The Beach Party Problem: An informal introduction to continuity problems in group choice *

Transcription

1 The Beach Party Problem: An informal introduction to continuity problems in group choice * by Nick Baigent Institute of Public Economics, Graz University nicholas.baigent@kfunigraz.ac.at Fax: Abstract This paper discusses the significance of results that show continuity to be incompatible with other desirable properties in the context of group choice. The discussion is based on the two person beach party problem which is the simplest setting in which these results may be expressed. It is argued that, while continuity may be justified in a class of special problems that include the beach party problem, it is not justified for more general interpretations of the beach party problem. *This paper was presented as the Presidential Address of the Central European Program in Economic Theory (CEPET) annual Workshop in Economic Theory (2002). I am grateful to all participants and to CISM for it generous hosting of the Workshop

2 1 Introduction The purpose of this paper is to provide a simple non technical introduction to the group choice problems that arise in the presence of continuity, and to discuss the significance of these problems. Simplicity is achieved by focussing on the simple case of the beach party problem, and by omitting all proofs. Significance is discussed in the context of different interpretations, variations and extensions of the beach party problem. It is argued that continuity plays an important role for group choice problems in which there must be direct preference information transfer from individuals to an entity that transforms this information into a group choice. However, it is also observed that this feature is absent in the central group choice problems in economics. Continuity was first introduced into social choice theory in Chichilnisky (1979, 1980 & 1982a), initiating that which has come to be known as topological social choice theory. However, interest in the area is confined to a small, though energetic and often enthusiastic, group of contributors. It is hoped that this paper goes some way to making this area accessible to a wider audience, whether or not they agree with the conclusions. The intuition underlying the proofs of the results in this paper may best be obtained by the diagramatic presentation in Chichilnisky (1982), and for the theorem in section 2, see the proof using integration in Lauwers (2002) or that in Baigent (2002) which, though quite long, has minimal prerequisites. An elegant proof drawing on Baigent (1984 & 1985) may be found in Candeal and Induráin (1994). Surveys of the area are available in Mehta (1997), Lauwers (2000) and Baigent (2002), and the issues are also discussed in Heal (1997). 2

3 The structure of the paper is as follows. Section 2 presents the beach party problem and a basic result. Variations and Extension of this result are given in section 3 and followed in section 4 by a discussion of the justification of continuity. Section 5 concludes the paper. 2 The beach party problem One person cannot have a party! Therefore assume two agents who wish to have a party on the shore of a perfectly circular lake. Let 1 S denote the shore of the lake. The superscript refers to the fact that this is a one dimensional set of points in the Euclidean plane 2. Later, but only briefly, it will be necessary to refer to a higher dimensional lake shore. Mostly, only the circle in figure 2.1 is required. f( s, s ) s s figure 2.1 In figure 2.1, s denotes the most preferred location of agent 1, s denotes the most preferred location of agent 2, and f( s, s ) denotes the best location for the group given these individually preferred locations. Thus, f : S S S aggregates individually preferred locations into a single group location. 3

4 One possible function would make the group location the same as the location most preferred by the first agent. That is, for all 1 1, f( s, s ) ( s, s ) S S = s. But this is a dictatorship (of agent 1) and therefore not satisfactory. If f : S S S is required to be symmetric, so that for all 1 1, f( s, s ) f( s, s ) ( s, s ) S S = then such dictatorial functions will be excluded. Such functions will be called Anonymous. Another possible function would assign the same group location to all possible lists of 1 individual preferences. That is, for a fixed s* S, for all 1 1, ( s, s ) S S f( s, s ) = s*. But this constant function is imposed in the sense that group choice does not depend in any meaningful way on individual preferences. Such functions are excluded by requiring that if all individuals have the same preferred location, then that must be the group choice. That is, for all s 1 S, f( s, s ) = s. Such functions are called Unanimous. These two simple properties have an immediate appeal, and for many problems are accepted without extensive justification. The remaining property is also widely used without recourse to extensive justification. Indeed, it is usually assumed for "technical" reasons (e.g. to permit differentiation or some other formal technique) and taken to be harmless. This property is continuity. In the present context, f : S S S is a function from a subset of one euclidean space to a subset of another euclidean space, and continuity is defined in the usual way for such functions. 1 Continuity however, is not completely innocuous, as shown by the following result. 4

