Research

We investigate a specific type of group manipulation in two-tier elections, which involves pairs of voters agreeing to exchange their votes. Two-tier elections are modeled as a two-stage choice procedure. In the first stage, voters are distributed into districts, and district preferences result from aggregating voters' preferences district-wise through some aggregation rule. Final outcomes are obtained in the second stage by applying a social choice function that outputs one or several alternatives from the profile of district preferences. Combining an aggregation rule and a social choice function defines a constitution. Voter preferences, defined as linear orders, are extended to complete binary relations by means of some extension rule. A constitution is swap-proof w.r.t. a given extension rule if one cannot find pairs of voters who, by exchanging their preferences get better off (w.r.t. their extended preference over sets). We consider four specific extension rules: Nehring, Kelly, Fishburn, and Gärdenfors. We establish sufficient conditions for the swap-proofness of a constitution w.r.t. each extension rule. Special attention is paid to majority constitutions, where both the aggregation rule and the social choice function are based on simple majority voting. We show that swap-proofness for majority constitutions pertains to a specific weakening of group strategy-proofness. Moreover, we characterize swap-proof majority constitutions w.r.t. each extension rule. Finally, we show that no constitution based on scoring methods is swap-proof.

Keywords: Representative Democracy, Vote Swapping, Vote exchange, Group Manipulation
JEL code: D71, C70

(DOI:10.1007/s00355-022-01445-z)

We consider collective choice problems where the set of social outcomes is a Cartesian product of finitely many finite sets. We assume that each individual is assigned a two-level preference, defined as a pair involving a vector of strict rankings of elements in each of the sets and a strict ranking of social outcomes. A voting rule is called (resp. weakly) product stable at some two-level preference profile if every (resp. at least one) outcome formed by separate coordinate-wise choices is also an outcome of the rule applied to preferences over social outcomes. We investigate the (weak) product stability for the specific class of compromise solutions involving q-approval rules, where q lies between 1 and the number I of voters. Given a finite set X and a profile of I linear orders over X, a q-approval rule selects elements of X that gather(s) the largest support above q at the highest rank in the profile. Well-known q-approval rules are the Fallback Bargaining solution (q=I) and the Majoritarian Compromise (q=⌈(I/2)⌉). We assume that coordinate-wise rankings and rankings of social outcomes are related in a neutral way, and we investigate the existence of neutral two-level preference domains that ensure the weak product stability of q-approval rules. We show that no such domain exists unless either q=I or very special cases prevail. Moreover, we characterize the neutral two-level preference domains over which the Fallback Bargaining solution is weakly product stable. 

Keywords: Compromise, Multidimensional voting, Consistency, Product stability
JEL code: D71

(DOI:10.1007/s00355-022-01387-6)

We consider seat-posted (or designated-seat) committee elections, where disjoint sets of candidates compete for each seat. We assume that each voter has a collection of seat-wise strict rankings of candidates, which are extended to a strict ranking of committees by means of a preference extension. We investigate conditions upon preference extensions for which seat-wise Condorcet candidates, whenever all exist, form the Condorcet winner among committees. We characterize the domain of neutral preference extensions for which the committee of seat-wise winners is the Condorcet winning committee, first assuming the latter exists (Theorem 1) and then relaxing this assumption (Theorem 2). Neutrality means that preference extensions are not sensitive to the names of candidates. Moreover, we show that these two characterizations can be stated regardless of which preference level is considered as a premise.

Keywords: Committee selection, Condorcet choice rules, Separability, Preference extensions, Lexicographic property
JEL code: D71
(DOI:10.1007/s10058-021-00260-9)

We consider two-tiers elections based on closed party-list proportional representation (PLPR), where party platforms involving multiple dichotomous issues are endogenously determined by their supporters’ preferences (via issue-wise simple majority voting). Assuming that voters compare platforms according to the criterion of the Hamming distance and provided a high enough number of voters and issues, we show that the outcome of PLPR may be Pareto dominated by the multiple referendum outcome defined as the issue-wise majority will in the whole electorate. We refer to this situation as the PLPR paradox. We characterize the set of party platforms for which the PLPR paradox is possible. We also investigate several restrictions upon voting situations that may be sufficient for avoiding the paradox.

Keywords: Proportional representation, Majority rule, Voting paradox
JEL code: D72
(DOI: 10.1007/s10726-020-09713-y)

Manipulation of indirect elections by vote pairing occurs when a group of voters in different electoral bodies secures a jointly preferred winner by performing pairwise exchanges of votes. We show that in elections involving a large enough number of districts, each with a large enough size, no reasonable constitution is immune to vote-pairing. 

Keywords: Representative democracy, Vote pairing, Group manipulation
JEL code: D71, C70
(DOI : 10.1016/j.econlet.2017.09.040)

We study a model of indirect elections where voters having weak orders as preferences over finitely many alternatives are distributed across a given set of districts. In each district preferences are aggregated into a district preference, and a voting rule selects one or several alternatives from the profile of district preferences. The referendum paradox holds at some profile and some distribution of voters across districts if the outcome of indirect elections does not coincide with the one of direct elections. We prove that whenever an indirect election procedure is separable, it is exposed to the referendum paradox if and only if it is exposed to a stronger version of the referendum paradox, where direct and indirect elections give different outcomes for any distribution of the voters across districts. We prove that many indirect elections based on a tournament solution are separable, whereas some based on a scoring rule are not. Finally, we show that all indirect elections based on a scoring rule are exposed to the strong referendum paradox.

