A note on the Condorcet Jury Theorem with supermajority voting rules

Social Choice and Welfare -     

On Ehrhart polynomials and probability calculations in voting theory

In voting theory, analyzing the frequency of an event (e.g. a voting paradox), under some specific but widely used assumptions, is equivalent to computing the exact number of integer solutions in a system of linear constraints. Recently, some algorithms for computing this number have been proposed in social choice literature by Huang and Chua (Soc Choice Welfare 17:143–155 2000) and by Gehrlein (Soc Choice Welfare 19:503–512 2002; Rev Econ Des 9:317–336 2006). The purpose of this paper is threefold. Firstly, we want to do justice to Eugène Ehrhart, who, more than forty years ago, discovered the theoretical foundations of the above mentioned algorithms. Secondly, we present some efficient algorithms that have been recently developed by computer scientists, independently from voting theorists. Thirdly, we illustrate the use of these algorithms by providing some original results in voting theory. Helpful comments by Philippe Clauss and his team are gratefully acknowledged.     

The probability of the paradox of voting for weak preference orderings

In this paper the probability of the voting paradox for weak orderings is calculated analytically for the three-voter-three-alternative case. It appears that the probability obtained this way is considerably smaller than in the corresponding case for linear orderings. The probability of intransitive majority relations for weak orderings in the 3 × 3 case is calculated as well, both with unconcerned and with concerned voters. Basic in the calculations are three theorems which are formulated in the field of domain conditions and restricted preferences. Received: 18 February 1997 / Accepted: 21 October 1997     

The probability of an undominated central voter in 2-dimensional spatial majority voting

Place a voter v in the plane and scatter additional voters at random about v. What are v's chances of being undominated? When the number of voters, n, is even, this probability is known to be nonzero [5]. Here we prove that the exact probability is 2^?(n?2).     

Yes-no voting

Yes-No (Y-N) voting is a voting method for choosing a governing coalition in a parliament after the seating of its members. Each member can designate a party to be Y (it must be included in the governing coalition), N (it must be excluded from the governing coalition), or neither (it may be either in or out of the governing coalition). The majority coalition acceptable to the most voters, because it includes all parties that these voters designate Y and no parties that they designate N, is given the first opportunity to form a government. Possible combinations of majority coalitions that a member might vote for are derived, including ones based on consistent and interval voting strategies. Examples illustrate a number of different phenomena, such as when a rational voter might not be loyal to his or her party by designating it Y.     

Monotonicity in Condorcet Jury Theorem

Consider a committee of experts dealing with dichotomous choice problem, where the correctness probabilities are all greater than We prove that, if a random subcommittee of odd size m is selected randomly, and entrusted to make a decision by majority vote, its probability of deciding correctly increases with m. This includes a result of Ben-Yashar and Paroush (2000), who proved that a random subcommittee of size m3 is preferable to a random single expert.Research supported in part by the Israel Science Foundation (Grant\#186/01)     

Fairness and voting

In this paper we present a two-period model in which we examine how concern about fairness might affect voter behavior. We show that in the first period politicians choose the median voter’s position even if this does not correspond to their bliss points and neither they nor the voters can commit to a particular action. Moreover, concern about fairness creates substantial incumbency advantages. Our results hold even if voters care very little about fairness.     

Jury theorems with multiple alternatives

I consider a game in which imperfectly informed jurors vote to select one of several possible choices when there is a natural ordering of the possibilities. Each juror votes for the largest alternative the juror would like to implement, and the alternative that is selected is the largest alternative supported by a given number of jurors. For non-unanimous voting rules, the probability of a mistaken judgment goes to zero as the number of jurors goes to infinity. I also give necessary and sufficient conditions to obtain asymptotic efficiency under unanimous voting rules, and show that unanimous rules may lead to a bias in which moderate outcomes are never chosen.     

Approval voting and positional voting methods: Inference,relationship, examples

Approval voting is the voting method recently adopted by the Society for Social Choice and Welfare. Positional voting methods include the famous plurality, antiplurality, and Borda methods. We extend the inference framework of Tsetlin and Regenwetter (2003) from majority rule to approval voting and all positional voting methods. We also establish a link between approval voting and positional voting methods whenever Falmagne et al.s (1996) size-independent model of approval voting holds: In all such cases, approval voting mimics some positional voting method. We illustrate our inference framework by analyzing approval voting and ranking data, with and without the assumption of the size-independent model. For many of the existing data, including the Society for Social Choice and Welfare election analyzed by Brams and Fishburn (2001) and Saari (2001), low turnout implies that inferences drawn from such elections carry low (statistical) confidence. Whenever solid inferences are possible, we find that, under certain statistical assumptions, approval voting tends to agree with positional voting methods, and with Borda, in particular.Michel Regenwetter thanks the National Science Foundation for funding this research through NSF grant SBR 97-30076. Both authors thank the Fuqua School of Business for financially supporting their collaboration. Most of this research was done while Regenwetter was a faculty member at Fuqua. We thank Prof. Steven Brams for his valuable comments as a discussant of a previous version of this paper, given at the 2002 Public Choice meeting, and Prof. Donald Saari for his helpful comments in conversations and on another draft. We also thank the editor in charge and a referee for their valuable comments. Tsetlin acknowledges the support of the Centre for Decision Making and Risk Analysis at INSEAD.     

