called an ultralter. A lter is called free or xed depending on whether the intersection of all its members is empty or non-empty. In the case that N is nite then by (F2) and (F3) any lter, and thus any ultralter, is xed. Lemma 2.3.2. (Kirman and Sondermann, 1972). If  : ON ! O is a BF and satises the weak Pareto property (P), then the family of decisive coalitions, D; satises (F1), F(2), F(3) and F(4). We shall prove this lemma below. Arrow's theorem follows from Lemma 2.3.2 since D will be an ultralter which denes a unique dicta- tor. This can be shown by the following three lemmas. Lemma 2.3.3. Let D be a family of subsets of N , which satises (F1), (F2), (F3) and (F4). Then if A 2 D, there is some proper subset B of A which belongs to D. Proof. Let B be a proper subset of A with B =2 D. By (F4), NnB 2 D. But then by (F3), A \ (NnB) = AnB 2 D. 2.3 Arrowian Impossibility Theorems 31 Hence if B  A, either B 2 D or AnB 2 D. Lemma 2.3.4. If D satises (F1), (F2), (F3) and (F4) then it is an ultra- lter. Proof. Suppose D1 is a lter which is strictly ner than D. Then there is some A;B 2 D1, with A 2 D but B =2 D. By the previous lemma, either AnB or A \ B must belong to D. Suppose AnB 2 D. Then AnB 2 D1. But since D1 is a lter (AnB) \ B =  must belong to D, which contradicts (F2). Hence A\B belongs to D. But by (F1), B 2 D. Hence D is an ultralter. Lemma 2.3.5. If N is nite and D is an ultralter with D = fAjg then \Aj = fig; where fig is decisive and consists of a single member of N . Proof. Consider any Aj 2 D, and let i 2 Aj . By (F4) either fig 2 D or Ajfig 2 D. If fig 2 D; then Ajfig 2 D. Repeat the process a nite number of times to obtain a singleton fig, say, belonging to D. Proof of Theorem 2.3.1. For N nite, by the previous four lemmas, the family of -decisive coalitions forms an ultralter. The intersection of all decisive coalitions is a single individual i, say. Since this intersection is nite, fig 2 D. Thus i is a dictator. Consequently any BF  : ON ! O which satises (P) must be dictatorial. Hence there is no BWF. Note that when N is innite there can exist a BWF  (Fishburn, 1970). However, its family of decisive coalitions still forms an ultralter. See Schmitz (1977) and Armstrong (1980) for further discussion on the exis- tence of a BWF when N is an innite society. The rest of this section will prove Lemma 2.3.2. The following den- itions are required. Denition 2.3.4. Let  be a BF,M a coalition, p a prole, x; y 2 W . (i) M is almost decisive for x against y with respect to p iff xpMy; ypNMx and x(p)y. 34 Chapter 2. Social Choice fashion: zp1xp1y xp2yp2z yp3zp2x yp4xp4z: Since A = V1 [ V2 2 D; we obtain x(p)y; B = V1 [ V3 2 D; we obtain z(p)x: By transitivity, z(p)y. Now zpV1y, ypNV1z and z(p)y. By Lemma 2.3.6, V1 2 D0(z; y) = D. Thus A \B 2 D. This lemma demonstrates that if  : TN ! T is a BF which satises (P) then D is a lter. However, if p 2 ON then p 2 TN , and if (p) 2 O then (p) 2 T , by Lemma 2.1.3. Hence to complete the proof of Lemma 2.3.2 only the following lemma needs to be shown. Lemma 2.3.9. If  : ON ! O is a BF and satises (P) then D satises (F4). Proof. SupposeM 2 D. We seek to show thatNnM 2 D. If for any f; there exist x; y 2 W such that yfMx and y(f)x, thenM would belong to D(x; y) and so be decisive. Thus for any f; there exist x; y 2 W; with yfMx and not (y(f)x) i.e., xR((f))y: Now consider g 2 ON ; with g = f on fx; yg and xgNMz; ygNMz and ygMz: By II; xR((g))y: By (P), y(g)z: Since (g) 2 O it is negatively transitive, and by Lemma 2.1.2, x(g)z: Thus NnM 2 D(x; z) and so NnM 2 D: 2.4 Power and Rationality Arrow's theorem showed that there is no binary social preference func- tion which maps weak orders to weak orders and satises the Pareto and non-dictatorship requirements when N is nite. Although there may ex- ist a BWF when N is innite, nonetheless “power” is concentrated in the sense that there is an “invisible dictator.” It can be argued that the require- 2.4 Power and Rationality 35 ment of negative transitivity is too strong, since this property requires that indifference be transitive. Individual indifference may well display intransitivities, because of just perceptible differences, and so may social indifference. To illustrate the problem with transitivity of indifference, consider the binary social preference function, called the weak Pareto rule written n and dened by: xn(p)y iff xpNy: In this case fNg = Dn . This rule is a BF, satises (P) by denition, and is non-dictatorial. However, suppose the preferences are zpMxpMy ypNMzpNMx for some proper subgroupM in N . Since there is not unanimous agree- ment, this implies xI(n)yI(n)z. If negative transitivity is required, then it must be the case that xI(n)z: Yet zpNx; so z(p)x: Such an ex- ample suggests that the Impossibility Theorem is due to the excessive rationality requirement. For this reason Sen (1970) suggested weakening the rationality requirement. Denition 2.4.1. A BF  : ON ! T which satises (P) and (ND) is called a binary decision function (BDF). Lemma 2.4.1. There exists a BDF. To show this, say a BF  satises the strong Pareto property (P) iff, for any p 2 BN ; ypix for no i 2 N; and xpjy for some j 2 N ) x(p)y: Note that the strong Pareto property (P) implies the weak Pareto property (P). Now dene a BF n, called the strong Pareto rule, by: xn(p)y iff ypix for no i 2 N and xpjy for some j 2 N: n may be called the extension of n; since it is clear that xn (p) y ) xn(p)y: Obviously n satises (P) and thus (P). However, just as n violates transitive indifference, so does n: On the other hand n satises transi- tive strict preference. 