intransitivities in multidimensional spatial voting: period three implies chaos

Intransitivities in multidimensional spatial voting: period three implies chaosAuthor(s): Diana RichardsSource: Social Choice and Welfare, Vol. 11, No. 2 (April 1994), pp. 109-119Published by: SpringerStable URL: http://www.jstor.org/stable/41106076 .

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Soc Choice Welfare (1994) 1 1 : 109-1 19 "

Social Choke adWdbre

© Springer-Verlag 1994

Intransitivities in multidimensional spatial voting: period three implies chaos Diana Richards

Department of Political Science, University of Minnesota, 267 19th Avenue South, 1414 Social Sciences, Minneapolis, MN 55455, USA

Received December 3, 1991 /Accepted August 16, 1993

Abstract. It is well known that multidimensional spatial voting can involve intransitivity and cycles, resulting in outcomes anywhere in the policy space. These results are typically referred to as the "chaos theorems". In this paper, I show that the connection between non-equilibrium spatial voting and "chaos" is not merely semantic, but is theoretic. Using symbolic dynamics, I show that if a three- cycle intransitivity among social choices exists, then cycles of all lengths greater than three are possible. This result is then used to establish the three conditions of sensitive dependence on initial conditions, topological transitivity, and dense periodic points, demonstrating the formal connection between multidimensional spatial voting and chaotic nonlinear dynamics.

1. Introduction

It is well known that social choices in a multidimensional spatial setting can involve intransitivity and cycles, resulting in outcomes anywhere in the policy space [2, 3, 8, 9, 16]. These results are typically referred to as the "chaos theorems", because of the negative interpretation that all order and stability is lost: agendas can be constructed between any two points, sequences of votes do not reach an equilibrium, and the final social choice may have little relationship to voter preferences.

In this paper, I show that the connection between non-equilibrium spatial voting and "chaos" is not merely semantic. Rather, multidimensional spatial voting and chaotic nonlinear dynamics have a theoretical connection. By viewing the social choice rule as a nonlinear correspondence, the tools of symbolic dynamics can be used to establish that the sequences generated in multidimensional spatial voting exhibit chaotic dynamics. The "chaos theorems" of multidimensional spatial voting can in fact be understood as chaotic nonlinear dynamics - implying that outcomes and agenda paths contain an implicit order in their complexity.



1 10 D. Richards

The general strategy is to use symbolic dynamics, in particular, the iterated inverse image approach [11], to establish Devaney's [4] three conditions defining a chaotic mapping. The paper begins by demonstrating that if a core does not exist for a configuration of voter preferences, then a three-cycle exists among some set of alternatives. The existence of a three-cycle is then shown to imply cycles of all lengths greater than three, which in turn leads to a specification of the dictionary of feasible sequences under the social choice rule. Using this dictionary, the conditions of sensitive dependence on initial conditions, topological transitivity, and dense periodic points can be easily established, demonstrating the formal connection between multidimensional spatial voting and chaotic nonlinear dynamics.

The paper is organized as follows. Section 2 outlines the assumptions and definitions of the multidimensional spatial voting model. Section 3 demonstrates the existence of three-cycles in classes of multidimensional social choice. Section 4 introduces and applies the iterated inverse image approach and Sect. 5 establishes the existence of chaotic dynamics in multidimensional spatial voting. Section 6 concludes .

2. Assumptions and definitions

I assume a set of η voters and an alternative space X which is Euclidean m space, i.e. X=Rm. Each voter ι has preferences over X in terms of a utility function UjiX-^R which I assume to be a monotone decreasing function of Euclidean distance from voters /' s ideal point, xr Preference is denoted as

x>iy**Ui(x)>Ui(y) (2.1)

xîyÛtixÛtiy) . (2.2)

Then, by the utility functions,

Χ>^''χ-χλ'<'^-χλ' . (2.3)

Let q be the number of votes needed to replace the status quo with a new alternative. Denote the social choice rule as q-majority rule. For example, for η

odd, q= denotes simple majority rule. Let / denote the ̂ -majority rule social choice correspondence. Denote group preference as x>q y<& C/,(x) > Ut{y) with | i | > q. Then / (x ) = { y ' y > q x} is the set of points in X that are preferred to χ by a minimum of q voters.

