No-show Paradox Research Articles

Transformation into anti-manipulation method in voting. Changes in properties

Abstract This paper examines the properties of the anti-manipulation method in voting. Such a method can be used by committees and similar bodies to ensure that votes reflect genuine preferences. The anti-manipulation method is based on the Borda Count and discourages strategic voting by excluding scores that deviate excessively from the mean. The method does not eliminate strategic voting but diminishes the motivation to apply it. We compare the properties of the Borda Count and the anti-manipulation method. The properties, which are most often found in the literature, were chosen for comparison. Thus, the following properties are examined: consistency, vulnerability to the no-show paradox, vulnerability to the subset choice condition, homogeneity, monotonicity, and vulnerability to the reversal bias paradox as well as the Condorcet winner and loser paradoxes. The anti-manipulation method fails to satisfy most of these properties. A real data example, the voting of a certain jury, is used as a counterexample in most cases.

A diagram for analyzing ordinal voting systems

I introduce a diagram for describing and analyzing single-winner elections in which voters rank the candidates—a class of voting systems including positional methods (e.g. plurality, Borda count, anti-plurality), Condorcet methods, and instant-runoff voting (i.e. ranked-choice voting or the alternative vote). The diagram shows how the outcome of an election depends on each candidate’s share of top rankings as a function of the voting system and the pattern of lower rankings. Using as examples two Brexit polls, a mayoral election in San Francisco, and the US’s first instant-runoff congressional election (all since 2018), I show how the diagram can concisely present preferences and results under different voting systems, identify Condorcet cycles, highlight system properties such as join-inconsistency and the no-show paradox, and illuminate strategic voting incentives.

Majority Judgment vs. Approval Voting

The two main criticisms by approval voting (AV) supporters have been that majority judgment (MJ) is not “Condorcet” and that it admits the no-show paradox. That MJ is not Condorcet consistent is a good property shared with AV: the domination paradox shows that majority rule may well err in an election between two. Whereas the no-show paradox is in theory possible with MJ, it is as a practical matter impossible. Moreover, it is proven that MJ with three grades does not admit the no-show paradox. In contrast, AV suffers from serious drawbacks. With AV, voters cannot express their opinions adequately; experiments show that Approve is not the opposite of Disapprove and that AV does not admit the no-show paradox—it admits the very closely allied no-show syndrome. Two grades are simply too few! The debate must concern three or more grades.

Strategic Abstention based on Preference Extensions: Positive Results and Computer-Generated Impossibilities

Voting rules allow multiple agents to aggregate their preferences in order to reach joint decisions. A common flaw of some voting rules, known as the no-show paradox, is that agents may obtain a more preferred outcome by abstaining from an election. We study strategic abstention for set-valued voting rules based on Kelly's and Fishburn's preference extensions. Our contribution is twofold. First, we show that, whenever there are at least five alternatives and seven agents, every Pareto-optimal majoritarian voting rule suffers from the no-show paradox with respect to Fishburn's extension. This is achieved by reducing the statement to a finite - yet very large - problem, which is encoded as a formula in propositional logic and then shown to be unsatisfiable by a SAT solver. We also provide a human-readable proof which we extracted from a minimal unsatisfiable core of the formula. Secondly, we prove that every voting rule that satisfies two natural conditions cannot be manipulated by strategic abstention with respect to Kelly's extension and give examples of well-known Pareto-optimal majoritarian voting rules that meet these requirements.

Welfare maximization entices participation

We consider randomized public good mechanisms with optional participation. Preferences over lotteries are modeled using skew-symmetric bilinear (SSB) utility functions, a generalization of classic von Neumann–Morgenstern utility functions. We show that every welfare-maximizing mechanism entices participation and that the converse holds under additional assumptions. As a corollary, we obtain a characterization of an attractive randomized voting rule that satisfies Condorcet-consistency and entices participation. This stands in contrast to Moulin's well-known no-show paradox (Moulin, 1988), which shows that no deterministic voting rule can satisfy both properties simultaneously.