36 Chapter 2. Social Choice Lemma 2.4.2. (Sen, 1970). n is a BDF. Proof. Suppose xn(p)y and yn(p)z: Now xn(p)y , xR(pi)y for all i 2 N and xpjy for some j 2 N: Similarly for fy; zg. By transitivity of R(pi); we obtain xR(pi)z for all i 2 N: By Lemma 2.1.2, xpjz for some j 2 N: Hence xn(p)z. While this seems to refute the relevance of the impossibility theorem, note that the only decisive coalition for n is fNg. Indeed the strong Pareto rule is somewhat indeterminate, since any individual can effec- tively veto a decision. Any attempt to make the rule more “determinate” runs into the following problem. Denition 2.4.2. An oligarchy  for a BF  is a minimally decisive coalition which belongs to every decisive coalition. Lemma 2.4.3. (Gibbard, 1969). If N is nite, then any BF  : ON ! T which satises P has an oligarchy. Proof. Restrict  to  : TN ! T . By Lemma 2.3.8, since  satises (P), its decisive coalitions form a lter. Let  = \Aj , where the intersection runs over all Aj 2 D: Since N is nite, this intersection is nite, and so  2 D: Obviously  fig 2 D for any i 2 : Consequently  is a minimally decisive coalition or oligarchy. The following lemma shows that members of an oligarchy can block social decisions that they oppose. Lemma 2.4.4. (Schwartz, 1986). If  : ON ! T and p 2 ON ; and  is the oligarchy for ; then dene x(p) = fi 2  : xpiyg y(p) = fj 2  : ypjxg: Then (i) x(p) 6=  and  = x(p) [ y(p)) not (y(p)x) (ii) x(p) 6= ; y(p) 6=  and  = x [ y ) xI((p))y: 2.5 Choice Functions 39 Denition 2.5.2. A choice function C : X BN ! X satises the weak axiom of revealed preference (WARP) iff wherever V  V 0, and p is dened on V 0, with V \ C(V 0; p) 6= , then V \ C(V 0; p) = C(V; pV ); where pV is the restriction of p to V . Note the analogue with (II). If we write C(V 0; p)=V for V \ C(V 0; p) when this is non-empty, then WARP requires that C(V; pV ) = C(V 0; p)=V: Denition 2.5.3. (i) A choice function C : X  BN ! X is said to be rationalized by an SF  : X BN ! B iff for any V 2 X and any p 2 BN ; C(V; p) = fx : y(V; p)x for no y 2 V g: (ii) A choice function C : X  BN ! X is said to be rationalized by a BF  : BN ! B iff for any p 2 BN ; and any x; y 2 W;x 6= y, C(fx; yg; p) = fxg , x(p)y: (iii) A choice function C : X  BN ! X is said to satisfy the binary choice axiom (BICH) iff there is a BF  : BN ! B such that for any V 2 X; any p 2 BN ; C(V; p) = fx 2 V : y(p)x for no y 2 V g: Say C satises BICH w.r.t.  in this case. (iv) Given a choice function C : X  BN ! X dene the induced BF C : B N ! B by C(fx; yg; p) = fxg , xC(p)y: (v) Given a BF  : BN ! B dene the choice procedure C : X BN ! X by C(V; p) = fx 2 V : y(p)x for no y 2 V g: 40 Chapter 2. Social Choice Note that C(V; p) may be empty for some V; p: Lemma 2.5.1. If C satises BICH w.r.t.  then  rationalizes C. Proof. (i) C(fx; yg; p) = fxg )not(y(p)x: If yI((p))x then not (x(p)y); so y 2 C(fx; yg; p): Hence C(fx; yg; p) = fxg ) x(p)y: (ii) x(p)y ) not (y(p)x): Hence C(fx; yg; p) = fxg: Another way of putting this lemma is that if C = C is a choice function then  = C : In the following we delete reference to p when there is no ambiguity, and simply regard C as a mapping from X to itself. Example 2.5.1. (i) Suppose C is dened on the pair sets ofW = fx; y; zg by C(fx; yg) = fxg; C(fy; zg) = fyg and C(fx; zg) = fzg: If C satises BICH w.r.t. ; then it is necessary that xyzx; so C(fx; y; zg) = : Hence C cannot satisfy BICH. (ii) Suppose C(fx; yg) = fxg C(fy; zg) = fy; zg: If C(fx; zg) = fx; zg; then xyI()z and xI()z; so C(fx; y; zg) = fx; zg: While C satises BICH w.r.t. ;  does not give a weak order, although  may give a strict partial order. 2.5 Choice Functions 41 Theorem 2.5.1. (Sen, 1970). Let the universal set,W; be of nite cardi- nality. (i) If a choice function C satises BICH w.r.t.; then  = C is a BAF. (ii) If  is a BAF then C; restricted to X  AN ; is a choice function. Proof. (i) By Lemma 2.5.1, if C satises BICH w.r.t.  then  rationalizes C; and so by denition the induced BF, C ; is identical to : We seek to show that  is a BAF, or that  : AN ! A: Suppose on the contrary that  is not a BAF. Since  is a BF and W is of nite cardinality, this assumption is equivalent to the existence of a nite subset V = fa1; : : : ; arg ofW; a prole p 2 A(V )N ; and a cycle a1(p)a2(p)


ar(p)a1: Let ar  a0: Then for each aj 2 V it is the case that aj1(p)j: SinceC satises BICHwith respect to ; it is evident thatC(V; p) = : By contradiction,  is a BAF. (ii) We seek now to show that for any nite set, V; if A(V )N and p 2 A(V )N and (p) 2 A(V ) then C(V; p) 6= . First, let I((p)) and R() represent the indifference and weak preference relations dened by (p): Suppose that V = fx1; : : : ; xrg: If x1I((p))x2 : : : xr1I((p))xr then C(V; p) = V: So suppose that for some a1; a2 2 V it is the case that a2(p)a1. If a2 2 C(V; p) then there exists a3; say, such a3(p)a2: If a1(p)a3; then by acyclicity, not(a2(p)a1): Since (p) is a strict preference relation, this is a contradiction. Hence not (a1(p)a3); and so a3 2 C(fa1; a2; a3g; p): By induction, C(V 0; p) 6= ) C(V 00; p) 6=  whenever jV 0j + 1 = jV 00j and V 0  V 00  W: Thus C(V; p) 6=  for any nite subset V ofW . Lemma 2.5.3. (Schwartz, 1976). A choice function C satises BICH iff 44 Chapter 2. Social Choice then V  C(W; p): Example 2.5.3. If majority rule is used with the prole given in Example 2.5.2, then there is a cycle z(p)w(p)x(p)y(p)z: So any choice function C which satises II P has to choose C(V; p) = V = fx; y; z; wg: However, zpNw; so the choice function can choose alternatives which are beaten under the weak Pareto rule (i.e., are not Pareto optimal). If one seeks a choice function which satises the strong consistency properties of WARP or EX, then choices must be made by binary com- parisons (BICH), and consequently the Arrowian Impossibility Theorems are relevant. If one seeks only IIP, then C  CC ; and again binary com- parisons must be made, so the Impossibility Theorems are once more relevant. The attraction of IIP is that it permits choice to be done by di- vision. Suppose a decision problem, V; is divided into components Vj; choice made from