A core point for the ̂ -majority correspondence / is any χ e X for which /(*) = 0. A core is structurally unstable if arbitrarily small changes in voters' preferences are sufficient to make the core empty. For example, the "Plott symmetry condition" is a structurally unstable core, since any change in the voters' ideal points results in a collapse of the core [10]. The existence of a structurally stable core depends on the number of voters, the dimension of the policy space, and the size of a minimally winning coalition [18]. The important aspect of the core from a dynamical approach is that, if a core point exists, then the iteration of the social choice rule reaches an equilibrium at the core.



Multidimensional spatial voting 1 1 1

3. Three-cycles in multidimensional voting

An alternative yeX can be reached from a point χ e X if it is possible to move from χ to y by iterations of /. Let fk(x) denote the kth iterate of the point χ via /. If an χ e X can be reached from itself by k iterations of / and x±fJ (x) for 1 <j < k, then a &-cycle exists. This section contains three lemmas. First, the relationship between the existence of a core and the number of voters, the dimension of the policy space, and the ̂-rule is clarified using the Nakamura number. Second, it is shown that every multidimensional spatial voting context without a core has at least one three-cycle among alternatives. Third, if a cycle of any length k > 3 exists, then a three-cycle also exists. These results are similar to previous findings (e.g., [2, 3, 15]), but are presented using concepts needed for the subsequent results.

Definition 1. The Nakamura number, N, is the cardinality of a smallest set of winning coalitions with the property that the intersection of its members is empty [18]. Lemma 1. Ν is the least integer greater than or equal to . The existence of a core is guaranteed if m<N~2.

Proof With a ^-majority rule, each winning coalition excludes n - q voters. In order for the intersection of the winning coalitions to be empty, all η voters must be excluded at least once. Since n - q voters are excluded with each winning coalition, it takes coalitions to make the intersection empty, where

n-q n n-q must be rounded up to the next integer in the cases when is not a whole

n-q number.1 Schofield [17] proves that the core is nonempty if N>m + 2. D

For example, consider some cases where a core is guaranteed. If voters' ideal points are collinear, then the expression in Lemma 1 is always satisfied and a core always exists (e.g., [1]). The expression of Lemma 1 is also satisfied in the case of four voters in a two-dimensional policy space and simple majority rule. Therefore, the question of the existence of a three-cycle is restricted to cases where a core may or may not exist depending on the configuration of voters' ideal points, as outlined by Lemma 1.

Lemma 2. The core is empty if and only if there exists a sequence of alternatives (a, b, c) such that b ef(a), c ef(b), and a ef(c).

Proof Suppose that a three-cycle exists among three alternatives a, b, and c. Then a, by and c form ajiondegenerate triangle in Rm. Label the three perpendicular bisectors of õZ>, ~bc, and ZÏÏ as Hab, Hbc, Hca, which must intersect in a space of m - 2. Denote H*b as the open halfspace which includes b and which by construction contains a ^-majority of voter ideal points. Denote Hbc and H*a similarly, such that each halfspace contains c and a, respectively and contains a ^-majority of voters. A cycle exists among three points a, b, and c if such a partitioning is possible (Fig. 1). This partitioning creates the "pinwheel" of social choice preferences such that bef(a), cef(b), and aef(c). Therefore, the question of the existence of a three-cycle for a given configuration of voters is

1 Andrew McLennan, personal communication 7/22/93



112 D.Richards

the question of whether voters' ideal points can be partitioned into the appropriate "pinwheel" of hyperplanes.

Assume voters' ideal points are located in the policy space such that a core exists. Denote the core point as v*. Then v* e / (x) for all χ Φ ν* and / (ν*) = 0 ; the agreement set of any winning coalition of voters must contain v* or there would be a winning coalition that could beat ι;*, violating the construction of v* as a core point. Therefore, in partitioning the voters into the sets H*b, Hbc, and H*a, it must be that v* is an element of H*b9 and of Hbe9 and of H*a, implying that H*b η Hbc η Η*α Φ 0 . However, for a three-cycle pinwheel partition to exist, the hyperplanes must be such that Hfb η Hbc Φ 0 and Hfb η Hbc η H?a = 0 (see Fig. 1). Therefore, if a core exists, a three-cycle partition is not possible.

