Distribution of coalitional power in randomized multi-valued social choice

This paper considers the distribution of coalitional influence under probabilistic social choice functions which are randomized social choice rules that allow social indifference by mapping each combination of a preference profile and a feasible set to a social choice lottery over all possible choice sets from the feasible set. When there are at least four alternatives in the universal set and ex-post Pareto optimality, independence of irrelevant alternatives and regularity are imposed, we show that: (i) there is a system of additive coalitional weights such that the weight of each coalition is its power to be decisive in every two-alternative feasble set; and (ii) for each combination of a feasible proper subset of the universal set and a preference profile, the society can be partioned in such a way that for each coalition in this partition, the probability of society's choice set being contained in the union of the best sets of its members is equal to the coalition's power or weight. It is further shown that, for feasible proper subsets of the universal set, the probability of society's choice set containing a pair of alternatives that are not jointly present in anyone's best set is zero. Our results remain valid even when the universal set itself becomes feasible provided some additional conditions hold. Received: 10 May 1999/Accepted: 18 June 2000 I would like to thank Professor Prasanta Pattanaik for suggesting to me the line of investigation carried out in this paper. I am solely responsible for any remaining errors and omissions.     

On probabilistic rationalizability

This paper examines the problem of rationalizability of a choice system without imposing restrictions on the universal set of alternatives and the family of feasible sets. A choice system is the probabilistic counterpart of a choice function which is basic to much of the theory of individual choice behaviour in economics. We provide a necessary and sufficient condition for a choice system to be induced by a probability measure on a set of utility functions. I would like to thank the editor and two anonymous referees for helpful comments.     

Contraction consistent stochastic choice correspondence

We model a general choice environment via probabilistic choice correspondences, with (possibly) incomplete domain and infinite universal set of alternatives. We offer a consistency restriction regarding choice when the feasible set contracts. This condition, ‘contraction consistency’, subsumes earlier notions such as Chernoff’s Condition, Sen’s α and β, and regularity. We identify a restriction on the domain of the stochastic choice correspondence (SCC), under which contraction consistency is equivalent to the weak axiom of revealed preference in its most general form. When the universal set of alternatives is finite, this restriction is also necessary for such equivalence. Analogous domain restrictions are also identified for the special case where choice is deterministic but possibly multi-valued. Results due to Sen (Rev Econ Stud 38:307–317, 1971) and Dasgupta and Pattanaik (Econ Theory 31:35–50, 2007) fall out as corollaries. Thus, conditions are established, under which our notion of consistency, articulated only in reference to contractions of the feasible set, suffices as the axiomatic foundation for a general revealed preference theory of choice behaviour.     

Minimal covering set solutions

We study necessary and sufficient conditions for a multi-valued solution S to be rationalized in the following sense: there exists a complete asymmetric relation T (a tournament) such that, for each feasible (finite) set, the solution set of S coincides with the minimal covering set of T restricted to that feasible set. Our characterization result relies only on properties relating S across feasible choice sets.     

Policy convergence in a two-candidate probabilistic voting model

We propose a generalization of the probabilistic voting model in two-candidate elections. We allow the candidates have general von Neumann–Morgenstern utility functions defined over the voting outcomes. We show that the candidates will choose identical policy positions only if the electoral competition game is constant-sum, such as when both candidates are probability-of-win maximizers or vote share maximizers, or for a small set of functions that for each voter define the probability of voting for each candidate, given candidate policy positions. At the same time, a pure-strategy local Nash equilibrium (in which the candidates do not necessarily choose identical positions) exists for a large set of such functions. Hence, if the candidate payoffs are unrestricted, the “mean voter theorem” for probabilistic voting models is shown to hold only for a small set of probability of vote functions.     