Vj; and then choice made from these. Then the resul- tant decision must be compatible with whatever choice would have been made from V: II P would seem to be a minimal consistency property of a choice procedure. Unfortunately it requires the selection of cycles, no matter how large these are. The next chapter examines the occurrence of cycles for arbirary voting rules. Since cycles, and particular non-Paretian cycles, will occur under such rules, there is a contradiction between implementability (or path independence) and Pareto optimally for general voting processes. Chapter 3 Voting Rules 3.1 Simple Binary Preference Functions The previous chapter showed that for a binary preference function  to satisfy certain rationality postulates it is necessary that the family of decisive coalitions obey various lter properties. A natural question is whether the previous restrictions on power, imposed by the lter prop- erties, are sufcient to ensure rationality. In general, however, this is not the case. To see this, for a given class of coalitions dene a new BF as follows. Denition 3.1.1. Let N be a xed set of individuals, and D a family of subsets of N: Dene the BF D : BN ! B by: xD(p)y , fi 2 N : xpiyg 2 D, whenever x; y 2 W: For a given BF  : BN ! B;D is dened to be its family of decisive coalitions. Consequently there are two transformations:  ! D and D ! D : In terms of these transformations, the previous results may be written: Lemma 3.1.1. If  is a BF which satises (P) and (i)  : ON ! O then D is an ultralter; (ii)  : TN ! T then D is a lter; (iii)  : AN ! A then D is a prelter. 45 46 Chapter 3. Voting Rules Lemma 3.1.2. (Ferejohn, 1977). If D is (i) an ultralter then D : ON ! O is a BF and satises (P); (ii) a lter then D : TN ! T is a BF and satises (P); (iii) a prelter then D : AN ! A is a BF and satises (P). However, even though D satises one of the lter properties,  need not satisfy the appropriate rationality property. The problem is that the transformation  ! D is “structure forgetting.” It is easy to see that for any x; y 2 W; p 2 BN ; xD(p)y ) x(p)y: Thus D  : For this reason it may be the case that, for some x; y; p;we obtain x(p)y but also not (xD(p)y): To see this consider the following example due to Ferejohn and Fishburn (1979). Example 3.1.1. Let N = f1; 2g;W = fx; y; zg and T be the cyclic relation xTy; yTz; zTx: Dene x(p)y , xp1y or [xI(p1)y and xTy]: It follows from this denition that D = ff1g; f1; 2gg is an ultralter. Obviously xD(p)y , xp1y: Hence D : ON ! O is dictatorial. On the other hand, if p is a prole under which f1g is indifferent on fx; y; zg then x(p)y(p)z(p)x; so (p) is cyclic. Denition 3.1.2. If f 1; 2 are two binary preference functions on W; and for all p 2 BN x1(p)y ) x2(p)y for any x; y 2 W; then say that 2 is ner than 1; and write 1  2: If in addition x2(p)y yet not (x1(p)y) for some x; y; then say 2 is strictly ner than 1; and write 1  2: If 2 is strictly ner than 1 then 1 may satisfy certain rationality 3.1 Simple Binary Preference Functions 49 D iff s(M)  q; (d)  = D : (ii) A simple weighted majority rule is written q(s); where q(s) = [q : s(1); : : : ; s(i); : : : ; s(n)]: If s(i) = 1 for each i 2 N; and q > n 2 ; then the voting rule is called the simple q-majority rule, or q-rule, and denoted q. (iii) Simple majority rule, written as m; is the q-rule dened in the fol- lowing way: if n = 2k + 1 is odd, then q = k + 1 = m; if n = 2k is even, then q = k + 1 = m: (iv) In the case q = n; this gives the weak Pareto rule n: Note that q is anonymous as well as simple. We shall often refer to a simple weighted majority rule as a q(s)-rule. Two further properties of a voting rule  are as follows. Denition 3.1.5. A voting rule  is (i) proper iff for any A;B 2 D; A \B 6= ; (ii) strong iff A 2 D then NnA 2 D: For example consider a q(w)-simple weighted majority rule, : Because q > s(N) 2 thenM 2 D implies that s(NnA) = s(N)nw(A) < s(N) 2 : Hence, if B  NnA then B 2 D: Thus  must be proper. On the other hand, suppose  is simple majority rule with jN j = 2k; an even integer. Then if jAj = k;A 2 D0 but jNnAj = k and NnA =2 D: Thus  is not strong. However, if jN j = 2k + 1, an odd integer, and jAj = k then A =2 D but jNnAj = k+1 and soNnA 2 D: Thus  is strong. Another interpretation of these terms is as follows. If A =2 D then A is said to be losing. On the other hand if A is such that NnA =2 D then call A 50 Chapter 3. Voting Rules blocking. If  is strong, then no losing coalition is blocking, and if  is proper then every winning coalition is blocking. Given a q(s)-rule, ; it is possible to dene a new rule ; called the extension of ; such that  is ner than : Denition 3.1.6. (i) For a q(w) rule, ; dene its extension  by x(p)y , s(Mxy)  q  s(Mxy) + s(Mxy) s(N)  where Mxy = fi : xpiyg and Myx = fj : ypjxg: Write q for the extension of the simple q-majority rule, q: Then xq(p)y , jMxyj  q  jMxyj+ jMyxj jN j  : (ii) The weak Pareto rule, n is dened by xn(p)y , jMxyj = n: (iii) The strong Pareto rule, n is dened analogously by xn(p)y , jMxyj  jMxyj+ jMyxj: That is to say jMyxj = : (iv) Plurality rule, written plur; is dened by: xplur(p)y , jMxyj > jMyxj: Lemma 3.1.4. The simple q-majority rules and their extensions are nested: 3.2 Acyclic Voting Rules on Restricted Sets of Alternatives 51 i.e., for any q; n=2 < q  n; n  n \ \ q  q \ \ m  m where as before 1  2 iff x1(p)y ) x2(p)y wherever x; y 2 W; p 2 BN : Notice that it is possible that xplur(p)y yet not[xm(p)y]: For exam- ple, if n = 4; and jMxyj = 2; jMyxj = 1; we obtain xplur(p)y: However, jMxyj <  3 4  [jMxyj+ jMyxj] so not[xm(p)y]: From Brown's result (Theorem 2.4.5) for a voting rule, ; to be acyclic it is necessary that there be a collegium : Indeed, for a collegial voting rule, each member i of the collegium has the veto power: xpiy ) not (y(p)x]: As we have seen n maps ON to T; and n maps TN ! T: However, any anonymous q-rule ; with q < n; is non-collegial, and so it is possible to nd a prole p such that  (p) is cyclic. The next section shows that such a prole must be dened on a feasible set containing a sufciently large number of alternatives. 