Sen's theorem: geometric proof, new interpretations - CiteSeerX

Soc Choice Welfare DOI 10.1007/s00355-007-0288-2 ORIGINAL PAPER

Sen’s theorem: geometric proof, new interpretations Lingfang (Ivy) Li · Donald G. Saari

Received: 7 June 2007 / Accepted: 23 November 2007 © Springer-Verlag 2007

Abstract Sen’s classic social choice result supposedly demonstrates a conflict between Pareto and minimal forms of liberalism. By providing the first direct mathematical proof of this seminal result, we underscore a significantly different interpretation: rather than conflicts among rights, Sen’s result occurs because the liberalism assumption negates the requirement that voters have transitive preferences. This explanation enriches interpretations of Sen’s conclusion by introducing radically new kinds of societal conflicts, by suggesting ways to sidestep these difficulties, and by providing insight into other approaches that have been used to avoid the difficulties.

1 Introduction Problems central to decision and social choice theory were aptly characterized with Amartya Sen’s comment (in his 1998 Nobel Prize lecture, also see Sen 1999) that a camel is a horse designed by a committee [because] a committee that tries to reflect the diverse wishes of its different members in designing a horse could very easily end up with something far less congruous, half a horse and half something else—a mercurial creation combining savagery with confusion. Expanding on his quote, a valued choice theory objective is to discover societal decision rules that avoid creating camels when horses are intended. Sen, a leader Our thanks to two referees for excellent suggestions. Saari’s research was supported by NSF grants DMI 0233798, 0640817, and DMS 0631362. L. (Ivy) Li CoB, Economics, University of Louisville, Louisville, USA D. G. Saari (B) Institute for Mathematical Behavioral Sciences, University of California, Irvine, CA 92697-5100, USA e-mail: [email protected]

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L. (Ivy) Li, D. G. Saari

in identifying subtle barriers that frustrate this objective, discovered a fundamental difficulty in his 1970 “Impossibility of a Paretian Liberal” (Sen 1970a,b, 1976, 1983, 1992). By demonstrating that it is impossible to satisfy even a surprisingly minimal aspect of liberalism when combined with the Pareto condition, Sen’s Theorem appears to identify a fundamental conflict between standard notions of rights, an assertion that has spawned an extensive literature. In fact, his result is much wider reaching as it captures central concerns across disciplines. To indicate this universality, we introduce Sen’s Theorem with an example of an academic department hiring a faculty member. A traditional way to analyze impossibility results from choice theory, such as Sen’s Theorem, is to modify the basic assumptions or impose profile restrictions until a positive conclusion emerges. Influential examples include Blau (1975); Breyer (1977); Breyer and Gigliotti (1980); Campbell and Kelly (1997); Harel and Nitzan (1987) and Seidl (1975, 1990) among many others. To more fully explore these results and the ongoing discussion of Sen’s result, it is necessary to go beyond knowing that Sen’s assumptions create a conflict to understand why they conflict. By doing so, we obtain the first general, direct proof of Sen’s assertion.1 Interesting corollaries follow from our proof. For instance, knowing why Sen’s theorem occurs enriches his seminal conclusion by identifying new, surprisingly different interpretations. Our proof shows that in order to successfully sidestep Sen’s fundamental difficulties, the intensity of other agents’ ordinal preferences must be considered. Notice how this comment differs from traditional approaches where by treating an individual’s choices as personal and private, it becomes irrelevant what others think. Also, by exhibiting the structure of Sen’s result, our proof sheds light on some of the clever ways others have proposed to avoid Sen’s conclusions. Then, by cataloguing all possible supporting profiles for a specified Sen outcome, the proof provides a template to explain the profile restrictions that are described in the literature and to even compare the likelihood of cycles created by Sen’s conditions with that of cycles coming from majority votes.

1.1 Sen’s Theorem To indicate how Sen’s result extends beyond individual rights, suppose a two-member search committee is charged with hiring one of Amy, Bill, or Cindy.2 Assume that 1 All proofs we have seen of Sen’s result use specific examples. Strictly speaking, a counterexample proves Sen’s assertion only in the specified situation; it does not address the infinite number of other possibilities involving any number of agents, alternatives, cycles, or decisive agents with any number of assigned pairs. Our proof, which handles all possible settings, appears to be the first general, direct verification of this result. 2 By identifying decisive agents with “expertise,” or “a division of labor,” or any setting where an individual makes a decision, it is easy to create a variety of such examples; indeed, for some time, one of us (DGS) has been using this as a “fun exercise” for students. Thus we were pleasantly surprised when a referee found this to be a novel break from the literature. We also thank this referee for informing us that Risse (2001) has a related example. Incidentally, by going even further, by identifying decisive agents with “forces of physics or nature,” Sen’s result can be extended to identify limitations of other processes such as multiscale analysis in engineering design.

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Sen’s theorem: geometric proof, new interpretations

each candidate’s citation index is examined leading to the following information list: Candidate Amy Bill Cindy

IQ Macro citations Micro citations Years from PhD 120 60 10 4 110 55 80 3 100 65 70 5

(1)

The rules for assembling the committee’s ranking are natural: • Unrestricted domain. Each committee member can rank the candidates in any desired manner as long as the ranking is complete and transitive. • Pareto. If everyone ranks a pair in the same manner, this common ranking will be the committee’s ranking. • Minimal Liberalism (ML), or division of expertise. Committee members were selected because of their expertise. As Garrett’s expertise is macroeconomics—an area in which both Amy and Bill prefer—it is natural to defer to Garrett’s knowledge by asserting that how he ranks Amy and Bill will be the committee ranking. Similarly, Sandy is an expert in microeconomics where both Bill and Cindy claim abilities: Sandy’s ranking of Bill and Cindy determines the committee ranking. (ML is defined in Theorem 1.) • No cycles. In order to make a decision, the committee ranking must be cycle free. By identifying this example with Sen’s Theorem, it follows that with three or more candidates and two or more committee members, no ranking rule can always satisfy these four conditions. Theorem 1 (Sen 1970a,b) Assume that each voter has a complete, binary, transitive preference ranking with no restrictions on the preferences. No decision rule that ranks the alternatives can always satisfy the following conditions: 1.

2. 3.

(Minimal Liberalism) Each of at least two agents is decisive over at least one assigned pair of alternatives. A decisive agent’s ranking of the assigned pairs of alternatives determines the societal ranking of the pairs. (Weak Pareto) If for any pair of alternatives all voters rank the pair in the same manner, then this unanimous ranking is the societal ranking of the pair. The outcome does not have any cycles.