5 Theorem 2.1 (Chichilnisky (1979, 1982 & 1982): There is no Continuous, Anonymous and Unanimous function f : S S S. 5

6 3 Variations and extensions Variations and extensions of the result given in previous section are presented in this section in order to provide a sufficiently wide context for the discussion in the following section. It will be seen that interest in the beach party problem is not limited to the planning of relatively frivolous social activities. From a purely formal point of view, the beach party problem is the aggregation of a finite list of points in a circle into a single point in a circle. The points in the circle have simple been interpreted as possible "locations" for a beach party. The following interpretations are closely related. Consider a point s in the circle. Such a point induces a preference ranking of all the points in the circle in the following way: s 1 S is ranked as least as high as s S 1 if and only if s is at least as close to s as s is to s. According to this interpretation, s denotes both a favorite location and a ranking of locations according to their distance from that location. Given this interpretation, the aggregation is over preferences rather than alternatives, and this is the usual sort of problem discussed in social choice theory. However, this is by no means the only possible way to associate a preference with a point on a circle. Another preference ranking of a circle's points induced by any particular point s has two indifference classes. According to such a dichotomous preference, s is strictly preferred to all other points between which there is indifference. Given the interpretation in the previous paragraph, it is clear that dichotomous preferences are a domain restriction. Extensive surveys of domain restrictions in general may be found in Gaertner (2000 & 2001). Dichotomous preferences were introduced in the context 6

7 of avoiding majority voting cycles in Inada (1969), and in the context of avoid manipulation in Brams and Fishburn (1978 & 1982).) However, it is clear that dichotomous preferences do not avoid the impossibility given in theorem 2.1. However, neither of these two preference aggregation interpretations of the problem seem to be closely related to the problems that attract most attention in economics, namely aggregations of preferences on commodity bundles. Nevertheless, there is third interpretation that achieves exactly that. Consider the case of two commodities so that the commodity space is 2 +, the non negative orthant of two dimensional real space, and preferences on this commodity space such that indifference curves are linear. Two cases of such linear preferences are illustrated in the left and middle diagrams of figure 3.1. figure 3.1 For any commodity bundle, assign an "arrow" to that point in the following way. The base of the arrow must be placed at the point representing the bundle, the direction of the arrow is perpendicular to the indifference curve and towards more preferred bundles, and the arrow is of unit length. 2 Since these preferences are linear, the arrows all have the same length and direction. It follows that one such arrow is sufficient to determine a linear preference. 7

8 Now take such an arrow from the left or middle diagrams of figure 3.1 and place its base at the center of the unit circle in the right diagram of figure 3.1. Thus, in the right diagram, the arrow pointing to the north east is obtained in this way from the left diagram and the arrow pointing to the north west is obtained from the center diagram. Alternatively, for every point on the unit circle in the right diagram, there is an arrow from the center of the unit circle to that point that determines a linear preference on the commodity space. Thus, the set of all points on the unit circle can be viewed as the set of linear preferences on commodity bundles. For this interpretation, the function f : S S S aggregates linear preferences on commodity bundles into a linear preference on commodity bundles. Now linear preferences are included in larger classes of preferences. Let P denote the set of non satiated preferences that have differentiable indifference curves. See figure 3.2 in which all arrows have the same unit length, but their directions vary in order to preserve perpendicularity with the indifference curves through their base points. 3 figure 3.2 Now consider aggregating two such preferences into group preferences by means of a function F: P P P such that, for all p, p P, F( p, p ) P is the group preference. Anonymity and Unanimity properties can be defined for functions 8

9 F: P P P in way that is similar to that used for functions f : S S S. Now if F: P P P is also continuous, then this would induce a function f : S S S that has all three properties that are known to be incompatible from theorem.2.1. Therefore the following result may be stated: Theorem 3.1 (Chichilnisky (1979, 1982, 1982) There is no continuous Anonymous and Unanimous function F: P P P. It is this theorem that has the strongest claims on the attention of economists. This section is concluded by noting two extensions of theorem 2.1 that may have interpretations of interest to economists. The first concerns the two dimensional boundaries 2 S of three dimensional lakes 3 D. Here 3 D is a ball in 3 and 2 S is its surface. Of course, this can be interpreted as a location problem on the surface of the earth. With appropriate definitions of the properties used in theorem 2.1, the incompatibility can readily be shows to hold for this case as well. It also holds for the following case. Let it is the set of points on or inside 2 D denote the punctured disk in 3. That is, 1 S with a single point removed, as in figure