3.2 Acyclic Voting Rules on Restricted Sets of Alternatives In this section we shall show that when the cardinality of the set of alter- natives is suitably restricted, then a voting rule will be acyclic. Let B(r)N be the class of proles, each dened on a feasible set of at most r alternatives, and let F (r)N be the natural restriction to a subclass dened by F  B: Thus A(r)N is the set of acyclic proles dened on feasible sets of cardinality at most r: 54 Chapter 3. Voting Rules (h 1)n: Consequently h  n nq ) j(D0)j = 0: But 1+ v(n; q) < n nq  2 + v(n; q): Thus h  2 + v(n; q) ) j(D0)j = 0: Hence v(q) = 2+v(n; q): If q = n; then q is collegial. When q  n1; clearly v(q) = 2 + v(n; q) < 2 + q n q  2 + q: Hence v(q) < 2 + q: (iii) By denition  is proper whenM1\M2 6=  for anyM1\M2 2 D: Clearly (D0) 6= when jD0j = 2 for anyD0  D and so v()  3: (iv) Majority rule is a q-rule with q = k+1 when n = 2k or n = 2k+1: In this case q nq = k+1 k = 1 + 1=k for n odd or k+1 k1 = 2 (k1) for n even. For n odd  3; k  2 and so v(n; q) = 1: For n even  6, k  3 and so 1 < q nq  2: Thus v(n; q) = 1 and v(m) = 3. Hence v(m) = 3 except for the case (n; q) = (4; 3): In this case, k = 2; so q nq = 3 and v(4; 3) = 2 and v(3) = 4: Comment 3.2.2. To illustrate the Nakamura number, note that if  is proper, strong, and has two distinct decisive coalitions then v() = 3: To see this suppose M1;M2 are minimal decisive. Since  is proper A = M1 \ M2 6= ; must also be losing. But then NnA 2 D and so the collegium of fM;M 0; NnAg is empty. Thus v() = 3: By Lemma 3.2.1, a q-rule mapsA(r)N ! A (r) iff r  v(n; q)+1: By Lemma 3.2.2, this cardinality restriction may be written as r  v(q)1: The following Nakamura Theorem gives an extension of the Ferejohn– Grether lemma. Theorem 3.2.3. (Nakamura, 1978). Let  be a simple voting rule, with Nakamura number v(): Then (p) is acyclic for all p 2 A(r)N iff r  v() 1: Before proving this theorem it is useful to dene the following sets. Denition 3.2.2. Let  be a BF, with decisive coalitions D and let p be a prole onW: 3.2 Acyclic Voting Rules on Restricted Sets of Alternatives 55 (i) For a coalitionM  N; dene the Pareto set forM (at p) to be Pareto(W;M; p) = fx 2 W : @y 2 W s.t. ypix8i 2Mg: IfM = N then this set is simply called the Pareto set. (ii) The core of (p) is Core(;W;N; p) = fx 2 W : @y 2 W s.t. y(p)xg : Thus Core(;W;N; p)  T [Pareto(W;M; p)]; where the intersection is taken over all M 2 D: If  is a voting rule, then this inclusion is an equality. (iii) An alternative x 2 W belongs to the cycle set, Cycle(;W;N; p); of (p) inW iff there exists a (p)-cycle x(p)x2(p) : : : (p)xr(p)x: If there is no fear of ambiguity write Core(; p) and Cycle(; p) for the core and cycle set respectively. Note that by Theorem 2.5.1, if (p) is acyclic on a nite alternative set W; so that Cycle(;W;N; p) is empty, then the Core(;W;N; p) is non-empty. Of course the choice and cy- cle sets may both be non-empty. We are now in a position to prove the sufciency part of Nakamura's Theorem. Lemma 3.2.4. Let  be a non-collegial, simple voting rule with Naka- mura number v() on the setW: If p 2 A(W )N andCycle(;W;N; p) 6=  then jW j  v(): Proof. Since Cycle(;W;N; p) 6=  there exists a set Z = fx1; : : : ; xrg  W and a (p)-cycle (of length r) on Z: x1(p)x2


xr(p)x1: Write xr  x0: For each j = 1; : : : ; r; let Mj be the decisive coalition such that xj1pixj for all i 2Mj:Without loss of generality we may sup- pose that allM1 : : : ;Mr are distinct and minimally decisive and jW j  r: 56 Chapter 3. Voting Rules Let D0 = fM1; : : : ;Mrg and suppose that (D0) 6= : Then there exists i 2 (D0) such that x1pix2


xrpix1: But by assumption, pi 2 A(W ): By contradiction, (D0) = ; and so, by denition of v(); jD0j  v(): But then r  v() and so jW j  v(): This proves the sufciency of the cardinality restriction of Theorem 3.2.3, since if r  v() 1; then there can be no (p)-cycle for p 2 A(r)N :We now prove necessity, by showing that if r  v() then there exists a prole p 2 A(r)N such that (p) is cyclic. To prove this we introduce the notion of a -complex by an example. First we dene the convex hull of set. Denition 3.2.3. (i) If x; y;2 Rw then Con[fx; yg] is the convex combination of fx; yg ; and is the set dened by Con[fx; yg] = fz 2 Rw : z = x+ (1 ) y where 2 [0; 1]g: (ii) The convex hull Con[Y ] of a set Y = fy1; : : : ; yvg is dened by Con[Y ] = fz 2 Rw : z = X yj2Y jyj; where X j = 1, all j  0g: Example 3.2.2. Consider the voting rule, ; with six players f1; : : : ; 6g whose minimal decisive coalitions are Dmin = fM1;M2;M3;M4g where M1 = f2; 3; 4g; M2 = f1; 3; 4g, M3 = f1; 2; 4; 5g; M4 = f1; 2; 3; 5g: Clearly v() = 4: We represent  in the following way. Since (DminnfMjg) = fjg; for j = 1; : : : ; 4; we let Y = fy1; y2; y3; y4g be the set of vertices, and let each yj represent one of the players f1; : : : ; 4g: Let
be the convex hull of Y in R3: We dene a representation  by (fjg) = yj and (Mj) = Con[Y nfyjg] for j = 1; : : : ; 4: Thus (Mj) is the face opposite yj: Now player 5 belongs to both M3 and M4; but not to M1 or M2; and so we place y5 at the center of the intersection of the faces corresponding toM3 andM4: Finally, since player 6 belongs to no minimally decisive coalition let (f6g) = fy6g; an isolated vertex. Thus the complex
 consists of the four faces of
together with fy6g: See Figure 3.1. 3.2 Acyclic Voting Rules on Restricted Sets of Alternatives 59 correspondence (or morphism)  : (D;\)! (
;\) between the coalitions in Dmin and the faces of