Notice that without the transitivity assumption, Sen’s conclusion would become obvious and immediate; it would lose all surprise and even content. After all, if cyclic preferences were allowed, we could use the extreme choice where all voters have the same cyclic preferences. According to Pareto, this unanimous cyclic choice must be the societal outcome. This comment underscores the reality that the assumption mandating the transitivity of individual preferences is crucial. Sen’s assertion ensures that situations exist where the search committee cannot satisfy the specified requirements. To see this with the table in Eq. (1), Garrett prefers Amy over Bill because of their relative performances in macroeconomics; as Cindy has the best macro performance, Garrett’s ranking is C A B. Based on citations of theory papers, Sandy ranks Bill above Cindy; as Amy is so poor in this area, Sandy’s

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ranking is B C A. Using a dash to represent irrelevant information (where the decision is determined by another member), all information needed to assemble the committee ranking is in the last three columns of Member Garrett Sandy

Ranking CAB BC A

{A, B} {B, C} {A, C} AB − CA − BC C A

(2)

Committee ranking A B B C C A Sen’s theorem is demonstrated by the cyclic outcome. Obvious modifications are possible; e.g., replace a committee member with a subcommittee where decisions are made by majority vote, etc.

1.2 Outline Sen (1970a,b) asserts that his result demonstrates the inconsistency of ML with the Pareto condition, but rather than a proof, it is tacitly assumed that the cyclic societal rankings are caused by this conflict. Therefore, if it turns out that all possible cyclic societal outcomes are caused by different features, there would be a need to re-examine the literature surrounding Sen’s theorem. The assumption is wrong. Saari (1998, 2001) proved that rather than a conflict between ML and Pareto, Sen’s seminal result occurs because these two conditions require the societal rankings to be made over pairs. As we show here, the problem caused by emphasizing pairs is that it forces all connecting information among pairs— including the transitivity of individual preferences—to be totally ignored. But if a rule cannot use Sen’s transitivity assumption, the de facto setting reverts to that of the paragraph following Theorem 1 where, rather than a surprise, cyclic outcomes must be anticipated. As our proof (Sects. 2, 5) makes clear, a ML–Pareto conflict of rights cannot be the basic problem because both can be replaced with any other conditions forcing attention strictly on individual pairs, such as an “expert opinion,” and the same conflict arises. Sen explicitly requires transitive preferences because without them and the ability of a decision rule to use this information, his theorem loses all substance. But by emphasizing only what happens with specific pairs, ML and Pareto prohibit the rule from using the intended information. More generally, almost any condition that strictly emphasizes individual pairs, such as ML and Pareto, forces the decision rule to treat certain transitive preferences as being cyclic preferences. As we prove, it is this misinterpretation, rather than any Pareto–ML conflict, that causes all possible Sen societal cycles. To demonstrate this phenomenon with the search committee example, all information needed by a Sen decision rule is in table in Eq. (2). Notice that each committee member’s full ranking is irrelevant; we do not know, and it does not matter, whether Garrett ranks {B, C} as C B, which creates a transitive ranking, or as B C, which creates cyclic preferences. By concentrating on individual pairs, ML and Pareto vitiate the crucial transitivity requirement.

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a

b

Fig. 1 Cube and rankings

As a comparison, the “Years-from-PhD” and “IQ” information table in Eq. (1) columns play no role in the committee outcome, so this irrelevant information can be safely removed without affecting the analysis. Similarly, once it is proved that transitivity of preferences plays no role in Sen’s result, the transitivity assumption can be removed from Sen’s Theorem. (It is not used in table in Eq. (2).) This removal makes it questionable whether Sen’s result captures a conflict between societal and individual rights. It can, but with very different interpretations (Sect. 3.3). To determine whether “transitivity” is, or is not, being ignored, we consider all possible profiles whether transitive or not. In this way we can determine whether Sen’s assumptions force the decision rule to ignore the individual rationality restriction. We prove that the rules ignore it. 2 Notions of the geometric proof Our proof of Sen’s Theorem is in Sect. 5, but we first describe the three alternatives {A, B, C} case to develop intuition about the subtle structure of Sen’s result.3 For a specified pair, say {A, B} in Fig. 1a, let the points {0, 21 , 1} on the line interval [0, 1] represent, respectively, B A (B is preferred to A), A ∼ B (indifference between A and B), and A B. To represent all three pairs, use the unit cube (Saari 1995) where the x axis represents the described {A, B} rankings, the y axis represents {B, C} rankings where y = 1 means B C, and the z axis represents the {A, C} rankings where z = 1 means C A. This is displayed in Fig. 1b where, for convenience, the vertices are labeled 1–8. The rankings associated with the eight vertices are Vertex Ver. no. Ranking (1, 1, 0) 1 ABC (1, 0, 0) 2 AC B (1, 0, 1) 3 CAB (1, 1, 1) 7 B C, C A, A B

Vertex Ver. no. (0, 0, 1) 4 (0, 1, 1) 5 (0, 1, 0) 6 (0, 0, 0) 8 B

Ranking CBA BC A B AC A, A C, C B (3)

3 A sketch for this proof first appeared as Li’s answer for an exam problem while she was a student in a

course taught by Saari. Li’s outline follows Saari’s proof (2001) of Arrow’s Theorem.

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Vertices 7 and 8 represent cyclic rankings while the other six represent transitive rankings. Notice that the two vertices on a cube edge represent rankings that differ only by the ranking of a single pair: e.g., the rankings assigned to vertices 1 ( A B C) and 2 (A C B) differ only in the {B, C} ranking: they must because the connecting edge is parallel to the y axis, which is the {B, C} ranking direction. So, opposing vertices on an edge reverse the indicated binary ranking. Also notice the geometric connections among transitive and cyclic rankings. Each cyclic vertex is adjacent (connected by an edge) to three transitive rankings; e.g., 7 is adjacent to 1, 3, and 5. Each transitive ranking is adjacent to a cyclic one; e.g., a vertex with an “odd” name is adjacent to vertex 7, a vertex with an “even” name is adjacent to vertex 8. A cube edge has at most one cyclic vertex. If an edge connects two transitive vertices, the binary ranking change involves adjacently ranked alternatives; e.g., vertices 1 and 2 differ in A (B C) and A (C B). But an edge connecting transitive and cyclic rankings changes a binary ranking for non-adjacent alternatives in the transitive ranking—they are separated by another alternative; e.g., vertices 3 and 7 differ in the {B, C} ranking where 3 has C A B, where A separates C and B. This adjacency geometry, which extends to any number of alternatives, plays a major role in our analysis.