Generic difference of expected vote share and probability of victory maximization in simple plurality elections with probabilistic voters

In this paper I examine single member, simple plurality elections with n ≥ 3 probabilistic voters and show that the maximization of expected vote share and maximization of probability of victory are “generically different” in a specific sense. More specifically, I first describe finite shyness (Anderson and Zame in Adv Theor Econ 1:1–62, 2000), a notion of genericity for infinite dimensional spaces. Using this notion, I show that, for any policy in the interior of the policy space and any candidate j, the set of n-dimensional profiles of twice continuously differentiable probabilistic voting functions for which simultaneously satisfies the first and second order conditions for maximization of j’s probability of victory and j’s expected vote share at is finitely shy with respect to the set of n-dimensional profiles of twice continuously differentiable probabilistic voting functions for which satisfies the first and second order conditions for maximization of j’s expected vote share.     

Uncovered set choice rules

I study necessary and sufficient conditions for a choice function to be rationalized in the following sense: there exists a total asymmetric relation T (a tournament) such that, for each feasible (finite) set, the choice set coincides with the uncovered set of T restricted to that feasible set. This notion of ‘maximization’ offers testable restrictions on observable choice behavior.     

Using elections to represent preferences

This paper takes voting theory out of the realm of mechanism design and studies elections as tools for representing preferences: every preference relation on a set of n elements is the outcome of pairwise voting by approximately 2 log₂ n voters with transitive preferences. Results like this one provide representation for preference relations not representable by utility functions. They also motivate definitions of the levels of intransitivity, nonlinearity and nonrepresentability (by utility function) of a preference relation. Received: 25 March 1999/Accepted: 19 June 2000     

On randomized rationality

In contrast to the traditional notion of rationalizability of stochastic choice which requires the preference relations to be strict, we propose a notion of rationalizability without requiring the preference relations to be strict. Our definition is based on the simple hypothesis of a two-stage choice process: stage (i) a preference relation R is drawn according to a probability assignment; stage (ii) an alternative is picked from each feasible set according to a uniform lottery over the R-greatest set in it. We provide a necessary and sufficient condition for rationalizability of stochastic choice. Since our framework is general enough to subsume the traditional case, our result also provides an alternative characterization of the traditional notion of rationalizability. We also show the equivalence between the two notions of rationalizability in a specific case.     

On the equilibrium of voting games with abstention and several levels of approval

We study the core of “(j, k) simple games”, where voters choose one level of approval from among j possible levels, partitioning the society into j coalitions, and each possible partition facing k levels of approval in the output (Freixas and Zwicker in Soc Choice Welf 21:399–431, 2003). We consider the case of (j, 2) simple games, including voting games in which each voter may cast a “yes” or “no” vote, or abstain (j = 3). A necessary and sufficient condition for the non-emptiness of the core of such games is provided, with an important application to weighted symmetric (j, 2) simple games. These results generalize the literature, and provide a characterization of constitutions under which a society would allow a given number of candidates to compete for leadership without running the risk of political instability. We apply these results to well-known voting systems and social choice institutions including the relative majority rule, the two-thirds relative majority rule, the United States Senate, and the United Nations Security Council.     

Hierarchy of players in swap robust voting games

Ordinarily, the process of decision making by a committee through voting is modeled by a monotonic game the range of whose characteristic function is restricted to {0, 1}. The decision rule that governs the collective action of a voting body induces a hierarchy in the set of players in terms of the a-priori influence that the players have over the decision making process. In order to determine this hierarchy in a swap robust game, one has to either evaluate a power index (e.g., the Shapley–Shubik index, the Banzhaf–Coleman index) for each player or conduct a pairwise comparison between players, whereby a player i is ranked higher than another player j if there exists a coalition in which i is more desirable as a coalition partner than j. In this paper, we outline an alternative mechanism to determine the ranking of players in terms of their a-priori power. This simple and elegant method uses only minimal winning coalitions, rather than the entire set of winning coalitions.     