; which is natural with respect to intersection. That is to say for any subfamily D0 of D;  ( (D0)) = T M2D0  (M) : Moreover,  can be extended over N : if for some i 2 N; Di = fM 2 D : i 2 Mg 6=  then (fig) =  ( ( (Di))) ; whereas if Di =  then (fig) is an isolated vertex in
: (ii) If there exists a representation (;
;D) of D then denote
by
(D) : (iii) If  is a BF with decisive coalitions D and (;
;D) is a represen- tation of D then write
as
 and say
 is the -complex which represents : Schoeld (1984a) has shown the following. Theorem 3.2.4. Let D be a family of subsets of N; with Nakamura num- ber v < 1: Let Dmin be a minimal non-collegial subfamily of D. Then there exists a simplex
(Y ); in Rv1; spanned by Y = fy1; : : : ; yvg and a representation  : (Dmin;\)! (
;\) where
is the natural complex based on the faces of
(Y ): Furthermore: (i) There exists a subset V = f1; : : : ; vg of N such that, for each j 2 V; (fjg) = yj; a vertex of
(Y ): (ii) After labelling appropriately, for eachMj 2 Dmin; (Mj) = F (j) the face of
(Y ) opposite yj: Proof. Each proper subfamily Dt = f::;Mt1;Mt+1; :: : t = 1; : : : ; vg of Dmin has a non-empty collegium, (Dt), and each of these can be identied with a vertex, yt of
: If j 2 (Dt); then j is assigned the vertex yt: Continue by induction: if j 2 (Dt \ Ds) (Dt) (Ds) then j is assigned the barycenter of [yt; ys]. By this method we assign a vertex to each member of the setN(Dmin) consisting of those individuals 60 Chapter 3. Voting Rules who belong to at least one coalition in Dmin: This assignment gives a representation (;
;Dmin): Corollary 3.2.1. Let  be a voting rule with Nakamura number v(): Then there exists a -complex
; of dimension v() 2 in Rw; for w  v() 1; which represents : Proof. Let v() = v: Let Dmin be the minimal non-collegial subfamily of D and let  : (Dmin;\) ! (
;\) be the representation constructed in Theorem 3.2.4. Extend  to a representation  : D !
(D) by adding new faces and vertices as required. Finally, the complex
 can be constructed so that, for any D0  D0 then  (D0) =  if and only if \ (Mj) = ; where the intersection is taken over allMj 2 D0: We are now in a position to prove the necessity part of Nakamura's The- orem. Corollary 3.2.2. Let  be a voting rule, with Nakamura number v() = v; on a nite alternative setW: If jW j  v() then there exists an acyclic prole p onW such thatCycle(;W;N; p) 6=  andCore(;W;N; p) = : Proof. Construct a prole p 2 A(W )N and a (p) cycle onW as follows. Each proper subfamily Dt = fM1; : : : ;Mt1;Mt+1; : : : ;Mv : t = 1; : : : ; vg of Dmin has a non-empty collegium, (Dt). By Corollary 3.2.1, each of these can be identied with a vertex, yt of
:Without loss of generality we relabel so that Y = fy1; : : : ; yt; : : : ; yvg  W: Let V = f1; : : : ; vg. We assign preferences to the members of these collegia on the set Y as follows. p1 : (D1) p2 : (D2) : : : pv : (Dv) y1 y2 yv y2 y3 y1 : : : : : : : : : yv y1 : : : yv1 : 3.2 Acyclic Voting Rules on Restricted Sets of Alternatives 61 To any individual j 2N(Dmin)who is assigned a position at the barycen- ter, (
(Y 0)); for a subset Y 0 = fyr; r 2 R  V g; we let pj = T j2R pr: It follows from the construction that every member j of coalition Mt has a preference satisfying T r 6=t fpr  pj: The prole so constructed is called a -permutation prole. It then follows that each j 2 Mt has the preference yt+1pjyt; where we adopt the notational convention that yv+1 = y1: We thus obtain the cycle y1(p)yv


(p)y2(p)y1: This prole can be extended over Y by assigning to an individual j not in N(Dmin) the preference of complete indifference. Obviously Cycle(;W;N; p) 6=  and Core(;W;N; p) = : The argument obviously holds whenever jW j > v(); again by assigning indifference to alternatives outside Y: Note that Corollary 3.2.1 does not require that the voting rule be sim- ple, since the construction holds for the simple rule D. In the same way, the corollary also holds for a BF, ; by applying the construction to D: Example 3.2.3. To illustrate the construction, consider the previous Ex- ample 3.2.2. The prole constructed according to the corollary is: 1 2 3 4 y1 y2 y3 y4 y2 y3 y4 y y3 y4 y1 y2 y4 y1 y2 y3: : Because y5 lies on the arc [y1;y2] in the gure, we dene p5 = p1 \ p2, so y2p5y3p5y4I5y1: 64 Chapter 3. Voting Rules the decisive coalitions for  are as given in the example. Let p be the permutation prole based on 1 2 3 4 y1 y2 y3 y4 y2 y3 y4 y1 y3 y4 y1 y2 y4 y1 y2 y3 with y2p5y3p5y4I5y1: If y4 2 C(p;W )whereW = fy1; y2; y3; y4y5g considerM4 = f1; 2; 3; 5g; and the manipulation p0 1 2 3 5 y3 y3 y3 y3 y1 y2 y4 y2 y2 y4 y1 y4; y1 y4 y1 y2 : Since M4 = f1; 2; 3; 5g 2 D, then if C is -compatible, we obtain fy3g = C(W; p0): But the preferences between y3 and y4 are identical in p and p0: Thus, if C were monotonic, we would obtain y4 2 C(p;W ): This contradiction implies thatC cannot be both monotonic and -compatible. In identical fashion, whichever alternative is selected by the choice func- tion, one of the four decisive coalitions may manipulate p to its advan- tage. Corollary 3.3.2. If jW j  n and jW  3j then for no monotonic choice function C does there exist a non-collegial binary social preference func- tion, ; such that C is compatible with : Proof. For any non-collegial voting rule, ; it is the case that v()  n: Thus ifW  n; Corollary 3.3.1 applies to every choice function. Ferejohn, Grether and McKelvey (1982) essentially obtained a ver- sion of Corollary3.3.1 in the case that  was a q-rule with q = n 1: In this case they said that a choice function that was compatible with  was minimally democratic. They then showed that a minimally democratic choice function could be neither monotonic nor implementable. 