Conversely, assume voters' ideal points are located in the policy space such that the core is empty. Then the halfplanes H*b, Hbc, H*a must be able to be placed such that q voters are contained in each H+ and such that H+b η Hbc η H?a = 0 . If this were not the case, then there would be a point that must be included in every winning coalition's win set and this point would be a core. Given that there is some Hca such that H„br'Hbcr'H+a = 0 , then there must be an Hca such that the three lines intersect in a single point. Either the three lines already intersect in a single point or H~a includes all of the cone Η 'ab η Hbc. In the second case, Hca has the flexibility to move towards the vertex of the cone (towards Hca's

' - ' region), since this can only result in possibly more, rather than less, voter ideal points. Therefore, Hca can be moved until it intersects Hab and Hbc in a single point at the base of the H*bnHb*c cone. Therefore, if there is no core then a three-cycle partition is possible. D

As an example, consider the case of five voters and two policy-dimensions. For <7 = 5 or # = 4, a core is guaranteed by Lemma 1. Note that in these cases, since each H+ must contain at least q = 4 or # = 5 voters, H*bnHb+cnH+a is always nonempty. For q = 3, a core exists if the voters' ideal points satisfy the Plott symmetry condition [10]. This condition requires that, for 2«+l voters, 2 « of the voters can be paired such that the point of indifference between each pair includes the unpaired voter's ideal point. Configure the voters in a Plott equilibrium and label the central voter (whose ideal point lies at the point of indifference between each of the 2« pairs of voters) as v*. Any winning subset of voters must, by the symmetry construction of the Plott configuration, contain υ*. Therefore, in attempting to partition the voters into sets H*b9 Hbc, and H*a, it must be that ü* is an element of H*b9 Hb+C, and H*a. So it must be that HabnHbcc'H?a Φ Çd. However, for a three-cycle pinwheel partition to exist, the hyperplanes must be placed so that H^bnHbcr'H^a is empty. In order to construct the necessary empty set, Hca must cross over v*f but this results in a switching of the ' + ' and ' - ' sides of Hca, destroying the three-cycle pinwheel.

Move one voter's ideal point by an arbitrary amount from the Plott symmetry condition. By breaking the symmetry condition by any amount, a line can be placed between v* and one of the pairs of voters such that v*e H~ . Therefore lines can be placed such that H^br'Hbc nH+a = 0. For example, place Hab between z;* and one of the pairs of voters such that v*eH~b. A line Hbc can be placed such that v* e Hbc and Hbc intersects Hab arbitrarily close to ι;* (or to the point where v* used to be located). A line Hca can be placed such that v*e H?a and such that Hca intersects Hab and Hbc in a single point, creating the necessary pinwheel and identifying a three-cycle.



Multidimensional spatial voting 113

*'. ♦/

' Λ / ^ Hbc

M + Γ ' + '- V

Hca Fig. 1

Lemma 3. If there is a k-cycle, k>3, such that a = fk(a), then there is some a' b', and c' such that b' =f(a')9 C = /(£'), and a' =f(c'), i.e. that a' = f'a'' Proof. The existence of a fc-cycle implies the core is empty [2, 16]. By Lemma 2, if the core is empty then a three-cycle exists. D

4. The iterated inverse image

Given a social choice rule in R™, one would like to characterize the permissible sequences of mappings of regions. This is accomplished by examining the iterated inverse image of the social choice correspondence. The policy space is partitioned into regions, and, by examining the nested sequences in terms of target regions, one can outline sequences that are permissible under the social choice correspondence / (see [11, 13]). This approach differs from the typical approach to spatial voting in two ways. First, rather then examining the mapping of single points - as in the win set formed by the indifference curves through a single point - this approach examines the mapping of regions into regions. It is in abandoning the "point precision" of previous approaches that one can characterize the general dynamics of the social choice rule [13]. Second, the iterated inverse approach differs from previous analyses of spatial voting by focusing not on the social choice correspondence, but on its inverse. Previous approaches examine the set of points that beat an existing status quo point, namely the correspondence f(x). The iterated inverse image approach reverses this logic and examines the target region of χ - namely the set of points that map to χ via the social choice rule: f~x (x). If y is in the target region of x, then χ can follow y in a social choice sequence. The iterated inverse image approach outlines which sequences of outcomes are possible under / by examining the iterations of target regions. Each permissible sequence describes the iteration of a single point under that /. Se- quences that are identical for the first η entries and only differ after the n+ 1st symbol designate initial points which are nearby; i.e. distance between points is measured by the extent to which their sequences match. This approach allows for a general cataloging of everything that can happen under the social choice rule in terms of permissible sequences of target regions.