Probabilistic Borda rule voting

An alternative voting system, referred to as probabilistic Borda rule, is developed and analyzed. The winning alternative under this system is chosen by lottery where the weights are determined from each alternatives Borda score relative to all Borda points possible. Advantages of the lottery include the elimination of strategic voting on the set of alternatives under consideration and breaking the tyranny of majority coalitions. Disadvantages include an increased incentive for strategic introduction of new alternatives to alter the lottery weights, and the possible selection of a Condorcet loser. Normative axiomatic properties of the system are also considered. It is shown this system satisfies the axiomatic properties of the standard Borda procedure in a probabilistic fashion.I thank Nic Tideman and Andrew Yates for helpful comments and especially appreciate detailed criticism from Keith Dougherty which greatly improved the exposition of the text. In addition, Don Saari, in his role as editor, suggested a number of clarifications and an important extension for which I am grateful. I accept full responsibility for any remaining shortcomings.     

Strategic voting and nomination

Using computer simulations based on three separate data generating processes, I estimate the fraction of elections in which sincere voting is a core equilibrium given each of eight single-winner voting rules. Additionally, I determine how often each rule is vulnerable to simple voting strategies such as ‘burying’ and ‘compromising’, and how often each rule gives an incentive for non-winning candidates to enter or leave races. I find that Hare is least vulnerable to strategic voting in general, whereas Borda, Coombs, approval, and range are most vulnerable. I find that plurality is most vulnerable to compromising and strategic exit (causing an unusually strong tendency toward two-party systems), and that Borda is most vulnerable to strategic entry. I use analytical proofs to provide further intuition for some of my key results.     

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.     

Inequality in voting power

This paper deals with the assessment of inequality in the distribution of voting power. Voting power is evaluated through a general concept of power measurement based on both the voting rule and the probability distribution over vote configurations. This general concept includes as particular cases the most usual power indices and other extensions of this traditional concept. Thus no particular power index is privileged in our approach. An inequality index over the class of all power profiles with a given number of voters generated by this general measure is then singled out by requiring reasonable properties. In order to compare profiles with different numbers of voters, two alternative principles are considered, which extend the said index consistently in two ways.Previous versions of this work were circulated under different titles. We gratefully acknowledge financial support from the Spanish Ministerio de Ciencia y Tecnología under the Ramón y Cajal programme and under projects PB96-0247 and BEC2000-0875; from the European Commission under the Training and Mobility of Researchers programme (contract FMRX - CT966-0055); and from the Instituto Valenciano de Investigaciones Económicas. We also thank two anonymous referees for their comments.2 October 2001     

Stability of voting games

This paper generalizes the result of Le Breton and Salles (1990) about stable set (far-sighted core of order 1) for voting games to far-sighted core of arbitrary order. Let m be the number of alternatives, n be the number of voters and G(n,k) be a proper symmetric simple game in which the size of a winning coalition is greater or equal to k. It is shown that the far-sighted core of order d for G(n,k) is nonempty for all preference profiles and for all n and k with n/(n–k)=v ¹ iff m(d+1)(v–1).This paper is part of my dissertation. I am grateful to my thesis advisor Leonid Hurwicz for his guidance and encouragement. I would like to thank Edward Green, Lu Hong, James Jordan, Andrew McLennan, Herve Moulin and Marcel Richter for their very helpful suggestions. Especially a referee and Maurice Salles made many good comments. Of course, any errors that remain are the sole responsibility of the author.     

Iterative elimination of weakly dominated strategies in binary voting agendas with sequential voting

This paper describes the correspondences between the results given by backward induction (BI) and iterative elimination of weakly dominated strategies (IEWDS) in binary voting agendas with sequential voting. When the voters have strict preferences over all candidates, the strategies that survive IEWDS all select the unique candidate selected by the BI strategy profiles. But if some voters are indifferent, this result no longer holds. However, when there are only two candidates, it is possible to demonstrate strong relationships between the results given by BI and IEWDS, even when some voters have indifferences.     

Majority rip-off in referendum voting

Many democracies complement a parliamentarian system with elements of direct democracy, where the electorate decides on single issues by majority voting. A well-known paradox states that in a sequence of referenda one can get from an arbitrary original income distribution to one in which one player gets almost all the cake. In this paper we design a three-player game modelling the sequential modification mechanism. The strategic analysis reveals that the paradox survives even with rational strategic voters and though the right to propose is allocated to each player once: the last player receives almost the entire cake. The result can be extended to the three-party n-voter case and is for some cases similar when we consider a random rather than fixed sequence of proposers.

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