Some Further Results on the Manipulability of Social Choice Rules

A typology of electoral environments in which strategic manipulation of voting rules can occur is proposed. From this typology, six different concepts of equilibria (or quasi-equilibria) are defined, including the usual ones. Then, focusing on three alternative elections, we derive some analytical representations for the vulnerability of a large number of voting rules to strategic manipulation by considering successively each type of equilibrium. Our results show that the hierarchy of the rules changes when the voting framework is modified. However, some rules are clearly dominated and some others (among which the Borda and the Nanson rules) appear to be uniquely the best in some specific contexts.     

Quaternary dichotomous voting rules

In this article, we provide a general model of “quaternary” dichotomous voting rules (QVRs), namely, voting rules for making collective dichotomous decisions (to accept or reject a proposal), based on vote profiles in which four options are available to each voter: voting (“yes”, “no”, or “abstaining”) or staying home and not turning out. The model covers most of actual real-world dichotomus rules, where quorums are often required, and some of the extensions considered in the literature. In particular, we address and solve the question of the representability of QVRs by means of weighted rules and extend the notion of “dimension” of a rule.     

Optimal voting rules for two-member tenure committees

A tenure committee first votes on whether to hire a candidate; if it does, it receives an informative performance signal, and then votes on whether to tenure the candidate; rejection at either stage returns the committee to a candidate pool, endogenising the value of the outside option. A candidate’s fate depends only on the behaviour of two ‘weather-vane’ committee members. Committee members may vote against favoured candidates if the weather-vane is opposed; enthusiastic assessments by one of these weather-vanes may harm a candidate’s chances by increasing others’ thresholds for hiring him; sunk time costs may lead voters who voted against hiring to vote for tenuring him, even after a poor probationary performance. For two member committees that are patient and perceptive, the optimal voting rule is a (weak) majority at the hiring stage and unanimity at the tenure stage; when such committees are impatient or imperceptive, the double (weak) majority rule is optimal. Perversely, the performance of a patient, imperceptive committee improves as its perceptiveness further declines. Consistent with practice, falling threshold rules are not optimal. Results on optimal voting rules are also presented in limit cases as committee members’ beliefs become more correlated. Finally, we compare the model to a discrete-time European options model.     

An extension of the Moulin No Show Paradox for voting correspondences

In this article, we analyse the possibility of extending the Moulin theorem to Condorcet voting correspondences. Moulin (1988) established that every Condorcet voting function suffers from the No Show Paradox, or Abstention Paradox, which means that in some voting situations some voters would achieve a better result by abstaining (in other words, could manipulate the election by abstaining). This problem is similar to that of extending the Gibbard–Satterthwaite theorem on voting manipulation through casting an insincere ballot to voting correspondences. The main result of the paper states that for every Condorcet voting correspondence there are situations in which every optimistic or pessimistic voter with some preferences could manipulate the election by abstaining. Another result states, by counterexample, that some Condorcet voting correspondences are free from the Abstention Paradox from the point of view of other types of voters.     

Polarization, Information Collection and Electoral Control

We identify the conditions under which voters can induce political parties to collect information and to select policies which are optimal from the representative voter’s point of view. We show that when parties are office motivated the voting rule should encourage parties to collect information. Voting rules that focus on the opposition party sometimes dominate voting rules that focus on the incumbent party. When parties are policy motivated, they also have to be motivated to select good policies. Generally, it is easier to stimulate policy motivated parties than office motivated parties to collect information. However, in contrast to office motivated parties, policy motivated parties will sometimes select policies that conflict with the representative voter’s interest.     

Self-selective social choice functions

It is not uncommon that a society facing a choice problem has also to choose the choice rule itself. In such situations, when information about voters’ preferences is complete, the voters’ preferences on alternatives induce voters’ preferences over the set of available voting rules. Such a setting immediately gives rise to a natural question concerning consistency between these two levels of choice. If a choice rule employed to resolve the society’s original choice problem does not choose itself, when it is also used for choosing the choice rule, then this phenomenon can be regarded as inconsistency of this choice rule as it rejects itself according to its own rationale. Koray (Econometrica 68: 981–995, 2000) proved that the only neutral, unanimous universally self-selective social choice functions are the dictatorial ones. Here we introduce to our society a constitution, which rules out inefficient social choice rules. When inefficient social choice rules become unavailable for comparison, the property of self-selectivity becomes more interesting and we show that some non-trivial self-selective social choice functions do exist. Under certain assumptions on the constitution we describe all of them.     