2.1 Creating an example To illustrate how to use the geometry to prove Sen’s Theorem, use the decisive choices from the search committee example but assume only that each agent ranks each pair; i.e., do not assume transitive preferences. Being decisive over {A, B}, Garrett’s choice forces the outcome to be either on the front (x = 1) face of the cube with vertices {1, 2, 3, 7} for the A B ranking, or on the back (x = 0) face with vertices {4, 5, 6, 8} for B A. With Sandy’s {B, C} choices, she can select the y = 1 side face representing B C, or the y = 0 side face representing C B. These choices determine the societal outcome for two pairs: as the societal outcome must be on both of the selected faces, it is represented by one of the two vertices on the edge between the two selected faces. There are four possible cases; for each, the two selected faces are connected by a vertical edge (representing the remaining pair {A, C}) that is identified by its two vertices. Garrett Sandy Edge vertices Garrett Sandy Edge vertices (4) x =0 y=0 4, 8 x =1 y=1 1, 7 x =1 y=0 2, 3 x =0 y=1 5, 6

According to the table in Eq. (4), two of the edges (bottom row) have a transitive societal outcome (with either vertex) independent of whether the agents have transitive or cyclic preferences and independent of the societal {A, C} ranking. The edges on the top row allow a cyclic outcome with vertex 7 or 8. What remains is to select an appropriate {A, C} ranking to obtain the cyclic conclusion.

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To ensure that the vertex 7 cyclic societal outcome accompanies the x = 1, y = 1 choices, introduce any condition4 that places the societal outcome on the z = 1 surface. Sen offers two choices: Pareto, or assign the {A, C} pair to a decisive agent. With the Pareto condition, the desired z = 1 societal ranking requires both Sandy and Garrett to have a z = 1 (C A) choice. With the second option, a z = 1 societal outcome requires only one person, the decisive agent, to have the z = 1 (C A) ranking. Similarly, for the vertex 8 societal conclusion, supplement the x = 0, y = 0 choices from decisive agents with any condition to force a z = 0 outcome. While the transitivity of individual preferences is not used, the geometry ensures there always is a supporting profile with transitive preferences for either the vertex 7 or 8 cyclic outcome. To see why, notice that the societal outcome is sole vertex on all three selected faces. Each agent’s preferences are on only those cube faces that the agent helped select. So, with two or more decisive agents, at most two specified faces are assigned to any agent; i.e., the agent’s admissible preferences are the two vertices on the edge that connects these two faces. But on any cube edge at least one vertex represents a transitive ranking; select this ranking for the agent. With Garrett and the vertex 7 outcome, his preferences are one of the vertices on the edge connecting the x = 1, z = 1 faces, or vertices 3 and 7 where vertex 3 represents transitive preferences. The geometry, then, always ensures a supporting transitive profile. The following table lists all supporting profiles when the z outcome is determined by the Pareto condition. Vertex outcome Garrett Garrett’s edge Sandy Sandy’s edge 7 x = 1, z = 1 3, 7 y = 1, z = 1 5, 7 8 x = 0, z = 0 6, 8 y = 0, z = 0 2, 8

(5)

According to the top row of table in Eq. (5), the cyclic societal outcome of vertex 7 is supported by four profiles where in each pair Garrett’s preference is listed first: {(3, 5), (3, 7), (7, 5), (7, 7)}. Only the (3, 5) choice endows each agent with a transitive preference. Similarly, all possible supporting profiles with a vertex 8 cyclic societal outcome come from the bottom row; they are {(6, 2), (6, 8), (8, 2), (8, 8)} where only (6, 2) represents transitive preferences. 2.2 Observations The full three-alternative proof of Sen’s result modifies the above to handle all choices for which agents are decisive, over what pairs, and whether the Pareto condition is used. A similar proof (Sect. 5) for any number of agents and alternatives uses the hypercube geometry. The only role of transitive preferences is that one can always be found among the supporting profiles; i.e., the individual rationality assumption is a bystander rather than a significant participant in Sen’s theorem. 4 For example, replace the Pareto condition with the majority vote over a pair, an “expert’s opinion,” or with a host of other possibilities. Indeed, for any number of alternatives and agents, a similar analysis shows that the Pareto condition is not needed to obtain a Sen-type conclusion. Any rule, which ranks a pair independent of the choices made by the decisive agents and preferences over other pairs, suffices. ML can be similarly replaced.

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The ML properties needed for the proof are that (1) rankings are assigned for specified pairs independent of how other pairs are ranked, and (2) each agent’s preferences are not involved in the ranking of at least one pair. (While not crucial, the second condition makes it easy to ensure a transitive supporting profile.) ML can be replaced with any requirement satisfying these properties and a Sen conflict will arise; i.e., these two properties and the geometry are the cause of Sen’s conclusion. Indeed, this structure is what permits the Batra and Pattanaik (1972) nice extension of Sen’s conclusion to group rights, and more is possible. Pareto is more restrictive by requiring everyone to have the same ranking, so replacing Pareto with other conditions, such as a decisive agent or a group’s choice, makes it even easier to generate cyclic outcomes. For other conclusions that come from the geometry, notice that if a pair’s societal ranking is determined by the Pareto condition, all agents have their preferences on this hypercube face. But an outcome determined by a decisive agent requires only this agent’s preferences to be on the face. Thus many other supporting profiles are permitted because the other agents’ preferences need not be on this face. This observation leads to a geometric proof (Sect. 5) of the following assertion. Theorem 2 (Saari 2001) Allow voter preferences to include all ways to rank pairs, rather than just transitive rankings. Any example of cyclic societal rankings created by decisive agents and the Pareto condition includes as a supporting profile the unanimity profile where each voter’s preference agrees with the cyclic societal ranking over these pairs. Recall the seemingly outlandish comment following Theorem 1 that if all voters had the same cyclic ranking, then the Pareto condition ensures that this cyclic ranking is the societal outcome. Theorem 2 proves that rather than an extreme comment, this setting of unanimously selecting a cyclic ranking must accompany all possible Sen cycles. In turn, this result now makes it easy to construct all possible examples illustrating Sen’s Theorem (see Saari 2001, 2008), including the interesting ones in (Salles 1997). Namely, for any collection of cycles, assign each agent’s preferences to be this desired societal outcome. For each cycle, assign pairs to decisive agents in a manner that reversing other voters’ rankings of this pair can make their preferences transitive. As this approach holds for all possible examples illustrating Sen’s result, Theorem 2 shows that Sen’s conditions force a decision rule to ignore the individual rationality assumption. Instead, the rule reflects the wishes of non-existent voters with cyclic preferences (Saari 2001, 2008; Saari and Petron 2006). 3 How often and why do societal cycles occur? To exploit our geometric proof, which identifies why Sen’s result holds, we now use this geometric structure to obtain other conclusions. For instance, as the geometry identifies all possible supporting profiles for any Sen outcome, it now is possible to discuss profile restrictions and to indicate why other attempts to sidestep Sen’s difficulties succeeded. A first direction for this discussion is motivated by the cycles that arise in pairwise voting; how serious are they? A way to address this question is to compute the likelihood of their occurrence. For the same reason, a natural question