3.4 Restrictions on the Preferences of Society 65 As we know from Lemma 3.2.2, v() = 3 for majority rule other than when n = 4: Thus even with three alternatives, any choice func- tion which is majoritarian is effectively manipulable. In a later chapter we shall show that Lemma 3.2.6 can be extended to show the existence of a permutation preference prole for a voting rule, ; in dimension v() 1: “Spatial” voting rules will therefore be manipulable, in the sense described above, even in dimension v() 1: In particular, majori- tarian rules will be manipulable in two dimensions. 3.4 Restrictions on the Preferences of Society The results of the previous section show that non-manipulability, of a non-collegial voting rule cannot be guaranteed without some restriction on the size, r; of the set of alternatives. It is, however, possible that while majority rule, for example, need not be “rational” for general n and r; it is “rational” for “most” preference proles. A number of authors have analyzed the probability of occurrence of voting cycles. Assume, for example, that each preference ranking on a setW with jW j = r is equally likely. For given (n; r) it is possible to compute, for majority rule, the probability of (a) an unbeaten alternative; (b) a permutation preference prole, and thus a voting cycle, containing all r alternatives. Niemi and Weisberg (1968) shows that for large n; the probability of (a) declined from about 0.923 when r = 3; to 0.188 when r = 40: By a simulation method Bell (1978) showed that the probability of total breakdown (b) increased from 0.084 (at r = 3) to 0.352 (at r = 15) to 0.801 (at r = 60): Sen (1970) responded to the negative results of Niemi and Weisberg and others with the comment that the assumption of “equi-probable” pref- erence orderings was somewhat untenable. The existence of classes in a society would surely restrict, in some complicated fashion, the variation in preferences. This assertion provides some motivation for studying re- strictions on the domain of a binary social preference function, ; which are sufcient to guarantee the rationality of the rule. These so-called ex- 66 Chapter 3. Voting Rules clusion principles are sufcient to guarantee the transitivity of majority rule. Denition 3.4.1. (i) A binary relation Q onW is a linear order iff it is asymmetric, tran- sitive and weakly connected (viz. x 6= y ) xQy or yQx): Write L(W ) for the set of linear orders onW: (ii) A preference prole p 2 B(W )N is single peaked iff there is a linear order Q on W such that for any x; y; z in W; if either xQyQz or zQyQx; then for any i 2 N who is not indifferent on fx; y; zg it is the case that (a) xR (pi) z ) ypiz; (b) zR (pi)x) ypix: The class of single peaked preference proles is written SN ; and the class of proles whose component preferences are linear orders is written LN : Example 3.4.1. Suppose N = f1; 2; 3g and the prole p on fx; y; zg is given by 1 2 3 x y y zy z z x x : Then dening Q by xQyQz it is easy to see the prole is single peaked. Arrow (1951), Black (1958) and Fishburn (1973) have obtained the re- sults given in Lemma 3.4.1. Lemma 3.4.1. (i) If p 2 SN \ TN then m(p) 2 T: (ii) If p 2 SN \ LN then m(p) 2 L: (iii) If p 2 SN \ ON ; n odd, and xQyQz and there is no individual indifferent on fx; y; zg ; then m(p) 2 O: Chapter 4 The Core In the previous chapter it was shown that a simple voting rule, ; was acyclic on a nite set of alternatives, W; if and only if the cardinality jW j of W satised jW j  v() 1: The same cardinality restriction was shown to be necessary and sufcient for the non-emptiness of the core. In this chapter an analogous result is obtained when the set of alter- natives, W , is a compact, convex subset of Rw.11 With this assumption on the set of alternatives, W  Rw; we shall show, for a simple voting rule, ; that the core of (p); for any convex and continuous preference prole p onW; will be non-empty if and only if the dimension ofW is no greater than v() 2:We say that v() 2 = v() is the stability dimension. Thus, if dim(W )  v() 1 then a prole can be con- structed so that the core is empty and the cycle set non-empty. Indeed, above the stability dimension v(); local cycles may occur, whereas below the stability dimension local cycles may not occur. In dimension v() + 1; local cycles will be constrained to the Pareto set. In dimen- sion above v() + 1; these local cycles may extend beyond the Pareto set, suggesting a degree of chaos. 4.1 Existence of a Choice We rst show the sufciency of the dimension restriction, by consider- ing the preference correspondence associated with (p): Let W be the set of alternatives and, as before, let X be the set of 11In fact, the same result goes through when W is a subset of a topological vector space. The denitions of a topological vector space, and other notions such as open- ness, compactness and continuity are given in a brief Appendix to this chapter. 69 70 Chapter 4. The Core all subsets of W: If p is a preference relation on W; the preference cor- respondence, P; associated with p is the correspondence which asso- ciates with each point x 2 W; the preferred set P (x) = fy 2 W : ypxg : Write P : W ! X or P : W  W to denote that the image of x under P is a set (possibly empty) in W: For any subset V of W; the restric- tion of P to V gives a correspondence PV : V  V; where for any x 2 V; PV (x) = fy 2 V : ypxg : Dene P1V : V  V such that for each x 2 V; P1V (x) = fy 2 V : ypyg : The sets PV (x); P1V (x) are sometimes called the upper and lower preference sets of P on V:When there is no ambiguity we delete the sufx V: The choice of P fromW is the set C(W;P ) = fx 2 W : P (x) = g : The choice of P from a subset, V; ofW is the set C(V; P ) = fx 2 V : PV (x) = g : If the strict preference relation p is acyclic, then say the preference corre- spondence, P; is acyclic. In analogous fashion to the denition of Section 2.5 call CP a choice function onW if CP (V ) = C(V; P ) 6=  for every subset V of W:We now seek general conditions on W and P which are sufcient for CP to be a choice function on W: Continuity properties of the preference correspondence are important and