Restricted Manipulation in Iterative Voting: Convergence and Condorcet Efficiency

We compare the Condorcet Efficiencies of the plurality rule, the negative plurality rule, and the Borda rule to examine what might be gained by using a voting rule that requires candidate rankings. Previous analyses have considered (1) voting situations for which the three rules determine different candidates as winners (and the Condorcet Efficiencies might actually differ across the three rules) as well as (2) refined measures of Group Mutual Coherence that assess the presence of underlying models of rational behavior that governs voter preferences. In this paper, we present the final step of this particular line of analysis by analyzing (1) and (2) simultaneously. We conclude that there is a considerable benefit on the basis of Condorcet Efficiency from asking voters to submit candidate rankings in three-candidate elections.

public choice theory to familiarize themselves with the masterworks in political theory. [6, 99-101] With this call for broader scholarship in mind, we were impressed in a recent reading of Aristotle's Politics with two aspects of his thought that, in our opinion, should be brought to the attention of modern scholars. First, we examine Aristotle's commentary on voting rules and propose a resolution of an important conflict in the literature interpreting his position on majority rule. Second, we examine Aristotle's general, but not unqualified, belief, presented mainly in analogies, that the collective wisdom of a people was superior to that of the lawgiver's. The main question that we address is whether this superior wisdom was to be embodied in the degree of restrictiveness in the voting rule or in the limits of suffrage. We show that under the former interpretation Aristotle hypothesized an important link between the restrictiveness of the voting rule and the amount of information that is brought to bear in making collective decisions. This is a link perhaps overlooked by modern writers on public choice, and we examine its effect in a model of the optimal voting rule, originally developed by J. M. Buchanan and G. Tullock, to observe its impact on the minimum cost decision rule. [7, 63-84]

Voting Behavior in One-Shot and Iterative Multiple Referenda

We study the effect on the participation rate of employing different voting rules in the context of the problem to allocate a fixed monetary budget to two different public projects. Specifically, we compare the mean rule according to which the average of the individually proposed allocations is implemented with the median rule which chooses the allocation proposed by the median voter as the social outcome. We report the results of a field experiment in which subjects (students of KIT) could allocate money to fund two different public projects, the student’s bike shop and a campus garden project. The treatment variable was the collective decision rule employed. While the mean and median rules have very different properties in theory, we found no significant treatment effect on the participation rate. Our results nevertheless shed important light on the use of different voting rules in the context of budget allocation in practice.

The standard two-stage voting rules Plurality Elimination and Negative Plurality Elimination are evaluated relative to Borda Rule on the basis of Condorcet Efficiency. Theoretical analysis shows that the Condorcet Efficiency of these two-stage voting rules consistently marginally dominate Borda Rule as the various measures of group mutual coherence change. However, it is also found that the probability that either of the two-stage rules will elect a different candidate than the winner by Borda Rule is quite small. Empirically-based analysis reinforces this observation to lead to the general conclusion that the additional effort that is required to use a two-stage voting rule, rather than simply using Borda Rule, is not likely to be worthwhile. Other benefits are considered that result from using Borda Rule rather than a two-stage voting rule.

In this paper, we attempt to explain the underlying strategic incentives confronting individuals when they must make a collective decision over a set of alternatives and each has information that is decision-relevant for others. A significant literature has emerged in formal political theory over the past several years that focuses on such problems, paying particular attention, first, to the extent to which voting can be expected to aggregate committee members' information and, second, to the role of communication among committee members prior to voting. Inter alia, this literature reveals a surprisingly subtle interaction between the voting rules used to make decisions and the incentives for committee members to share information prior to voting.