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is to compare the likelihoods of majority vote cycles and cycles coming from Sen’s assumptions. 3.1 Likelihoods Table in Eq. (4) identifies all supporting profiles for all possible societal conclusions, so it serves as a template to construct examples that support any societal outcome, whether it is a transitive or cyclic ranking. By making assumptions about the likelihood of each binary choice, we can determine the likelihood of each societal outcome. The conclusion is that societal cyclic outcomes are distinctly less likely to occur than transitive ones, even when voters have cyclic preferences. Theorem 3 With three alternatives and n agents (n ≥ 2), suppose there are two decisive agents where each is decisive over precisely one designated pair. Let the voters independently rank each pair, but not necessarily in a transitive manner, where each of the pair’s two rankings is equally likely. If the Pareto condition is satisfied for the pair not assigned to decisive agents, the conditional likelihood of a transitive ranking is 34 , and that of a cyclic outcome is 41 . If each voter’s preferences are given by strict transitive rankings, where each is selected independently and with equal likelihood, then, subject to the condition that the Pareto condition is satisfied for the one pair, the likelihood of a transitive societal outcome is 89 while the likelihood of a cyclic outcome is 19 . With three decisive agents, each decisive over a different pair, the likelihood of a cyclic outcome is 41 ; this is true whether the voters can only rank pairs, or have transitive rankings. We found it surprising that, for individual transitive rankings, cycles become more than twice as likely by replacing Pareto with a decisive agent; this result reflects the restrictive role of Pareto. Another surprise is that, even with cyclic preferences, cyclic behavior is not prominent. Nevertheless, the 25%, and 11.1% levels, significantly exceed the 5.6–8.8% likelihood of cycles with majority voting (Riker 1982; Jones et al. 1995). (The reason is that ML imposes a more relaxed constraint on profiles than majority votes, so it permits larger likelihoods to arise.) Similar statements holds for any number of alternatives and decisive voters. The proof of Theorem 3 is in Sect. 5; intuition why these assertions hold follows from the table in Eq. (4). Our main goal is to understand what causes Sen’s result. Because our proof identifies all supporting profiles, a way to accomplish this goal is to characterize all settings with transitive and with cycle outcomes. We do so by using level sets; i.e., the set of all profiles supporting a specified societal outcome. To indicate can be learned by this approach, notice that if cyclic preferences are not in the level set, then cyclic preferences do not influence that kind of outcome. Conversely, if cyclic preferences dominate the level set, then it is arguable that they play a major role. So a way to analyze our assertion that ML and Pareto undercut the assumption of transitive preferences is to determine which level sets contain cyclic preferences.

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Proposition 1 With three alternatives and two agents, assume each is decisive over a pair and they agree on the remaining pair’s ranking. For two of the six possible transitive outcomes, all supporting preferences are transitive. For four of the six transitive outcomes, half of the supporting profiles have both agents with transitive preferences, and half have one with transitive and the other with cyclic preferences; i.e., a transitive societal outcome cannot be supported by a profile consisting strictly of cyclic preferences. With a cyclic outcome, three-fourths of the supporting profiles include a voter with cyclic preferences, and both rankings are cyclic for one profile. Similar results holds for any number of alternatives, etc., but the approach developed next is an easier way to understand why an emphasis on pairs causes Sen’s result. 3.2 Interpretation As Proposition 1 shows, transitive societal outcomes can be associated with transitive profiles because 13 (1) + 23 ( 21 ) = 23 of the time both decisive voters have transitive preferences; for the remaining one-third, at least one decisive voter has transitive preferences. A transitive societal outcome, then, appears to be related to transitive preferences. As we prove below, this is the case. The situation changes with a cyclic outcome: only “outlier profiles” (one-fourth of all possible supporting scenarios) endow all agents with transitive preferences. In contrast, half of the supporting profiles have at least one cyclic voter and one-fourth require both voters to have cyclic preferences. These comments suggest that Sen’s cyclic outcomes occur because, by ignoring the individual rationality assumption, the rule reflects the views of non-existing cyclic voters rather than the actual transitive ones. This interpretation differs significantly from that found in the literature, so it must be made precise, and this is done next. To support these assertions, we modify notions that Saari and Sieberg (2001) developed to explain all pairwise voting paradoxes. Saari and Sieberg proved that majority votes over pairs accurately reflects the average of all supporting profiles, but not necessarily the actual profile. This means that “paradoxes” and voting cycles occur when the specified profile is an outlier; i.e., the rule essentially ignores those profiles that deviate from the average of all supporting profiles. Similarly, as we show, each Sen rule outcome represents the average of all supporting profiles. This means that a Sen cycle is where this average treats the actual profile as a non-representative “outlier.” To introduce our modification of the Saari and Seiberg argument, let vertex 1 be the societal outcome with Garrett’s decisive A B choice and Sandy’s B C: both have the A C ranking. Thus for a decisive agent, the societal outcome specifies the rankings for two of the three pairs. This agent’s set of supporting preferences, then, consists of the two vertices on the edge connecting the Pareto face and the face the agent selected; these vertices represent the two choices the agent can select for the remaining pair. Recall, the flexibility arises because this pair’s societal outcome is determined by another agent’s decisive choice. For instance, Garrett’s preference can be either vertex 1 or 2 that are on a {B, C} edge as the {B, C} societal outcome is determined by Sandy; Sandy’s preference is from {1, 6} along an {A, B} edge. All of this is indicated by the bullets in Fig. 2a, which is the bottom

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a

b

Fig. 2 Explaining Sen’s outcomes

face of the Fig. 1b cube. Subdivide the square into four smaller squares; let a point’s ranking be that of the closest vertex; i.e., points in the bottom right-hand small square have the ranking of vertex 1. Of the four possible profiles, (1, 1), (1, 6), (2, 1), (2, 6), the societal outcome is conclusively supported by (1, 1) with its unanimous support of vertex 1. Only one voter in each of the next two profiles prefers vertex 1, but the outcome remains reasonable. The outlier (2, 6) profile, however, might raise doubts about the vertex 1 (A B C) conclusion. For instance, why not have the outcome of vertex 8? To handle a voter’s unknown preferences, replace them with the average of all possible choices. In Fig. 2a, Garrett’s average for B C and C B is B ∼ C; this is denoted by the triangle on the Fig. 2a bottom edge. Similarly, averaging Sandy’s two choices for the {A, B} ranking leads to A ∼ B—the triangle on the right edge of Fig. 2a. The “average of the averaged rankings” is the dark square; it averages the “averaged points” (the triangles) of both agents. Using coordinates, Garrett’s Fig. 2a coordinates are (1, 0, 0) and (1, 1, 0), so the averaged value is 21 (1, 0, 0) + 21 (1, 1, 0) = (1, 21 , 0); Sandy’s averaged coordinates are ( 21 , 1, 0). The averaged outcome of the two averaged rankings is 21 (1, 21 , 0) + 21 ( 21 , 1, 0) = ( 43 , 43 , 0), which is the A B, B C, A C rankings of vertex 1; this is, of course, the average of all supporting profiles; i.e., ( 43 , 43 , 0) = 41 [(1, 0, 0) + (1, 1, 0) + (1, 1, 0) + (0, 1, 0)]. As indicated by Fig. 2b, for any societal outcome, the same geometric analysis can be used to identify all supporting profiles and their average. While the geometric shape of the set of supporting profiles remains unchanged, differences in the role played by transitive preferences arise with changes in the names of vertices; e.g., with a cyclic outcome, which happens by reversing “1” and “8” in Fig. 2a, the “doubtful outlier” is the only profile with transitive preferences. The next definition extends this construction to any number of agents, pairs, assignment of decisive agents to pairs, etc.; the Fig. 1b cube is replaced by a hypercube to represent all pairs. Set K j represents assigned profiles for the jth agent; they will be all profiles that support a societal outcome. The rest of the comments capture the “average of the averages” statement. Definition 1 For n specified pairs, the orthogonal cube in Rn is the cube defined by the 2n vertices with coordinate entries all 0’s or 1’s. A coordinate direction designates the rankings for a particular pair; 0 and 1 represent the pair’s two strict rankings. The