so we require the set of alternatives to be a topological space. For simplicity, we can just assume that W is a subset of Rw, with the usual Euclidean topology (as dened in the Appendix to this chapter). Denition 4.1.1. LetW;Y be two topological spaces. A correspondence P : W  Y is (i) Lower hemi-continuous (lhc) iff, for all x 2 W; and any open set U  Y such that P (x) \ U 6=  there exists an open neighborhood V of x inW; such that P (x0) \ U 6=  for all x0 2 V: (ii) Upper hemi-continuous (uhc) iff, for all x 2 W and any open set U  Y such that P (x)  U; there exists an open neighborhood V of x inW such that P (x0)  U for all x0 2 V: (iii) Lower demi-continuous (ldc) iff, for all x 2 Y; the set P1 (x) = fy 2 W : x 2 P (y)g is open (or empty) inW . (iv) Upper demi-continuous (udc) iff, for all x 2 W; the set P (x) is open 4.1 Existence of a Choice 71 (or empty) in Y (v) Continuous iff P is both ldc and udc. We shall use lower demi-continuity of a preference correspondence to prove existence of a choice. In some cases, however, it is possible to make use of lower hemi-continuity. For completeness we briey show that the former continuity property is stronger than the latter. Lemma 4.1.1. If a correspondence P : W  Y is ldc then it is lhc. Proof. Suppose that x 2 W with P (x) 6=  and U is an open set in Y such that P (x) \ U 6= : Then there exists y 2 U such that y 2 P (x) : By denition x 2 P1 (y) : Since P is ldc, there exists a neighborhood V of x inW such that V  P1 (y) : But then, for all x0 2 V; x0 2 P1 (y) or y 2 P (x0) : Since y 2 U; P (x0) \ U 6=  for all x0 2 V: Hence P is lhc. We shall now show that if W is compact, and P is an acyclic and ldc preference correspondence P : W  W; then C(W;P ) 6= : First of all, say a preference correspondence P : W  W satises the nite maximality property (FMP) on W iff for every nite set V in W; there exists x 2 V such that P (x)\V = :Note that P is acyclic onW then P satises FMP. To see this, note that if P is acyclic onW then P is acyclic on any nite subset V of W; and so, by Theorem 2.5.1, C(V; P ) 6= : But then there exists x 2 V such that P (x) \ V = : Hence P satises FMP. Lemma 4.1.2. (Walker, 1977). If W is a compact, topological space and P is an ldc preference correspondence that satises FMP onW; then C(W;P ) 6= : Proof. Suppose on the contrary that C(W;P ) = : Then for every x 2 W; there exists y 2 W such that y 2 P (x) ; and so x 2 P1 (y) : Thus fP1 (y) : y 2 Wg is an open cover for W: (Note that since P is ldc, each P1 (y) is open.) Moreover, W is compact and so there exists a nite subset, V; of W such that fP1 (y) : y 2 V g is an open cover for W: But then for every x 2 W there exists y 2 V such that x 2 P1 (y) ; and so y 2 P (x) : Since y 2 V; y 2 P (x) \ V: Hence P (x) \ V 6=  for all x 2 W and thus for all x 2 V: Thus P fails FMP. By contradiction C(W;P ) 6= : 74 Chapter 4. The Core core.12 We shall obtain a generalization of Greenberg's result, by show- ing that if  is a general non-collegial voting rule with Nakamura number, v(); then if dim(W )  v() 2; and certain continuity, convexity and compactness properties are satised then  has a core. To make use of Theorem 4.1.1, we need to show that when individual preference correspondences are ldc then so is social preference. Suppose, therefore, that p = (p1; : : : ; pn) is a preference prole for society. Let  be a voting rule, and D be the family of decisive coalitions of : Let P = (P1; : : : ; Pn) be the family of preference correspondences dened by the prole p = (p1; : : : ; pn): Call P a preference (correspon- dence) prole. For any coalitionM  N dene PM : W  W by PM(x) = T i2M Pi(x): For a family D of subsets of N; dene PD : W  W by PD(x) = S M2D PM(x): If  is a BF, with D its family of decisive coalitions then clearly x 2 PD(y)) x(p)y; whereas when  is a voting rule, then x 2 PD(y), x(p)y: In this latter case we sometimes write P for the preference correspon- dence PD; where D = D:We have dened the core of (p) by x 2 Core(;W;N; p) iff /9y 2 W such that y(p)x: Thus, if  is a BF with D its family of decisive coalitions, then Core(;W;N; p)  C(W;PD): with equality in the case of a voting rule. Theorem 4.2.1. (Schoeld, 1984a; Strnad, 1985). Let W be admissible and let  be a voting rule with Nakamura number v(): If dim(W )  12Here dim(W ) can be identied with the number of linear independent vectors that spanW: Thus we can regard w as the smallest integer such thatW  Rw: 4.2 Existence of the Core in Low Dimension 75 v() 2 and p = (p1; : : : ; pn) is a preference prole such that for each i 2 N; the preference correspondence Pi : W  W is ldc and semi- convex. Then Core(;W;N; p) 6= : Note that the theorem is valid in the case  is collegial, with Nakamura number 1: We prove this theorem using the Fan Theorem and the fol- lowing two lemmas. For convenience we say a prole P = (P1; : : : ; Pn) satises a property, such as lower demi-continuity, iff each Pi; i 2 N; satises the property. Lemma 4.2.1. If W is a topological space and P = (P1; : : : ; Pn) is an ldc preference prole then PD : W  W is ldc, for any family D of coalitions in N: Proof. We seek to show that P1D (x) is open. Suppose that y 2 P1D (x): By denition x 2 PD(y) and so x 2 PM(y) for some M 2 D. Thus x 2 Pi(y) for all i 2 M: But P1i (x) is open for all i 2 N; and so there exists an open set Ui  W such that y 2 Ui  P1i (x) for all i 2M: Let U = T i2M Ui. Then for all z 2 U; it is the case that z 2 P1i (x); 8i 2 M: Hence x 2 Pi(z)8i 2 M; or x 2 PM(z); so x 2 PD(z): Thus U  P1D (x) and is open, so PD is ldc. Comment 4.2.1. Note that if P = (P1; : : : ; Pn) is a lhc prole on W then it is not necessarily the case that the preference correspondence PM : W  W is lhc. For this reason we require the stronger continuity property of lower demi-continuity rather than lower hemi-continuity. An easy example in Yannelis and Prabhakar, 