1 14 D. Richards

' a -4 ' 3j-

a k r Fig. 2

Rl »2

To illustrate the intuition of the existence of cycles, consider three alternatives a, b, and c that are cyclic under /. Then a maps to b, b maps to c, and c maps back to a, as in Fig. 2. Let Rx = [α,ό) and Α2 = [Ζ>, c]. Interval 1^ maps only to the interval Λ2, but R2 maps to intervals 7?χ and R2. The possibility of cyclic sequences arises because of the feedback region of R2. By iterating within R2, cycles of all lengths are possible ; the mapping back to R{ allows for the completion of the cycle. The rule on the mapping is simply that -R2 must be followed by R2; R2 can be followed by either R1 or R2.

For example, a four-cycle sequence is possible: (R^R^R^R^R^R^...). Finding this four-cycle is not obvious with the "forward" approach used in choice theory. The iterated inverse image approach [13], on the other hand, identifies where the cycle must appear. First, find all points in Rx that map to the first target region of Tx = R2; this is given by/~1(r,) = /""1 (R2) η Rl . However, the goal is not to map to any point in R2, but to the subset of points in R2 where the next iterate stays in R2. This means we need to refine the starting region of Rl to the refined target of T2 = f~l(R2)r'R2c:Tx. These points f~'T2) c /

- l (Tj ) are all the points that start in Rl , go to R2, and on the second iterate,

remain in R2. The same argument ensures that the third iterate is in R2. Here the refined

target region is T3 = / " *

(/ ~ l

(R2) η R2) c T2, and

/-'(^/-■(rjc/^aOcÄ!.

At each stage, the set of initial points is refined to include only those points that hit the refined target - this is the region that ensures that more of the steps of the proposed cycle occur. The final stage, TA = f~l(...(f~l(Rl))czT39 defines the set of inverse iterated points /

~ l (ΤΛ) c Rx . This is the set of all points in Rx

where the image of the fourth iterate is in Rl . Trivially, there is at least one point xe Rx so that fA{x)~x. This is the period four orbit.

Unlike Sarkovskii's theorem, the iterated inverse image approach is not restricted to continuous one-to-one functions. Although the previous example was a one-to-one function, a similar logic applies to the social choice correspondence in multidimensional spatial voting. As shown in Lemma 2, non-equilibrium multidimensional spatial voting results in a three-cycle from an alternative ato b, b to c, and c to a. The existence of a three-cycle creates regions of the policy space that feed back to other regions. By dividing the policy space into regions each containing a, b, or c, the rules on the iteration of regions can be specified. As in the example above, the existence of regions feeding back into other regions allows for cycles of many lengths and for the demonstration of chaos.




The first step is to define some regions. Let a, b, and c denote three alternatives in a three-cycle, with Hab, Hbc, and Hca the respective perpendicular bisectors (as in Lemma 2). Label a closed region Β bounded by a hyperplane parallel to Hab and such that beB. Define a set R (B) as the open set of all points defined by reflecting across Hab (see Fig. 3). By construction a e R (B)9 since Hab is the perpendicular bisector and beB. Regions C, R(C), A, and R(A) are defined similarly, such that ceC.be R(C), aeA, and c e R(A), as in Fig. 3. Then the following lemma holds. Lemma 4. Sequences of the type {Α,Β,Β,.,.,Α,...} are permissible under a q- majority voting rule.

Proof. Assume a sequence {Α,Β,Β,.,.,Α,...}. First we need to find those points in A that map to Τλ=Β in the first iterate. This is given by the cone f~l(7') = R(B)nA. Now we need to refine /

" 1 (7] ) to only those points that remain in Β in the second iterate. That is, we want to find the points in A that map to the refined target region of T2 = f~l(B)nBc:T1. The open cone BnHbc e f

~ 1 (B) since all points in this cone map to Β by approaching the Hbc hyperplane, so T2 = BnH^c. The same argument ensures that the third iterate is in B. Here the refined target region is T3=-f~l(f~l(B)nB)czT2. At each iteration, the set of initial points is refined to include only those points that remain in B. Then Tt_ x = /