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rankings assigned to a point in the orthogonal cube are those of the cube’s closest vertex. Let K j be a set of k vertices of the cube; K j represents k possible strict rankings of the n pairs for the jth agent. The agent’s “averaged preference ranking” is the average of the k vectors. If a vector representing a ranking, or an averaged ranking, is assigned to each of a agents, then the “averaged ranking over the agents” is the average of the a specified vectors. As indicated in Fig. 2b, the Fig. 2a geometric argument explains all societal outcomes—even a cyclic one. Rather than a ML and Pareto conflict, a Sen outcome must always agree with the average over all possible supporting profiles (Theorem 4; Proof in Sect. 5). Consequently a Sen cycle means that the rule essentially ignores the specified profile with transitive preferences; instead, it reports the averages of all supporting profiles where most are cyclic. Theorem 4 For any number of alternatives, any specified number of decisive agents where each is assigned a specified number of (distinct) pairs, let S represent the portion of a decision rule’s societal outcome that is determined by the choices made by the decisive agents and the Pareto condition. For any societal outcome, one supporting profile is the unanimity profile (which need not be transitive) where each voter’s preference (for the S portion of the rankings) agrees with the societal outcome. This S portion of the societal outcome also coincides with the ranking determined by the average of all supporting profiles (with no assumption of transitivity). The message is that a Sen rule behaves precisely the same with a transitive as with a cyclic outcome; in all cases, the rule merely reports averages. Thus Theorem 4 means that the decision rule does not involve the crucial assumption of individual rationality; with a cyclic outcome the actual profile with transitive preferences is statistically dismissed as an unlikely outlier because the outcome is determined by the cyclic preferences of non-existent (by Sen’s assumption) voters. 3.3 Individual liberties? The source of cyclic outcomes is that the structure of ML and Pareto forces each pair’s outcome to be determined independent of the outcomes for other pairs and to ignore the explicit individual rationality assumption. Thus ML and Pareto can be replaced with any number of other conditions, which need not involve “rights,” and reach the same conclusion. For instance, each decisive agent could be replaced with a decisive committee with outcomes determined by majority vote. Nevertheless, Sen’s conditions reflect actual behavior: people do make individual decisions and there is a tendency to respect the Pareto condition. As “rights” form a special but important case of a broader behavior, it is worth reinterpreting ML and Pareto in the light of the above analysis. In doing so, Saari and Petron (2006) found surprisingly different interpretations of Sen’s result: rather than individual liberties, at times Sen’s cycles more accurately model a contentious, maybe dysfunctional, society where everyone is in strong conflict with someone else. Thus it is interesting to

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determine when a Sen outcome captures individual liberties or a contentious society. Our argument uses the following definition. Definition 2 (Saari 2001; Saari and Petron 2006) In a transitive ranking where alternative X is ranked above Y, let [X Y, α] represent the relative ranking where α designates the number of alternatives in the transitive ranking that separate X and Y . A strict societal ranking for a pair of alternatives, (X, Y ) determined by a decisive agent imposes a strong negative externality on another agent if that other agent’s transitive preference ranks the pair in the opposite manner, say [Y X, α], and the alternatives are separated by at least one other alternative, that is α > 0. Sue’s decisive B C does not create a strong negative externality for Ann’s C B D A preferences as Ann’s [C B, 0] ranking has α = 0. But Jane, with a C A B D ranking, or [C B, 1], does experience a strong negative externality. In the search committee example, Garrett’s [C B, 1] ranking displays strong disagreement with Sandy’s B C choice; Sandy’s [B A, 1] ranking demonstrates the strong negative externality caused by Garrett’s A B choice. As shown next, with all possible Sen cycles, everyone strongly disagrees with someone else. The second part of Theorem 5 is in (Saari 2001) and (Saari and Petron 2006) but a new geometric proof is given here. The first part, which we need to interpret transitive societal outcomes, is new. For other views on negative externalities, see Sen (1970a,b); Fine (1975) and Salles (2000). Theorem 5 For any number of agents with transitive preferences, let two or more agents be decisive where each is assigned at least one (distinct) pair of alternatives. If the S part of a societal ranking is transitive, there are transitive supporting profiles where no agent suffers a strong negative externality due to choices made by the decisive agents. If S has cycles, for each S cycle, each agent suffers a strong negative externality for a choice made by a decisive agent. With a transitive societal outcome, then, there is no apparent reason to question whether minimal liberalism models individual liberties. Either all agents agree, or disagreements need not be severe because nobody needs to suffer a strong negative externality, and the average of all profiles coincides with the transitive outcome (Theorem 4). But with a cyclic societal outcome, everyone must suffer a strong negative externality imposed by someone else, so we must wonder whether this contentious society is caused by decisive agents abusing their absolute freedom of choice. This universal suffering of strong negative externalities allows very different, new interpretations. This notion of strong preferences captures some of the structure behind Theorem 5 and Proposition 1, and it suggests how to design profile restrictions. For instance, prohibiting profiles with strong negative externalities prevents cycles. Of particular interest is the information we obtain about how different agents view the alternatives. To explain, in designing an agent’s profile, the choices are vertices on an edge where the edge direction is determined by a different decisive agent; e.g., with our hiring example, once a societal outcome is selected, Garrett’s only remaining choice is to select a ranking for the {B, C} pair over which Sandy is decisive. One of these vertices