Keywords: Voting paradox, Referendum paradox, Representative democracy, Gerrymandering
JEL code: D72
(DOI : 10.1007/s11135-016-0361-y)

Selecting a committee that meets diversity and proportionality criteria is a challenging endeavor that has been studied extensively in recent years. This task becomes even more challenging when some of the selected candidates decline the invitation to join the committee. Since the unavailability of one candidate may impact the rest of the selection, inviting all candidates at the same time may lead to a suboptimal committee. Instead, invitations should be sequential and conditional on which candidates invited so far accepted the invitation: the solution to the committee selection problem is a query policy. If invitation queries are binding, they should be safe: one should not query a candidate without being sure that whatever the set of available candidates possible at that stage, her inclusion will not jeopardize committee optimality. Assuming approval-based inputs, we characterize the set of rules for which a safe query exists at every stage. In order to parallelize the invitation process, we investigate the computation of safe parallel queries and show that it is often hard. We also study the existence of safe parallel queries with respect to proportionality axioms such as extended justified representation.

(link to full version)

We consider the elections of a seat-posted committee, and investigate the propensity of seat-wise majority voting to choose a committee that fulfills the majority will with respect to preferences over committees. Voters have seat-wise preferences and preferences over committees are derived from seat-wise preferences by means of a neutral preference extension. Neutrality means that the names of candidates do not play any role. The majority committee paradox refers to a situation where a Condorcet winner exists for each seat, and a Condorcet winner committee also exists but does not coincide with the combination of seat-wise Condorcet winners. The majority committee weak paradox refers to a situation where the combination of seat-wise Condorcet winners is not a Condorcet winner among committees. We characterize the domains of preference extensions immune to each of the paradoxes.

Keywords: Committee election, Voting paradoxes, Majority voting, Separable preferences
JEL code: D71
(Link)

We investigate group manipulation by vote exchange in two-tiers elections, where voters are first distributed into districts, each with one delegate. Delegates’ preferences result from aggregating voters’ preferences district-wise by means of some aggregation rule. Final outcomes are sets of alternatives obtained by applying a social choice function to delegate profiles. An aggregation rule together with a social choice function define a constitution. Voters’ preferences over alternatives are extended to partial orders over sets by means of either the Kelly or the Fishburn extension rule. A constitution is Kelly (resp. Fishburn) swapping-proof if no group of voters can get by exchanging their preferences a jointly preferred outcome according to the Kelly (resp. Fishburn) extension. We establish sufficient conditions for swapping-proofness. We characterize Kelly and Fishburn swapping-proofness for Condorcet constitutions, where both the aggregation rule and the social choice function are based on simple majority voting.

Keywords: Representative democracy, Vote swapping, Vote exchange, Group manipulation
JEL code: D71,  C70
(Link)

New version coming soon!

In this paper, we theoretically analyze the effect of the wealth levels and underlying bankruptcy rules on investment decisions in a bankruptcy game setting. There is considerable literature studying the axiomatic properties of different bankruptcy rules and answering how bankrupt values should be divided. However, very few studies focus on the relation between the underlying bankruptcy rule to be used and the investment decisions, and this study aims to contribute to this literature. More specifically, we develop a bankruptcy model that enables us to study the wealth effect, using DARA (Decreasing Absolute Risk Aversion) as investors' utility function. This utility function specification enables us to include the uninvested portion of their wealth in the utility equation and reach intuitive equilibrium behavior. By using Nash Equilibrium as the solution concept, investment levels are reviewed under different rules and parameters. We focus on the three commonly studied bankruptcy rules in the literature, namely, Proportional (PRO), Equal Awards (EA), and Equal Losses (EL). These rules are examined separately and as combinations to see which rule(s) leads to higher total investment levels. It is shown that an agent's equilibrium investment is affected by her own wealth and the wealth of the other agents. There is a two-agents case for computational and illustrative purposes in the last part of the paper to complement the theoretical part.

Keywords: Bankruptcy Rules, Wealth Effect, Decreasing Absolute Risk Aversion (DARA), Total Investment
JEL code: C72, D78, G33

We study a game of political competition where two parties adopt ambiguous strategies. Each voter identifies each party with a single policy, but both parties can adopt ambiguous platforms, where each policy x is given a weight p(x) in [0; 1], which gives the probability that each voter considers x as the promoted policy. Assuming that party payoffs are defined by the probability to win elections, we prove that, in the two-policy case, there exists a unique Nash equilibrium when elections take place in a single district, where both parties unambiguously promote the majority will in the electorate. Equilibria may fail to exist in the case of multiple-district elections, regardless of whether the referendum paradox holds or not. We also show that, if there are more than two policies, the same results hold for the game where p(x) defines the proportion of voters identifying x as the party policy, and where payoffs are defined by the sign of net expected pluralities (Laslier, 2000).

Keywords: Electoral competition, Representative democracy, Referendum paradox, Ambiguity
JEL code: D71, C70