10 removed point figure 3.2 Some, though not all, countries are topologically 2 dimensional disks. The UK and the USA are not, but the Czech Republic is. Call the location at the puncture "the President's House", and consider the problem of locating a garbage/toxic waste facillity. Since theorem 2.1 is easily extended to cover this problem as well, it follows that there is no continuous Anonymous and Unanimous way to locate a noxious facility when one possible location is excluded. Interestingly, if the President's House is a possible location, then procedures with all three properties do exist, though it does not follow from this that the noxious facility must always be located at the President's House! This section has shown that there exist a fairly rich set of interpretations of the beach party problem. These will provide the basis for discussing the justification of continuity. 10

11 4 Justifications of Continuity Justifications are interpretation specific. That is, a properties my be justified given one interpretation but not given other interpretations. See Sen (1977). This section seeks to clarify the type of interpretations for which continuity is justified, and those for which it is not. Consider first the way that continuity is often imposed routinely for technical reasons, such as to permit the use of formal techniques that require it. When imposed in this way, it is usually thought to be harmless. At least theorems 2.1 and 3.1 serve as a warning that continuity may be anything but harmless. 5 In some aggregation problems it rules out the joint use of two properties, Anonymity and Unanimity, that have strong substantive justifications. What about a substantive justification for continuity? Chichilnisky (1982) points out that continuity "... makes mistakes in identifying preferences less crucial". See also Kelly (1987). It is a "practical" justification. The effectiveness of this justification will depend on the interpretation of the aggregation problem. Consider the beach party interpretation with a little more interpretive structure. Suppose that there is a "party organizer" that is either a real person or a machine. Assume that the party organizer has the purely administrative role of receiving information about individually preferred locations and determining the party location using that information. Chichilnisky's practical justification depends on the possibility of imperfections in the transfer of information from agents to the party organizer. Note however, that the transfer of this information is background interpretation and not an explicit part of the formal model. 11

12 One could certainly imagine situations in which the unfortunate party organizer - administrator - does have difficulty in obtaining good clear information on where the partygoers want to whoop it up. But one could also imagine an organizer (machine or process) for which there would be no error dealing with rational partygoers. After all, given the circumference of the lake, the partygoers only need a single number (the direction of an arrow) to the party organizer who then puts this information into the formula for f : S S S. In that case, there is no practical purpose for continuity to serve. On the other hand, if for example, errors in s are possible, then the significance of mistakes will indeed be reduced in the case of continuous aggregation. Now consider a more general, and more abstract, interpretation of the beach party problem that draws heavily on the theory of mechanism design. The framework is illustrated in figure 4.1. Beginning in the top left corner, L k = P denotes the set of all lists of individual preferences, one preference for each of the finite number, k, of individuals, k > 1. The function f assigns a group preference to all such lists of individual preferences. It is very important to emphasize that this function is evaluative in the sense that the group preferences that it gives are the preferences that the group ought to have given the preferences in the list L. The function β determines a unique best alternative, tie breaking if necessary, in the set of all possible alternatives X, for all possible group preferences. Thus, the composition β f will give what alternative ought to be the case, for all possible lists of individual preferences. Thus, the entire top of the diagram in figure 4.1 illustrates outcome evaluation. In contrast, the lower left and right parts of the diagram in figure 4.1 illustrate outcome determination. 12

13 Lists of preferences: L f Group preferences: P β Outcomes: X α m m Actions: A figure 4.1 Outcomes are determined by the actions of individuals. This transformation of individual actions into social outcomes is sometimes called a mechanism, but also an institution, outcome function or game form. In any case, the function m determines an outcome for all actions in the set A of all possible actions of all individuals. 6 These actions are chosen by individuals on the basis of their preferences and in view of the function m. The function that does this is α m. That is, the composition m αm of αm and m determines what the outcome will be for all possible lists of individual preferences. This framework makes clear the dual role of individual preferences, in outcome evaluation and outcome determination. What might be called classical social choice theory, in the tradition established by Arrow and Sen, is concerned with the consistency of properties that might be imposed on f. The theory of mechanism design views f as fixed and the design issue then is the determination of a function m, that for some basis of behavior α m, brings outcome determination into line with outcome evaluation so that the diagram in figure 4.1 commutes. That is, m implements f relative to α m if and only if β f = m α. 13 m