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must be the societal outcome, which lists the binary choices of the decisive agents and Pareto. If the societal outcome is a cycle, changing that {B, C} ranking makes the Garrett’s full set of binary rankings transitive. But recall the edge geometry described in Sect. 2; reversing this pair’s ranking to make it transitive requires separating the {B, C} alternatives in Garrett’s ranking by another alternative, which is A; this separation creates the strong negative externality. In contrast, the two alternatives in the {A, B} pair over which Garrett is decisive are not separated. This leads to the interesting observation about three alternatives that with a cyclic outcome, the decisive agent always has weaker feelings (i.e., α = 0) about his assigned pair than the agent suffering a strong negative externality. A distinctly different relationship occurs with a transitive societal ranking where the other edge vertex is transitive. Here the reversed pair has adjacently ranked alternatives in both rankings; e.g., A (B C) changing to A (C B). The interesting feature arises by changing the ranking of the pair that is different from that assigned to the decisive agent; it creates a ranking where the alternatives in the decisive agent’s assigned pair now become separated; i.e., the decisive agent now is the one with a strong opinion (α > 0) about the assigned pair. In other words, with a transitive societal outcome, some decisive agents can have “strong” feelings about their decisive pair (e.g., Garrett has [A B, 1]) that can never arise with cyclic outcomes, while the other agents need not have strong feelings about the decisive agent’s decision. With transitive outcomes, then, either the profile represents unanimity, or at least one decisive agent has strong feelings about her assigned pair. Similar results hold for any number of decisive agents, etc., but as the geometry replace edges with faces, the extra vertices introduce more options. 4 Sidestepping Sen-type consequences The mathematical structure identifies three ways to avoid Sen’s conclusions. 1. Reintroduce the lost individual rationality information into the analysis. 2. For each cycle, find a reason to change one or more of the binary outcomes to make that portion non-cyclic. 3. The geometry captures differences in the structures of transitive preferences with a transitive and with a cyclic outcome of the kind described in the paragraphs preceding this section; use these differences to develop profile restrictions. The next two easily proved results are intended to illustrate this structure. Theorem 6 If a decisive voter’s ranking of a pair imposes a strong negative externality on another agent, replace the decisive voter’s choice with a tie ranking. The portion of the societal outcome due to this modified form of decisive voters and the Pareto condition does not create cycles. This result reintroduces aspects of the transitivity of preferences (#1) by using the strong negative externality condition, which involves the opinions of others, and it selectively chooses which binary outcome to change (#2). While this approach ensures a non-cyclic outcome, it need not be transitive; e.g., with the search committee example, Garrett’s A B choice creates a strong negative externality for Sandy with her

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B C A preferences, so replace Garrett’s choice with A ∼ B. Similarly, replace Sandy’s B C choice with B ∼ C. The societal ranking of C A, A ∼ B, B ∼ C is quasi-transitive, but it leads to the hiring of Cindy. The next theorem also combines #1 and 2 by developing criteria to determine which binary ranking to change; it selects the decisive agent whose action creates the most egregious violation of others. Each agent suffering a negative externality [X Y, α] casts a ballot listing the α value. Arguably, the decisive agent with the largest tally causes the strongest problem, so restrict this agent’s rights. Theorem 7 If a decision rule satisfying Sen’s conditions creates a societal cycle with the actions of decisive agents and the Pareto condition, then the actions of at least two decisive agents are involved. For the choice made by each decisive agent in the cycle, sum the α values for all voters with a negative externality [X Y, α]. Identify the decisive agent with the largest tally, where ties are broken in some manner. For the societal outcome of that pair, replace that agent’s ranking with indifference. The societal outcome will not have a cycle. To illustrate with an example, suppose the information is Person Preferences {A, B} {B, C} Anne ABC D AB BC AB − Barb C D AB AB − Connie C D AB Outcome AB BC

{C, D} {A, D} CD − CD DA CD − CD DA

(6)

The {B, C} choices have both Barb and Connie with [C B, 2], so the level of discontentment with Anne is 4. For the {A, D} choices, Anne has [A D, 2], but Connie agrees with the choice. Anne had the highest level of complaints, so, in determining the societal ranking of {B, C}, replace Anne’s preferred outcome of B C with B ∼ C to obtain the non-cylic societal A B, B ∼ C, C D, D A outcome. An explanation why we use indifference rather than reversing the pair’s ranking is given below. While this Theorem 7 procedure may never be used as specified, it reflects what happens in practice. For sound abatement laws to be enforced, for instance, someone must complain; e.g., making a complaint involves a cost, so reporting the noise is a strong negative externality. If complaints are lodged against several individuals, enforcement priorities need to be established. Theorem 7 suggests acting against the agent who received the most and loudest complaints. Some clever results in the literature that were discovered by discussing “rights” also satisfy these characteristics. For instance, Sen (1976) introduced a “rights-respecting” individual, Breyer and Gigliotti (1980) created an “empathy” condition, and Blau (1975) designed a “non-meddlesome” condition; all involve #1 and 2. Seidl (1990) (also Seidl 1975) accomplishes much of #3 with interesting trading mechanisms; e.g., with a cyclic outcome, each agent has a weak preference over his assigned pair but strong preferences over the other agent’s pair making a trade feasible; the stronger preferences over new pairs ensure a transitive outcome. Another result capturing #3 is Breyer’s (1977) extreme liberalism condition requiring a person to have

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“overwhelming concern for the issue he or she controls.” To define an “overwhelming concern,” notions of the Definition 2 kind are needed, which reintroduces some of the transitivity of voter preferences. From a geometric perspective, recall from the description following Theorem 5 that with a cyclic outcome, an agent’s preferences for his or her decisive pair never can be extreme, but they can for a transitive outcome. In general, to sidestep Sen’s problems, an approach must reflect that Sen’s problems occur because over the societal rankings determined by decisive agents and the Pareto condition, the decision rule never uses information about the transitivity of individual preferences. All of the above conditions reintroduce, even if unintentionally, levels of transitivity back into consideration; some do so by including information about what “others” think of the decisive agents’ actions. It remains to explain why Theorems 6 and 7 use indifference rather than reversing the ranking of a pair. The reason, which uses the explanation for the loss of transitivity, is that determining a strict societal ranking of a pair independent of what happens with other pairs need not resolve the difficulty. Even worse, a transitive societal ranking could be converted into a cyclic one! For an example, let Ann, Barb, and Carl be decisive over, respectively, {A, B}, {B, C}, and {A, C} where everyone except David, has the A B C preference ranking; David prefers C B A. The transitive societal outcome of A B C imposes negative externalities of [C B, 0], [B A, 0] and [C A, 1] for David. If a rule reversed an outcome should someone suffer a strong negative externality, then disgruntled David’s choices would convert the transitive A B C into the cyclic A B, B C, C A. Our final point, then, is that to avoid societal cycles, a pair’s societal ranking must be coordinated with what happens with other pairs.