Characterizing best–worst voting systems in the scoring context

An increasing body of theoretical and empirical work on discrete choice considers a choice design in which a person is asked to select both the best and the worst alternative in an available set of alternatives, in contrast to more traditional tasks, such as where the person is asked to: select the best alternative; select the worst alternative; rank the alternatives. Here we consider voting systems motivated by such “best–worst” choice; characterize a class of “best–worst” voting systems in terms of a set of axioms in the context of scoring rules; and discuss briefly possible extensions to approval–disapproval systems.     

A probabilistic synopsis of binary decision rules

Decision rules for Yes–No voting systems are placed in a probabilistic framework. Selfdual and permutationally invariant distributions are introduced. Under such distributions, the mean success margin of the majority rule and of the unanimity rule are shown to bound the mean success margin of all other decision rules. For bloc decision rules in the Penrose/Banzhaf model, a product formula for the voters’ influence probabilities is derived. Other indices and the Shapley/Shubik model are also discussed.     

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.     

The reinforcement axiom under sequential positional rules

The reinforcement axiom roughly states that when an alternative is selected by two different constituencies, it must also be selected by their union. Hare and Coombs rules are special cases of sequential positional voting rules, which are known to violate this axiom. In this article, we first show that reinforcement can be violated by all such rules. We then evaluate, by the use of Monte Carlo simulations and the Fishburn–Gehrlein technique, the proportion of profiles at which this phenomenon occurs.     

MANIPULATING VOTING PROCEDURES

A voting procedure can be manipulated if, by misrepresenting his preferences, some individual can secure an outcome which he prefers to the outcome he gets when he is honest.
This is an expository paper on the theory of voting manipulation. Section I is an historical sketch of the contributions of Condorcet, de Borda, Arrow, and others. Section II provides a set of examples of manipulation: of plurality voting, of majority voting, of exhaustive voting, of the single transferable vote procedure, and of approval voting. It also contains an example of a nonmanipulable random voting scheme. Section HI provides a simple proof of the Gibbard-Satterthwaite manipulation theorem.     

How frequently do different voting rules encounter voting paradoxes in three-candidate elections?

We estimate the frequencies with which ten voting anomalies (ties and nine voting paradoxes) occur under 14 voting rules, using a statistical model that simulates voting situations that follow the same distribution as voting situations in actual elections. Thus the frequencies that we estimate from our simulated data are likely to be very close to the frequencies that would be observed in actual three-candidate elections. We find that two Condorcet-consistent voting rules do, the Black rule and the Nanson rule, encounter most paradoxes and ties less frequently than the other rules do, especially in elections with few voters. The Bucklin rule, the Plurality rule, and the Anti-plurality rule tend to perform worse than the other eleven rules, especially when the number of voters becomes large.     

Pairwise Strategy-Proofness and Self-Enforcing Manipulation

“Strategy-proofness” is one of the axioms that are most frequently used in the recent literature on social choice theory. It requires that by misrepresenting his preferences, no agent can manipulate the outcome of the social choice rule in his favor. The stronger requirement of “group strategy-proofness” is also often employed to obtain clear characterization results of social choice rules. Group strategy-proofness requires that no group of agents can manipulate the outcome in their favors. In this paper, we advocate “effective pairwise strategy-proofness.” It is the requirement that the social choice rule should be immune to unilateral manipulation and “self-enforcing” pairwise manipulation in the sense that no agent of a pair has the incentive to betray his partner. We apply the axiom of effective pairwise strategy-proofness to three types of economies: public good economy, pure exchange economy, and allotment economy. Although effective pairwise strategy-proofness is seemingly a much weaker axiom than group strategy-proofness, effective pairwise strategy-proofness characterizes social choice rules that are analyzed by using different axioms in the literature.     