We have observed that some recent results cast serious doubts uponthe applicability of majority rule as a device for forming a group-willout of individual wills. On the other hand, majority rule has manyproperties to recommend it. Especially, the decisiveness propertyshould please anyone interested in efficiency considerations. How-ever, when it comes to the second major criterion, consistency, major-ity rule with pairwise comparison of alternatives often leads to diffi-culties. Indeed, we have seen that there is typically no core or Con-dorcet winner when the policy space is multidimensional real space.And when there is no core, the situation looks grim indeed for major-ity rule: literally any outcome may become the majority winner. Somerelief of this very strong result is provided by the fact that no majoritycycles can go through the outcomes that belong to the M2 bargainingset. This finding, of course, solves the problem facing majority rule onlyinsofar as a method can be devised by which the M2 bargaining set out-comes can be determined in a reliable way. Here, thus, we are con-fronted with the sincerity problem. The absence of the core can lead to chaotic trajectories of votingoutcomes in multidimensional space. However, there are ways of over-coming the difficulties if a sufficient amount of flexibility is allowedin institutional design. If we consider the various nonequilibrium solu-tion concepts, and the minimax set in particular, we can always designa decision rule which (under mild restrictions) makes the vote-maximiz-ing outcome trajectory enter the solution set. The flexibility needed isgreat, indeed: the decision rule, λ-majority, must be alterable for eachvoting situation. The determination of the winner in a pair-wise comparison of alter-natives is not the main source of the difficulties surrounding majorityrule. As a matter of fact, the method of building the agenda is the mostcrucial part of the problem of cyclical majorities. In multicandidatecontests the occurrence of cyclical majorities has motivated the studyof other.decision procedures in addition to the majority comparisonof alternatives. The most recent proposal, approval voting, is morestrategyproof than its competitors in the class of single-ballot, non-ranked voting systems. At the moment the relationships betweenapproval voting and majority rule are not clear. It seems that furtherstudy of the interrelationships of these procedures is called for. Thefeatures in which approval voting is superior to majority comparisonof pairs of alternatives, however, are not, strictly speaking, procedural,but are external to the method of making collective decisions.

Consider a three-alternative election with n voters and assume that preferences are single-peaked. LetC(λ,n) be the Condorcet efficiency of the rule that assigns 1,λ and 0 points (respectively) to each first, second and third place vote. An exact representation is obtained forC(λ,∞). This relation shows that Borda rule (λ=1/2) is not the most efficient rule. In addition to this result, exact closed form relations are provided forC(0,n),C(1/2,n) andC(1,n). All these relations are obtained by assuming that every admissible configuration of preferences is equally likely to occur.

The Condorcet efficiency of Borda Rule with anonymous voters

Results from voting theory are increasingly used when dealing with collective decision making in computational multiagent systems. An important and surprising phenomenon in voting theory is the No-Show Paradox (NSP), which occurs if a voter is better off by abstaining from an election. While it is known that certain voting rules suffer from this paradox in principle, the extent to which it is of practical concern is not well understood. We aim at filling this gap by analyzing the likelihood of the NSP for six Condorcet extensions (Black's rule, Baldwin's rule, Nanson's rule, MaxiMin, Tideman's rule, and Copeland's rule) under various preference models using Ehrhart theory as well as extensive computer simulations. We find that, for few alternatives, the probability of the NSP is rather small (less than 4% for four alternatives and all considered preference models, except for Copeland's rule). As the number of alternatives increases, the NSP becomes much more likely and which rule is most susceptible to abstention strongly depends on the underlying distribution of preferences.

A Theory of Corporate Boards with Endogenous Information Collection, Optimal Compensation and Strategic Voting: When Do Independent Boards Dominate Rubberstamping Ones?

Chapter 14 - Voting and Agenda Setting in Political Science and Economics

Proof of Stake (PoS) protocols rely on voting mechanisms to reach consensus on the current state. If an enhanced majority of staking nodes, also called validators, agree on a proposed block, then this block is appended to the blockchain. Yet, these protocols remain vulnerable to faults caused by validators who abstain either accidentally or maliciously.To protect against such faults while retaining the PoS selection and reward allocation schemes, we study weighted voting in validator committees. We formalize the block creation process and introduce validators’ voting profiles which we update by a multiplicative weights algorithm relative to validators’ voting behavior and aggregate blockchain rewards. Using this framework, we leverage weighted majority voting rules that optimize collective decision making to show, both numerically and analytically, that the consensus mechanism is more robust if validators’ votes are appropriately scaled. We raise potential issues and limitations of weighted voting in trustless, decentralized networks and relate our results to the design of current PoS protocols.

The existence of equilibrium in incomplete markets and the objective function of the firm