On the selection of the same winner by all scoring rules

Different scoring rules can result in the selection of any of the k competing candidates, given the same preference profile, (Saari DG 2001, Chaotic elections! A mathematician looks at voting. American Mathematical Society, Providence, R.I.). It is also possible that a candidate, and even a Condorcet winning candidate, cannot be selected by any scoring rule, (Saari DG 2000 Econ Theory 15:55–101). These findings are balanced by Saari’s result (Saari DG 1992 Soc Choice Welf 9(4):277–306) that specifies the necessary and sufficient condition for the selection of the same candidate by all scoring rules. This condition is, however, indirect. We provide a sufficient condition that is stated directly in terms of the preference profile; therefore, its testability does not require the verdict of any voting rule.     

Symmetry extensions of “neutrality”

For n3 candidates, a system voting vector W ⁿ specifies the positional voting method assigned to each of the 2 ⁿ –(n+1) subsets of two or more candidates. While most system voting vectors need not admit any relationships among the election rankings; the ones that do are characterized here. The characterization is based on a particular geometric structure (an algebraic variety) that is described in detail and then used to define a partial ordering among system voting vectors. The impact of the partial ordering is that if W ⁿ ₁ W ⁿ ₂, then W ⁿ ₂ admits more kinds of (single profile) voting paradoxes than W ⁿ ₁. Therefore the partial ordering provides a powerful, computationally feasible way to compare system voting vectors. The basic ideas are illustrated with examples that completely describe the partial ordering for n=3 and n=4 candidates.This reearch was supported in part by NSF Grant IRI-9103180.     

Behavioral evidence for framing effects in the resolution of the doctrinal paradox

The aggregation of individual sets of judgments over interconnected propositions can yield inconsistent collective sets of judgments, even when the individual sets of judgments are themselves consistent. A doctrinal paradox occurs when majority voting on a compound proposition (such as a conjunction or disjunction) yields a different result than majority voting on each of the elements of the proposition. For example, when most individuals accept proposition X; most individuals accept proposition Y ; but only a minority of individuals accept the compound proposition ‘X and Y’. Conducting two elemental votes would lead to accept X and Y , but conducting one compound vote would lead to reject X and Y . In such a situation, do people manifest a stable preference as to which voting procedure should be applied? This research reports the results of two behavioral experiments using a within-participant design, which show that procedural preferences can be upturned by framing either positively or negatively the set of judgments to be aggregated. This shift in procedural preference leads to large swings in the final collective judgment endorsed by participants.     

Majority rule in multi-dimensional spatial models

A sufficient condition is derived for a policy y to be a Condorcet winner, when the set of feasible policies is some subsety ofR ^N with non-empty interior. Voters are assumed to differ in some scalar characteristic w. The sufficient condition refers only to voters' preferences over the set of preferred policies (of the various other voters). This set is a one-dimensional curve inR ^N> . The condition is that the indifference curves of each typew through the preferred policyy ^* (v) of any typev all be collinear. If the condition holds, then the preferred policyy ^* (m) of the median type of voter will be a Condorcet winner. If there are only three types of voter, then the above condition is also necessary for the existence of a Condorcet winner.This paper is a revision of the first part of my earlier Majority Rule with Multidimensional Economic Choices. I thank Ted Bergstrom, David Donaldson, and Greg Dow for very valuable comments on versions of that earlier paper, without implicating them in this current incarnation.     

The Borda count and agenda manipulation

A standard objection to the Borda count, as an actual voting procedure, is that it is subject to agenda manipulation. The classical example is the introduction, in order to favour a candidate or option y, of a new option z ranked on every voter's preference scale immediately below y; y may as a result obtain the highest Borda count, although, if z had not been introduced, a different option would have done so. Strategic use of this device is not greatly to be feared, but it does point to a defect in the system: equating the distance on each voter's preference scale between any option and the next imposes a penalty for dissimilarity. The corrective is to vary the distances by introducing a measure of dissimilarity, as perceived by the voters and revealed by their preference scales. Two possible ways of doing this are described, yielding revised and adjusted Borda scores. It is shown by examples that these will often, but not always, undo the `agenda manipulation' effect, and questioned whether it is desirable to undo it altogether. Received: 22 August 1996/Accepted: 31 October 1996     

In quest of the banks set in spatial voting games

The Banks set (1(4):295–306, 1985) is one of the more important concepts in voting theory since it tells us about the sophisticated outcomes of standard amendment voting procedures commonly in use throughout the English speaking world (and elsewhere as well). While the properties of the Banks set for finite voting games have been extensively studied, little is known about how to find members of this set for majority rule spatial voting games involving possibly infinite agendas. We look at this question for two-dimensional games where voters have Euclidean preferences, and offer a variety of new results that delimit areas of the space that can be shown to lie within the Banks set, such as the Schattschneider set, the tri-median set, and the Banks line set—geometric constructs which we show to be nested within one another.     