1983, shows that an lhc prefer- ence correspondence need not be ldc, although, as Lemma 4.1.1 showed, an ldc correspondence must be lhc. Lemma 4.2.2. Let W be admissible and P = (P1; : : : ; Pn) be a semi- convex prole. If D is a family of subsets of N with Nakamura number v (D) = v, and dim (W )  v 2; then PD : W  W is semi-convex. Proof. Suppose, on the contrary, that for some z 2 W; it is the case that z 2 ConPD (z) : By Caratheodory's Theorem (Nikaido, 1968) there exists x1; : : : ; xw+1 2 PD (z) ; where w = dim(W ); 76 Chapter 4. The Core such that z 2 Con(fx1; : : : ; xw+1g): Let V = f1; : : : ; w + 1g: For each j 2 V; xj 2 PD(z) and so there exists Mj 2 D such that xj 2 PMj (x): Let D0 = fMj : j 2 V g: Observe that D0  D and jD0j  w+1  v 1: By denition of the Nakamura number  (D0) 6= : Thus there exists i 2 N such that i 2 Mj for all Mj 2 D: Hence xj 2 Pi (z) for all j 2 V: But z 2 Con(fxj : j 2 V g)  ConPi(z): Thus contradicts semi-convexity of Pi: Thus z 2 ConPD(z) for no z 2 W: Hence PD is semi-convex. Lemma 4.2.3. If  is a non-collegial voting rule, W is admissible with dim(W )  v() 2; and P = (P1; : : : ; Pn) is semi-convex, then C(W;P) 6= : Proof. Since  is a voting rule, P = PD: Since v() = v(D) and dim(W )  v()2; by Lemma 4.2.2, P : W  W is semi-convex. By Lemma 4.2.1 P is also ldc. By Theorem 4.1.1, C(W;P) 6= : Note that the result also holds when  is collegial. If (D) 6= , then M 2 D implies (D) M so PM(x) = T i2M Pi(x)  P(D): Thus PD : W  W satises PD(x)  P(D)(x): Just as in the proof of Lemma 4.2.3, P(D) will be semi-convex, and so there is a choice C(W;P(D)): Clearly if P(D)(x) =  then C(PD(x) =  so C(W;P(D))  C(W;PD): For a voting rule C(W;P) = Core(;W;N; p); and so Lemma 4.2.3 gives a proof of Theorem 4.2.1. As a further corollary, we obtain Green- berg's result. As a reminder, note that v(n; q) = h q nq i ; for q < n; is the largest integer strictly less than the bracketed term, q nq . 4.3 Smooth Preference 79 4.3 Smooth Preference From now on we consider preference proles that are representable by smooth utility functions. As in Denition 2.1.2, the preference relation pi is representable by a utility function ui : W ! R iff, for any x; y 2 W; xpy , u(x) > u(y): A smooth function ui : W ! R has a continuous differential dui : W ! L(Rw;R); where L(Rw;R) is the topological space of continuous linear maps from Rw to R: A smooth prole, u; for a society N is a function u = (ui; : : : ; un) : W ! Rn; where each component ui : W ! R is a smooth utility function repre- senting i's preference. We shall use the notation U(W ) for the class of smooth utility functions on W and U(W )N for the class of smooth pro- les for the society N onW: Just as with preferences, we write uM(y) > uM(x) whenever ui(y) > ui(x), for all i 2 M , forM  N: Because we use calculus techniques, we shall assume in the following discussion that W is either an open subset of Rw;with full dimension, w, or that W is identical to Rw: In the notation that follows, we shall delete the reference toW and N when there is no ambiguity. The Pareto set Pareto(M;u) for coalitionM  N is Pareto(M;u) = fx 2 W : uM(y) > uM(x) for no y 2 Wg: When  is a binary social preference function, we write (u) for the preference relation (p); where p 2 O(W )N is the underlying prefer- ence prole represented by u: In this case we shall write Core(; u) = Core(; p): When  is a voting rule, with the family of -decisive coalitions, D, then Core(; u) = Core(D; u) = \ D Pareto(M;u): With smooth preferences we make use of the critical and local “approxi- mations” to the global optima set. 80 Chapter 4. The Core Instead of regarding a preference prole p = (p1; : : : ; pn) as a primi- tive, we dene the notion of a prole of “direction gradients,” as follows. Denition 4.3.1. LetW  Rw; u 2 U(W )N ; and  be a voting rule with decisive family D. (i) For each i 2 N; let pi[u] : W ! Rw be dened such that pi[u](x) is the direction gradient of ui at x with the property that for all v 2 Rw; dui(x)(v) = (pi[u](x)  v) where (pi[u](x)  v) is the scalar product of pi[u](x) and v: Let p[u] : W ! (Rw)n be the prole (of direction gradients) dened by p[u](x) = (p1[u](x); : : : ; pn[u](x)): (ii) For each i 2 N; and each x 2 W let Hi (x) = fy 2 W : pi[u](x)  (y x) > 0g be the critical preferred set of i at x. Call Hi : W  W the critical preference correspondence of i: (iii) For each coalitionM  N; dene the criticalM -preference corre- spondence HM : W  W by HM(x) = \ i2M Hi(x): (iv) Dene the “critical” preference correspondence HD : W  W of (u) by HD(x) = [ M2D HM(x): (v) Dene the criticalM -Pareto set by (M;u) = fx 2 W : HM(x) = g: 4.3 Smooth Preference 81 (vi) Dene the critical core of (u) by (; u) = (D; u) = fx 2 W : HD(x) = g = \ M2D (M;u): (vii) Dene the local M -Pareto set, L(N; u); by x 2 L(N; u) iff there exists a neighborhood V of x such that for no y 2 V is it the case that uM(y) > uM(x): (viii) Dene the local core of (u) by LCore(; u) = LCore(D; u) = \ M2D L(M;u): Comment 4.3.1. We use the symbols  and L to stand for critical and local Pareto sets, to distinguish them from the Pareto set. Note also that the direction gradient for i (given the prole u) may be written pi[u](x) =  @ui @xi ; : : : ; @u1 @xw  x where x1; : : : ; xw is a convenient system of coordinates for Rw: Thus the prole p[u](x) 2 (Rw)n of direction gradients at x may also be repre- sented by the n by w Jacobian matrix at x: J [u](x) =  @ui @xj  i=1;:::;n j=1;:::;w : For convenience, from now on we drop reference to u and write p(x) for p[u](x) and pi(x) for pi[u](x): Note that p : W ! (Rw)n is a continuous function with respect to the usual topologies onW and (Rw)n. Standard results in calculus give the following. Lemma 4.3.1. Let u 2 U(W )N and let  be a simple voting rule with decisive coalitions D. Then the following sets are closed and are nested as indicated below.