Voting paradoxes and referenda

In representational democracies the referenda constitute an additional way for the voters to express their opinions. At the same time they are accompanied by problems of agenda manipulation and interpretation of results. In this context various voting paradoxes and their interrelationships are of considerable interest. In this article particular attention is paid to opinion aggregation paradoxes in referendum institutions. The limits and interrelationships of paradoxes are discussed. Some ways of avoiding paradoxical situations are also outlined. Received: 11 September 1995 / Accepted: 7 January 1997     

The vulnerability of four social choice functions to coalitional manipulation of preferences

The vulnerability of a social choice function to coalitional manipulation of preferences is defined as the proportion of voting situations in which the social choice function can be manipulated by a coalition of individuals. In this paper, we provide exact relations giving the vulnerability of four specific social choice functions (plurality rule, anti-plurality rule, plurality with runoff, anti-plurality with runoff) in three-alternative elections. Plurality with runoff appears to be less vulnerable than the other three rules.     

Single-Crossing, Strategic Voting and the Median Choice Rule

This paper studies the strategic foundation of the Representative Voter Theorem (Rothstein in: Pub Choice 72:193–212, 1991), also called the “second version” of the Median Voter Theorem. As a by-product, it also considers the existence of strategy-proof social choice functions over the domain of single-crossing preferences. The main result shows that single-crossing constitutes a domain restriction over the real line that allows not only majority voting equilibria, but also non-manipulable choice rules. In particular, this is true for the median rule, which is found to be group strategic-proof over the full set of alternatives and over every nonempty subset. In addition, the paper also examines the relation between single-crossing and order-restriction. And it uses this relation together with the strategy-proofness of the median rule to prove that the outcome predicted by the Representative Voter Theorem can be implemented in dominant strategies through a simple mechanism. This mechanism is a two-stage voting procedure in which, first, individuals select a representative among themselves, and then the winner chooses a policy to be implemented by the planner.     

Partially efficient voting by committees

On the separable preference domain, voting by committees is the only class of voting rules that satisfy strategy-proofness and unanimity, and dictatorial rules are the only ones that are strategy-proof and Pareto efficient. To fill the gap, we define a sequence of efficiency conditions. We prove that for strategy-proof rules on the separable preference domain, the various notions of efficiency reduce to three: unanimity, partial efficiency, and Pareto efficiency. We also show that on the domain, strategy-proofness and partial efficiency characterize the class of voting rules represented as simple games which are independent of objects, proper and strong. We call such rules voting by stable committee.The author is deeply indebted to William Thomson for many helpful discussions on an earlier draft. The current version is greatly benefited from detailed comments of an anonymous referee. Thanks are also due to Jeffrey Banks, Salvador Barberà, Marcus Berliant, Ryo-ichi Nagahisa, Takehiko Yamato, and participants in a seminar at Rochester in 1992, the 1992 Midwest Conference at Michigan State, and the 1993 Summer Meeting of Econometric Society at Boston University for conversations and suggestions.     

On the Average Minimum Size of a Manipulating Coalition

We study the asymptotic average minimum manipulating coalition size as a characteristic of quality of a voting rule and show its serious drawback. We suggest using the asymptotic average threshold coalition size instead. We prove that, in large electorates, the asymptotic average threshold coalition size is maximised among all scoring rules by the Borda rule when the number m of alternatives is 3 or 4, and by -approval voting when m ≥ 5.     

Voting power: an information theory approach

In recent years, there has been increasing awareness of the importance of formal measures of voting power and of the relevance of such measures to real life political issues. Nevertheless, existing measures have been criticized, especially because of their dependence on the unrealistic assumption that different coalitions have equal probabilities. In this paper we show that the classical problem of measuring voting power can be naturally embedded in information theory. This perspective on voting power allows us to extend measures of voting power to cases in which there are dependencies among voters. In doing so, we distinguish between two different notions of a given voter’s power—‘control’ and ‘informativeness’—corresponding, respectively, to the average uncertainty regarding the outcome of a vote that remains when all others have voted and the average uncertainty that is eliminated when only the given voter has voted. This distinction settles a number of well-known paradoxes and enables the study of voting power on the basis of actual political behavior at all levels.