Probabilistic allocation rules and single-dipped preferences

We consider the problem of allocating an infinitely divisible endowment among a group of agents with single-dipped preferences. A probabilistic allocation rule assigns a probability distribution over the set of possible allocations to every preference profile. We discuss characterizations of the classes of Pareto-optimal and strategy-proof probabilistic rules which satisfy in addition replacement-domination or no-envy. Interestingly, these results also apply to problems of allocating finitely many identical indivisible objects – to probabilistic and to deterministic allocation. Received: 23 November 1998/Accepted: 20 October 2000     

Limiting distributions for continuous state Markov voting models

This paper proves the existence of a stationary distribution for a class of Markov voting models. We assume that alternatives to replace the current status quo arise probabilistically, with the probability distribution at time t+1 having support set equal to the set of alternatives that defeat, according to some voting rule, the current status quo at time t. When preferences are based on Euclidean distance, it is shown that for a wide class of voting rules, a limiting distribution exists. For the special case of majority rule, not only does a limiting distribution always exist, but we obtain bounds for the concentration of the limiting distribution around a centrally located set. The implications are that under Markov voting models, small deviations from the conditions for a core point will still leave the limiting distribution quite concentrated around a generalized median point. Even though the majority relation is totally cyclic in such situations, our results show that such chaos is not probabilistically significant.We acknowledge the support of NSF Grants #SOC79-21588, SES-8106215 and SES-8106212.     

Two-stage discrete aggregation: the Ostrogorski paradox and related phenomena

Motivated by certain paradoxa that have been discussed in the literature (Ostrogorski paradox), we prove an impossibility theorem for two-stage aggregation procedures for discrete data. We consider aggregation procedures of the following form: The whole population is partitioned into subgroups. First we aggregate over each subgroup, and in a second step we aggregate the subgroup aggregates to obtain a total aggregate. The data are either dichotomous (1 — 0; yes-no) or take values in a finite ordered set of possible attributes (e.g., exam grades A, B,...F). Examples are given by multistage voting procedures (indirect democracy, federalism), or by the forming of partial grades and overall grades in academic examinations and similar evaluation problems (sports competitions, consumer reports). It is well known from standard examples that the result of such a two-stage aggregation procedure depends, in general, not only on the distribution of attributes in the whole population, but also on how the attributes are distributed across the various subgroups (in other words: how the subgroups are defined). This dependence leads to certain paradoxa. The main result of the present paper is that these paradoxa are not due to the special aggregation rules employed in the examples, but are unavoidable in principle, provided the aggregators satisfy certain natural assumptions. More precisely: the only aggregator functions for which the result of a two-stage (a fortiori: multi-stage) aggregation does not depend on the partitioning are degenerate aggregators of the following form: there exists a partial order (dominance) on the set of possible attributes such that the aggregate over any collection of data is always equal to the supremum (w.r.t. dominance) of the attributes occurring in the data, regardless of the relative frequnencies of these occurrences. In the voting context, degeneracy corresponds to the unanimity principle. Our theorem is true for arbitrary partitionings of arbitrary (finite) sets and generalizes the results of Deb & Kelsey (for the matrix case with dichotomous variables and majority voting) to general two-stage aggregation procedures for attributes belonging to a finite ordered set. The general result is illustrated by some examples.This paper was completed during a visit to the University of Bielefeld. I am much indebted to the Faculty of Economics there for its hospitality; in particular I should like to thank Gerhard Schwödiauer and Walter Trockel for their support.