5 Proofs General proof of Theorem 1: three alternatives A general proof of Sen’s result requires handling all possible choices for which agents are decisive, over what pairs, and whether the Pareto condition is used. Sen’s assumptions for three alternatives and any finite number of agents require each of at least two agents to be decisive over separate pairs. (If decisive over the same pair, it would be trivial to create a contradiction.) Start with two agents each decisive over a pair. Using the geometry of the cube, for each assigned pair, the decisive agent selects one of two opposing cube faces. The societal outcome must be on each selected face, so it is one of the two vertices on the edge connecting the two faces selected by the decisive agents. The four possible pairs of faces define four edges that are parallel to each other. Each edge is defined by two vertices: each vertex represents the outcome depending on how the remaining pair of alternatives is ranked. In any coordinate direction, the cube has four parallel edges; e.g., there are four cube edges parallel to the x-axis. Each of the cube’s eight vertices is on precisely one edge. Thus, each cyclic vertex is on one of the four described edges. As the cyclic vertices are diametrically opposite one another, both cannot be on the same edge. For each decisive agent, select a ranking for his assigned pair of alternatives (equivalently,

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for each agent, select the appropriate cube surface) so that the pair of selected faces defines an edge with a cyclic vertex. Next, determine the societal ranking for the remaining pair; i.e., select one of the two vertices of the identified edge. If one of the two agents (or a third agent) is decisive over the remaining pair, select the agent’s ranking for this pair so that it requires the cyclic vertex. If no agent is decisive over the remaining pair, then use the Pareto condition: select each agent’s ranking for the remaining pair to force the cyclic vertex ranking. It remains to find transitive preferences that support this societal outcome. If an agent is decisive, the societal outcome is represented by one of the vertices on the cube face she selected. Likewise, if a ranking for the societal outcome is determined by the Pareto condition, each agent’s ranking is on that face. An important point is that at most two conditions are imposed on any agent’s preferences; e.g., if an agent is decisive over two pairs, then the above construction specifies his preferences for two pairs (i.e., for two faces). Thus his preferences must be one of the two vertices along the edge defined by these particular cube faces. Similarly, if an agent is decisive over one pair and shares a common (Pareto) ranking with another pair, these choices define two surfaces, so her preferences are represented by one of the two vertices on the defined cube edge. Finally, for a non-decisive agent, either there are no restrictions on his preferences (when others are decisive over all three pairs), or his preferences are restricted to the four vertices on the cube face mandated by the Pareto condition where everyone agrees on the ranking of a pair. These vertices catalogue all possible supporting profiles. The societal outcome is the vertex surviving the intersection of the specified cube faces, so one of the choices for each agent is the cyclic vertex representing the societal outcome. But the choices for each agent includes an edge, and, according to the geometry of the cube, each edge emanating from a cyclic vertex has a transitive vertex. Select this vertex. This completes the proof. (Notice, of the four choices of profiles for the two decisive agents, only one is transitive.) pairs of alterProof for n ≥ 4 alternatives. The n alternatives define n2 = n(n−1) 2 natives. For each pair, (X, Y ), represent its three rankings X Y, X ∼ Y, Y X , by n the points 0, 21 , 1 on a particular R(2) coordinate axis. Over all pairs, the integer points n define the vertices of a unit hyper-cube in R(2) : each vertex defines the rankings over all pairs of alternatives. We will use a lower k-dimensional cube. Suppose k pairs are assigned to decisive agents; with at least two decisive agents 2 ≤ k ≤ n2 . Take two of the pairs that are assigned to two different decisive agents, say (X 1 , Y1 ) and (X 2 , Y2 ). The pairs cannot be the same, so at most one alternative is in both pairs, or they are disjoint. In the first case, assume that Y1 = X 2 and add the pair (Y2 , X 1 ). As this setting reduces to the three alternative setting already analyzed, the proof is completed. (For pairs outside of this triplet, let everyone agree on the ranking.) In the latter case, add the pairs (Y1 , X 2 ) and (Y2 , X 1 ). The four pairs define a hyper-cube in the four dimensional space R4 with 24 = 16 vertices. There are two cyclic vertices: one is given by X 1 Y1 , Y1 X 2 , X 2 Y2 , Y2 X 1 and the other reverses the rankings. As a 0 in a vertex coordinate (i.e., representing the ranking of one pair) for one cycle is replaced by a 1 for the other cycle, these vertices

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are diametrically opposite each other. By being four-dimensional, four edges emanate from each vertex of this cube. As each edge coming from a cyclic preference reverses the ranking of precisely one pair, this change will make the ranking transitive (with respect to the specified pairs). Thus, as in the three-dimensional setting, the cyclic rankings have a special geometric relationship with transitive rankings. The rest of the construction mimics the three-alternative case. Select a cycle. Let the rankings for the two pairs determined by decisive agents coincide with the selected cycle. For the remaining two pairs, if assigned to a decisive agent, select the ranking according to the chosen cycle. If not, use the Pareto condition by requiring all voters to have the specified ranking. Each selection corresponds to selecting a cube face. By construction, the specified cyclic vertex is on all cube faces, so it defines the societal outcome for these pairs. To find transitive preferences for the voters, notice that with two (or more) decisive agents, the above construction mandates for each agent the choice of preferences for at most three pairs (at most three cube faces). Thus the selection of an agent’s preference is one of the vertices on the agent’s selected faces. By construction, the choice of vertices always includes the two vertices on at least one edge coming out of the cyclic vertex. One of the vertices must be transitive: transitive preferences can always be found. While the above proves the theorem, notice that the argument generalizes to any number of pairs involving any number of cycles. Note that a cycle must involve at least two decisive agents. If this were not true, then some agent’s choice must agree with each ranking in the cycle (through the Pareto condition and by being decisive), so this agent must have cyclic preferences. (This is the only place the transitivity assumption occurs: it shows the need for at least two decisive agents.) Second, a cycle involving k pairs can be represented by a vertex on a k-dimensional cube: each vertex on an edge with the cyclic vertex is a transitive ranking. Proof of Theorem 2. The proof follows immediately from the comments made prior to the statement of the theorem. The societal outcome is the intersection of all faces of a hypercube determined by the decisive agents and the Pareto condition. Each agent’s preferences are given by the vertices of the object that result from the intersection of some (but not all) faces. Consequently, the societal outcome must be one of these vertices. Proof of Theorem 3. The assumptions on preferences makes this a counting exercise. When the voters can rank each pair independently, then the condition that all voters rank one pair in the same manner means that there are four remaining choices of rankings for each voter. The ranking of the other two pairs, however, are determined by the two decisive voters, so the choices of the n −2 voters do not matter. Their choice define four parallel edges; each is equally likely. On each edge, the Pareto condition mandates that only one vertex can be selected. Thus, the outcomes are one of the four vertices on a cube face; each is equally likely. But on each cube face, three of the vertices are transitive and one is cyclic. This completes the proof for the first part. Now assume that the voters can select only transitive preferences, and that they agree on the ranking for one pair. Thus each voter has three choices for the transitive ranking as determined by whether the remaining alternative is ranked above, below, or between the specified pair. There are nine possible rankings for the two decisive voters.