The No-Show Paradox Under a Restricted Domain

The no-show paradox occurs whenever a group of identically-minded voters is better off abstaining than by voting according to its preferences. Moulin’s (J Econ Theory 45:53–64, 1988) result states that if one wants to exclude the possibility of the no-show paradox, one has to resort to procedures that do not necessarily elect the Condorcet winner when one exists. This paper examines ten Condorcet-consistent and six Condorcet-non-consistent procedures in a restricted domain, viz., one where there exists a Condorcet winner who is elected in the original profile and the profile is subsequently modified by removing a group of voters with identical preferences. The question asked is whether the no-show paradox can occur in these settings. It is found that only two of the ten Condorcet-consistent procedures investigated (Maximin and Schwartz’s procedure) are not vulnerable to the no-show paradox, whereas only two of the six non-Condorcet-consistent ranked procedures investigated (Coombs’ and the Negative Plurality Elimination Rule procedures) are vulnerable to this paradox in the restricted domain. In other words, for a no-show paradox to occur when using Condorcet-consistent procedures it is not, in general, necessary that a top Condorcet cycle exists in the original profile, while for this paradox to occur when using (ranked) non-Condorcet-consistent procedures it is, almost always, necessary that the original profile has a top cycle.

Geometry of run-off elections

Abstract We present a geometric representation of the method of run-off voting. With this representation we can observe the non-monotonicity of the method and its susceptibility to the no-show paradox. The geometry allows us easily to identify a novel compromise rule between run-off voting and plurality voting that is monotonic.

Revisiting the connection between the no-show paradox and monotonicity

We investigate the relation between monotonicity and the no-show paradox in voting rules. Although the literature has established their logical independence, we show, by presenting logical dependency results, that the two conditions are closer than a general logical independency result would suggest. Our analysis is made both under variable and fixed-size electorates.

No-show paradox in Slovak party-list proportional system

Abstract The phenomenon of the paradoxes of the largest remainders methods has been studied by numerous authors. Nevertheless, the examples presented in their studies do not deal with the case where a party’s possible additional votes can directly lead to a loss in the party’s number of representatives. This paradox, which can be called the no-show apportionment paradox, has not previously been mentioned in the literature. It is based on the assumption that a voter’s favourite party may lose a seat if the voter votes honestly but get more seats if he or she abstains. The article gives simple examples illustrating the no-show paradox in a party-list proportional representation electoral system using the quota method of apportionment. In these situations it would be better for the party if some of their voters abstained, otherwise it would lose one or more seats.

Optimal bounds for the no-show paradox via SAT solving

One of the most important desirable properties in social choice theory is Condorcet-consistency, which requires that a voting rule should return an alternative that is preferred to any other alternative by some majority of voters. Another desirable property is participation, which requires that no voter should be worse off by joining an electorate. A seminal result by Moulin (1988) has shown that Condorcet-consistency and participation are incompatible whenever there are at least 4 alternatives and 25 voters. We leverage SAT solving to obtain an elegant human-readable proof of Moulin’s result that requires only 12 voters. Moreover, the SAT solver is able to construct a Condorcet-consistent voting rule that satisfies participation as well as a number of other desirable properties for up to 11 voters, proving the optimality of the above bound. We also obtain tight results for set-valued and probabilistic voting rules, which complement and significantly improve existing theorems.

Revisiting the Connection between the No-Show Paradox and Monotonicity

We investigate the relation between monotonicity and the no-show paradox in voting rules. Although the literature has established their logical independence, we show, by presenting logical dependency results, that the two conditions are closer than a general logical independency result would suggest. Our analysis is made both under variable and fixed-size electorates.

What Happened in Burlington?

Three visualization techniques illustrate the distribution of electoral preferences over a candidate triple. Two of them, introduced here, concern an IRV tally. The conditions that may allow the “pushover strategy” and the “No-Show Paradox”, are identified, and the practical consequences discussed. The controversial mayoral election of Burlington, Vermont, in 2009 is background. We see the IRV method in a legal and in a political context, presenting aspects of a judgment in the Minnesota Supreme Court 2009 and of the UK referendum over IRV in 2011. IRV is the single-seat version of STV. Both may achieve proportional voter influence with a designed disproportional distribution of seats in a legislature, e.g. as part of a potentially viable modus vivendi in ethnically divided societies.