" l (...(/ ~ * (Β) η Β)); one can stay in Β as many times

as necessary since the boundary Hbc is open. The final stage defines the set of inverse iterated points f~l (7])) a A - i.e. the set of points in A where the image of the /th iterate returns to A. This means the final target region must be refined to Ti - Hbcr'f~l{A). This is an open cone and is non-empty. Therefore one can construct a sequence that begins in A, iterates within Β for any number of iterations, then returns to Α. Π

Note, however, that the existence of a three-cycle does not imply the existence of an equilibrium point, or "one-cycle." Although region Β can follow region B9 in fact, subsequent points in Β cannot return to previous points. As seen in the proof of Lemma 4, mappings within Β must approach the open boundary of Hbc. Therefore, there is no periodic point of one. Similarly, a refinement of region A demonstrates that there is no two-cycle periodic point. The social choice correspondence is unlike the functions included in Sarkovskii's theorem [4], where a three-cycle implies the existence of cycles of all lengths. Proposition 1. If a three-cycle among alternatives exists, then there are cycles among alternatives of all periods kfor which k> 3.

Proof Lemma 4 established that sequences of the type {Α,Β,Β,.,.,Α,...} are permissible under this social choice rule. Then it is straightforward to see that cycles of all length greater than or equal to 3 can be constructed by varying the number of iterates within Β from 2 (a three-cycle) to i? = i (an i + 1 -cycle). D The following two corollaries are not new (e.g., [2, 8, 9, 16]), but illustrate how previous results follow in a straightforward way from the iterated inverse image approach. Corollary I. If a cycle exists, then cycles of all lengths k>3 exist. Proof The result follows from Lemma 3, that the existence of any cycle implies the existence of a three-cycle, and from Proposition 1 , that a three-cycle implies cycles of all lengths greater than three. D



116 D.Richards

Hab '

/

"■·, VX / /

Xlrn ' / ' ' / '

Fig. 3

Corollary 2. If a three-cycle exists, then the cycling covers the entire policy space Rm.

Proof. Construct Β as in Proposition 1 but set the boundary of Β as Hab rather than as a hyperplane parallel to Hab. Construct regions A and C in the same manner as region B. By enlarging the regions A, B and C in this manner, the cone regions are expanded and cover Rm. D

Proposition 1 focused on the lengths of cycles by cataloging permissible sequences of the type {R, RC9...,RC, R}. However, in fact, all sequences of symbols A,B, and C are possible under nonequilibrium ^-majority rule. The sequences created by the iterations among the regions are referred to as words. Then for the social choice correspondence, certain sequences are permissible under the social choice rule and other sequences are not permissible. The set of permissible sequences is referred to as the dictionary. If all words are permissible, then the dictionary is the universal set.

Corollary 3. For the partitions A, B and C, and the nonequilibrium q-majority rule social choice correspondence, the dictionary of feasible sequences is the universal set.

Proof Proposition 1 established the three components necessary for the dictionary of sequences of A9 B, and C to be the universal set. First, each region maps to the region that follows it in the three-cycle - i.e. A maps to B. Second, each region maps to a region of itself - i.e. Β maps to B. Third, each region maps against the direction of the three-cycle to the previous region - i.e. Β maps back to A although the three-cycle direction is a->b->c. Combining these permissible mappings outlines the general rules in creating permissible sequences with a ̂ -majority social choice rule :

(i) if the nth symbol is A, then the η + 1st symbol can be A, B, or C; (ii) if the nth symbol is B, then the η + 1st symbol can be A, B, or C; (Hi) if the nth symbol is C, then the η + 1st symbol can be A, B, or G

Obviously these mapping rules imply that all sequences are possible and therefore the dictionary is the universal set. D




5. Chaotic dynamics

A symbolic dynamics approach not only addresses the question of cycling, but, more importantly, allows for the straightforward demonstration of chaotic dynamics. The following definition of chaos in a dynamical system is from Devaney [4: ρ 50], with the critical terms defined in the subsequent two definitions:

Definition 2. Let V be a set. f: V-> V is chaotic on V if (i) f has sensitive dependence on initial conditions; (ii) f is topologically transitive; (Hi) periodic points are dense in V.