14 The role of individual preferences in outcome evaluation is hypothetical. "If individual preferences are given by the list l L, then the group preference ought to be f( l ) and the social outcome ought to be β ( f( l)) ". In this hypothetical, or conditional, use of individual preferences, there is no transfer of preference information to anyone, and therefore mistakes cannot arise. The role of preferences in outcome determination is that individuals "use" their own preferences to choose actions. Given such self-use, there is no transfer of preference information to anyone who then determines a group outcome using this information. However, suppose that m implements f relative to α m. It follows that Now consider a special case in which actions are declared, that is revealed, β f = m α. preferences. That is, let L= A and consider a mechanism µ = β f. Assume further that α µ is the identity function on L, so that behavior always reveals correct m preferences. 7 In the theory of mechanism design, this is often described in fictional terms as though the mechanism µ is a "planner" who receives preference information messages from individuals from which the planner determines a social outcome using µ = β f. According to this story, we do have transfer of preference information from individuals to a planner. However, this story is merely a convenient fiction. Given all of the assumptions in the previous paragraph, µ = β f can be regarded as a reduced form of outcome determination that is really described by m αm. The planner is a fiction, only useful for expository story telling and, in that story, there cannot be mistakes because the underlying reality summarizes in reduced form a reality in which there are no transfers of preference information. 14

15 5 Conclusion After introducing the beach party problem (section 2), and discussing some of its variations and extension (section 3), the justification of continuity was discussed (section 4). It was argued that the justification given by Chichilnisky is only successful for problems in which there is some preference information transfer. It was concluded, however, that while preference transfer is a feature of some group choice problems, it is not a feature of those that are of central interest to economists. This does not mean that continuity cannot be justified. It just means that it requires more than currently exists in the literature and the significance of this area of social choice theory remains tentative. 1 More specifically, the topologies on the domain and range of f are induced by the usual Euclidean topologies. 2 Formally, the preference is represented by a constant unit vector field on the commodity space. 3 Since only ordinal preferences matter, the lengths of all arrows are normalized to one. 4 The indifference curves are locally linear, so the induced function approximates F at a point. 5 I owe this argument to the late Mike Martin of Essex University. 6 More specifically, A is the Cartesian product of the sets of possible individual actions. 7 In more specific models, this is proved and known as the Revelation Principle. References: Baigent, N. (2002): Topological Theories of Social Choice, forthcoming in Handbook of Social Choice and Welfare vol 2, edited by K. Arrow, A. K. Sen and K. Suzumura, New York: North-Holland. Baigent, N., A reformulation of Chichilnisky's impossibility theorem. Economics Letters 16, Baigent, N., Anonymity and continuous social choice. Joumal of Mathematical Economics 14, 1-4. Brams, S.J. and P.C. Fishburn, 1978, "Approval Voting"; American Political Science Review; Vol. 72, No. 3; September;

16 Brams, S.J. and P.C. Fishburn; 1982, Approval Voting; Boston; Birkhäuser. Candeal, J.C., Induráin, E., 1994a. The Moebius strip and a social choice paradox. Economics Letters 45, Chichilnisky, G., On fixed point theorems and social choice paradoxes. Economics Letters 3, Chichilnisky, G., Social choice and the topology of spaces of preferences. Advances in Mathematics 37, Chichilnisky, G., 1982, Social aggregation rules and continuity. Quarterly Journal of Economics 97, Gaertner, W., 2000, Domain Restrictions in vol. 1, forthcoming in Handbook of Social Choice and Welfare vol 1, edited by K. Arrow, A. K. Sen and K. Suzumura, New York: North-Holland. Gaertner, W., 2001, Domain Conditions in Social Choice Theory, Cambridge, Cambridge University Press. Heal, G., 1997a. Social choice and resource allocation: a topological perspective. Social Choice and Welfare 14, Inada, K.I., 1969, The Simple Majority Decision Rule, Econometrica, 37, Kelly, Jerry S.; 1987, Social Choice Theory: An Introduction; Berlin; Springer Verlag. Lauwers, L., Topological Social Choice, Mathematical Social Sciences, 40, Lauwers, L., 2002, A note on Chichilnisky's social choice paradox, Theory and Decision, 52(3), Mehta, P., Topological methods in social choice: an overview. Social Choice and Welfare 14,

17 Sen, A. K. (1977): Social Choice Theory: A Re-examination, Econometrica, 45,