Priorities in the location of multiple public facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We characterize the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among “interest groups”. We characterize the subclasses of priority rules that respectively satisfy anonymity, avoid the no-show paradox, strategy-proofness and population-monotonicity. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Any such rule can also be viewed as a collection of generalized peak-selection median rules, that are linked across populations, in a way that we describe.

No-Show Paradoxes under Parliamentary Agendas

It may happen that the favorite candidate of a voter loses an election if that voter votes honestly, but gets elected if the voter abstains. Such a situation is known in the social choice literature under the name of no-show paradox. This paper is devoted to the evalution of the likelihood of that phenomenon under parliamentary voting rules.

A responsive voting system

To say that a democratic system of government ought to be responsive to the opinions of the citizens sounds like a truism which could hardly be expected to generate much controversy. Yet, the notion of responsiveness turns out to be open to several nonequivalent interpretations: Condorcet-consistency, monotonicity, invulnerability to the no-show paradox, to name the most important ones. Several results have been achieved to establish incompatibilities between these properties. We shall review some of these and discuss some similar notions and their relationships with those just mentioned.

Monotonicity and its Cognates in the Theory of Choice

The standard requirement of monotonicity of a voting procedurestates that an improvement in the ranking of the winningalternative, ceteris paribus, should not make itnon-winning. A concept apparently closely linked tomonotonicity is known as the participation axiom whichrequires that it should never be advantageous for a voter toabstain rather than to vote according to his/her preferences.Situations in which a group of voters may end up with a betteroutcome by not voting at all than by voting according to theirpreferences are called instances of the no-show paradox. Astrong version of the paradox occurs when the abstainers endup with their most preferred outcome by abstaining. A thirdrelated concept is invulnerability to preference truncation.This is satisfied by such procedures that make itadvantageous for voters to always reveal their entirepreference rankings. The fourth concept, Maskin monotonicity,plays an important role in mechanism design literature. Wediscuss these requirements in the context of votingprocedures. Particular attention is paid to the plausibilityof conditions guaranteeing various forms of monotonicity.

The voter who wasn?t there: Referenda, representation and abstention

We analyze single binary-choice voting rules and identify the presence of the No-Show paradox in this simple setting, as a consequence of specific turnout or quorum conditions that are included in actual rules. Since these conditions are meant to ensure a representative outcome, we formalize this concern and reach our main result: no voting rule can ensure representation if abstention is possible, unless restrictive assumptions are made on the preference domain of abstainers. We then focus on the main referendum systems and show that appropriate restrictions do make them compatible with representation. The main purpose of our paper is, however, to provide a tool for referendum design: rather than imposing arbitrary restrictions on the preference domain of non-voters, we recommend instead that a conscious choice be made on how abstention is to be interpreted and that this choice be used to derive the corresponding referendum rule.

Non-monotonicity does not imply the no-show paradox

Non-monotonicity does not imply the no-show paradox

Scoring run-off paradoxes for variable electorates

Summary. A no-show paradox occurs each time a single voter or a group of voters can manipulate the outcome by not participating to the election process. Among other voting procedures, the scoring run-off methods, which eliminate progressively the alternatives on the basis of scoring rules, suffer from this flaw. We here estimate how frequent this paradox is for three candidate elections under the classical Impartial Culture and Impartial Anonymous Culture assumptions, for different population sizes. The conditions under which this paradox occurs are also described, as well as the relationships with manipulations for a fixed number of voters.

To vote or not to vote: What is the quota?

The observation of a new type of perverse behavior of voting rules—Brams and Fishburn's “no-show paradox”—led Moulin to introduce the Participation Axiom (PA). It requires that an elector's failure to vote should never result in the election of a candidate whom he/she prefers to the one elected if he/she votes sincerely. The present paper examines PA in the context of Condorcet-type conditions. For a given quota q, 1 2 ≤q≤1 , the q-Core Condition ( qCC) requires that whenever there exists a candidate such that no other candidate is preferred to him/her by a fraction of q or more of the voters, the elected candidate should have this property. It is shown here that PA and qCC are consistent iff q≥ (m−1) m or m ≤3, where m is the number of candidates. This essentially confirms a conjecture of Moulin and extends his original result for q= 1 2 .