Definition 3. fiJ-*J has sensitive dependence on initial conditions if there exists δ > 0 such that, for any χ e J and any neighborhood Ν of x, there exists y e Ν and τι > 0 such that 'fn(x)-fn(y)' >δ. Definition 4. /:/->/ is topologically transitive if for any pair of open sets [/, Fc/ there exists k > 0 such that fk(U)nV*0. A periodic point is a point-* such that fk(x) = x for some k. Periodic points are dense if for any periodic point there is another periodic point arbitrarily close by. Proposition 2. If a social choice cycle exists among any alternatives under q-majority rule, then the social choice rule exhibits chaotic dynamics.

Proof. Consider the three conditions in terms of the rules on mappings between regions A, B, and C for the ̂-majority social choice rule. A function exhibits sensitive dependence on initial conditions if two points that are arbitrarily close separate by a distance δ by the iteration ofthat function. Let the distance between subsets of regions A and Β be δ. One must demonstrate that two points that are arbitrarily close will separate by a distance δ by the iteration of the social choice correspondence. This requires that one can construct two sequences of A, B, and C symbols which are identical for η entries and that differ on the n+ 1st entry. Assume that the nth entry of these two sequences is A, then the η + 1st entry can be A for one sequence and Β for the other, separating the points by δ and demonstrating sensitive dependence on initial conditions.

Topological transitivity requires that the mapping has points that move from one arbitrarily small neighborhood to another arbitrarily small neighborhood. Then any point that begins in A and ends in Β satisfies this condition. Such a point does exist by the permissible sequences outlined in Proposition 1.

The condition of dense periodic points requires that given any periodic point, there is another periodic point arbitrarily close by. Recall that the distance between the iteration of two points is evident by the extent to which two sequences match. Given a sequence of arbitrary length, it is possible to find another sequence that matches the beginning η entries of the first sequence, demonstrating the existence of two periodic points arbitrarily nearby. D

6. Chaos and social choice

A chaotic mapping implies that any outcome is possible. However, as seen in the rules governing permissible sequences of symbols under the social choice correspondence, this does not imply that outcomes are "arbitrary" or random in the



118 D. Richards

sense that they are unrelated to preferences. Additional constraints on outcomes become apparent as the symbolic regions are further refined.

The fact that the social choice rule is chaotic implies that there is a sensitive dependence on initial conditions. In the context of multidimensional voting, this implies that outcomes are highly sensitive to slight changes in the preference profile, rather than the dynamic interpretation in terms of forecasting sequential voting in real time. When chaotic dynamics are present, a slight change in individual-level information can radically affect the group-level outcome.

Chaotic systems are also indecomposable - i.e. the problem cannot be solved by dividing it into a hierarchy of smaller problems [4]. In the context of voting, this implies that the search for stability does not point to a division of the voting problem into smaller sub-groups, such as a hierarchy of committees. For example, although Shepsle's [19] structure-induced equilibrium divides the voters into sub- groups using a committee system, the emergence of stability depends on the jurisdictional and germaneness rules.

The value of interpreting the social choice function using chaotic dynamics ultimately depends on the insight it provides into substantive questions. For example, the use of symbolic dynamics, illustrated in Sect. 4 for the purpose of establishing chaos, can be refined to address the question of the likelihood of outcomes, extending experimental and simulation work that has noticed patterns in the distribution of outcomes (e.g., [5-7]). In addition, by cataloging the extent of instability in a mapping, one can compare the stability of different social choice rules (see e.g., [11, 12, 14] on the Borda Count). Finally, understanding the dynamic properties of the social choice mapping is the first step towards understanding the theoretical reasons for instability in multidimensional voting. Multidimensional spatial voting is only one example in a larger class of processes that aggregate micro-level information into macro-level outcomes, a category that includes voting, market mechanisms, and the behavior of organizations. A systematic theoretically-driven search for ways to reduce instability can only be accomplished after understanding the theoretical reasons why instability can emerge when individual actions or preferences are aggregated into a group outcome.

Acknowledgements. The author is indebted to Donald Saari for identifying errors in the original proof and outlining an alternative approach. Thanks also to Andrew McLennan for a very careful reading. Any remaining errors are the responsibility of the author. This project was begun while the author was a postdoctoral fellow at the Irvine Research Unit in Mathematical Behavioral Sciences at University of California-Irvine. Earlier versions of this paper appeared as technical reports IRU-MBS 91-34 and 92-05.

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intransitivities in multidimensional spatial voting: period three implies chaos

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