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As demonstrated in the proof for three alternatives, for each way the societal ranking of a pair can be determined by the Pareto condition, there is precisely one transitive profile for the two decisive agents that gives rise to the cyclic outcome. Consequently, the likelihood of a cyclic outcome is 19 and of a transitive outcome is 89 . With three decisive agents, each agent must select the societal ranking for one pair. It is immaterial how they rank the other pairs. Therefore, there are eight choices for the selected rankings (cube faces); each is equally likely. Of the eight choices, two define cyclic vertices and six define transitive ones. Thus, the likelihood of a cyclic outcome is 41 . Proof of Proposition 1. According to the construction of any example and the hypothesis, each decisive agent’s preferences are given by the two vertices on the edge connecting the face determined by the Pareto condition and the face selected by the decisive agent. Whatever face (see Fig. 1b), is selected by Pareto, it has three transitive and one cyclic vertex. Without loss of generality select one of them, say the y = 1, or B C, face with vertices {1, 5, 6, 7}. Each decisive agent’s preferences is either the societal outcome or the vertex on the edge coming from the outcome in the agent’s assigned direction (via the assignment of a pair); i.e., one is in the x, and the other in the z direction. The proof follows the geometry of the cube (Fig. 1b). With societal outcome vertex 6, both edges have only transitive outcomes; thus a third of the transitive outcomes have only transitive preferences. For vertices 5 and 1, one edge direction includes vertex 7; all other vertices are transitive. With vertex 5, for instance, one voter’s preferences are from vertices {5, 6} and the other from {5, 7}; the four possible profiles are (5, 5), (5, 7), (6, 5), (6, 7). Thus, two-thirds of the transitive outcomes have two supporting profiles with transitive preferences and two with mixed—one transitive and one cycle. The remaining choice is the cyclic vertex 7. The choices of preferences are {1, 7} for one agent and {5, 7} for the other; as the four possible profiles are (1, 5), (1, 7), (7, 5), (7, 7), the conclusion follows. This completes the proof. Proof of Theorem 4. When the outcomes for k pairs are determined by Pareto and decisive agents, the outcome is a vertex of a k-dimensional cube with components that are either 0 or 1. By relabeling endpoints of edges if needed, we can assume that the societal outcome is given by (1, . . . , 1). Namely, if the societal outcome for a pair is X Y , select the names of the endpoints for the {X, Y } edge so that 1 represents X Y. All possible supporting preferences available to an agent are given by all vectors where the component has value 1 in a direction where the agent contributes to the outcome (by being decisive or through Pareto), and either 0 or 1 for all remaining directions. Thus, unanimity is an admissible profile. The components for the average of the agent’s vectors are either 21 or 1. Therefore, when finding the average of these averages, each component is the average of values that are either 21 or 1 where some values must be unity (from Pareto or the decisive agent). Trivially, each component value is greater than 21 , so the average of all possible supporting profiles is closest to (1, . . . , 1) vertex representing the societal outcome. This proves the theorem. Proof of Theorem 5. For the first part of the theorem, we know from Theorem 4 that one choice of preferences is where the preferences of all agents agree with the tran-

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sitive societal outcome. With complete agreement, the choices made by the decisive agents do not cause a negative externality. There are n2 edges coming out of this vertex: n − 1 of them connect to another transitive ranking—these edges represent a pair of alternatives that are adjacently ranked in a ranking representing the vertex. (For instance, for A B C D, these would be the edges for {A, B}, {B, C}, {C, D}.) By being adjacent, reversing the ranking of a pair does not affect transitivity; while a change in these rankings creates it is not a strong negative a negative externality, edges, the pairs are not adjaexternality. For the remaining n2 − (n − 1) = (n−1) 2 cently ranked in the original ranking, so a change in the indicated pair creates a cycle. (With A B C D, reversing the {A, C} ranking creates a cycle.) For a cyclic ranking, changing the ranking of any pair defines a transitive ranking. The changed alternatives cannot be adjacently ranked in the transitive ranking—if they were, then reversing them again to return to the original setting would create a transitive rather than a cyclic ranking. Thus, each voter suffers a strong negative externality. Proof of Theorem 6. With a cycle, only one ranking needs to be changed to convert it into a transitive ranking. Changing one of these rankings into indifference converts the ranking into a quasi-transitive ranking because, by removing an offending strict ranking, the remaining strict rankings are transitive. Proof of Theorem 7. If a cycle created by decisive agents and the Pareto condition involved only one decisive agent, this agent’s preferences would have to agree with the cyclic outcome. This violates the assumption of transitive preferences. (Again, this is about the only place the assumption of transitivity is used.) The remainder of the theorem follows because a rule is specified to convert the ranking of one pair to indifference, and that eliminates the cyclic ranking. References Arrow KJ (1950) A difficulty in the concept of social welfare. J Polit Econ 58(4):469–484 Batra RN, Pattanaik PK (1972) On some suggestions for having non-binary social choice functions. Theory Decis 111:1–11 Blau JH (1975) Liberal values and independence. Rev Econ Stud 42:395–401 Breyer F (1977) The liberal paradox, decisiveness over issues, and domain restrictions. J Econ 37:45–60 Breyer F, Gigliotti GA (1980) Empathy and respect for the rights of others. J Econ 40:59–64 Campbell DE, Kelly JS (1997) Sen’s theorem and externalities. Economica 64:375–386 Fine B (1975) Individual liberalism in a paretian society. J Polit Econ 83:1277–1282 Harel A, Nitzan S (1987) The liberatarian resolutioon of the paretian liberal paradox. J Econ 47:337–352 Jones B, Radcliff B, Taber C, Timpone R (1995) Condorcet winners and the paradox of voting. Am Polit Sci Rev 89:137–144 Riker WH (1982) Liberalism against Populism. W. H. Freeman and Co., San Francisco Risse M (2001) What to make of the liberal paradox? Theory Decis 50:169–196 Saari DG (1995) Inner consistency or not inner consistency: a reformulation is the answer. In: Barnett W.A., Moulin H., Salles M., Schofield N. (eds) Social choice, welfare, and ethics. Cambridge University Press, Cambridge pp 187–212 Saari DG (1995) Basic geometry of voting. Springer, New York Saari DG (1998) Connecting and resolving Sen’s and Arrow’s theorems. Soc Choice Welf 15:239–261 Saari DG (2001) Decisions and elections; explaining the unexpected. Cambridge University Press, New York Saari DG (2008) Disposing dictators; demystifying voting paradoxes. Cambridge University Press, New York

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