Tacit co-operation in three-alternative non-cooperative voting games: A new model of sophisticated behaviour under the plurality procedure

Two models, one due to Farquharson and the other to Niemi-Frank, attempt to account for sophisticated voting behaviour when the voters' preference orderings are common knowledge and communication among Voters is impossible. Having subjected these two models to experimental testing, we have found them lacking. Hence, we propose a new model of sophisticated voting for 3-alternative n-person non-cooperative games under the plurality procedure, which can be extended to other voting procedures and more than three alternatives.The model assumes that voters whose first preference is (one of) the Condorcet winner(s) will (tacitly) co-ordinate their strategies and vote for their first preference, and specifies the conditions under which voters whose second preference is (one of) the Condorcet winner(s) will vote for their second (rather than their first) preference. Consequently, our model predicts that: (i) if there is a single Condorcet winner he or she will be elected; (ii) if there is more than one Condorcet winner the final outcome will be a tie between them; and (iii) when there are cyclical majorities with a single maximin alternative, this alternative will be elected.     

President Perot or Fundamentals of Voting Theory Illustrated with the 1992 Election

Different voting systems can lead to different election outcomeseven when voter preferences are held constant. Using the 1992 election as anexample, it is shown how the outcome of every positional votesystem can be found. Similarly, every possible cumulative andapproval vote outcome is shown. Multiple vote systems, likeapproval and cumulative voting, have disturbing properties. Usingthe 1992 election as illustration, it is shown how a candidate whowins under every positional vote system, who wins every pairwisevote (i.e. is the Condorcet winner), and who has the most firstplace and least last place votes may nevertheless lose underapproval or cumulative voting. Similarly, it is shown how acandidate who loses under every positional system, who loses everypairwise vote (i.e. is the Condorcet loser), and who has the leastfirst place and most last place votes may nevertheless win underapproval or cumulative voting.     

Vote trading: An experimental study

This essay reports the results of ninety 3-person and 5-person bargaining experiments using several alternative vote trading scenarios. These experiments are designed to test: (1) Riker and Brams' controversial hypothesis that vote trading can yield inferior outcomes as against the alternative hypothesis that vote trading induces ‘market-like’ efficiency in voting bodies; (2) the relative adequacy of several game theoretic solution concepts for vote trading games without a Condorcet winner (core); and, (3) the adequacy of the core itself. First, on the basis of eighteen experiments with binding commitments, we find some support for Riker and Brams' hypothesis: in nine trials, subjects first trade to a Pareto dominated outcome. In five of these trials, however, these outcomes are eventually displaced by Pareto efficient ones. Without binding commitments, however, we find little support for the ‘paradox of vote trading’ hypothesis. Specifically, while six of seven trials of a 3-person game yield ‘equitable’ outcomes, seventeen trials of several 5-person games without binding commitments strongly support the competitive solution as a cooperative game solution concept, and suggest that the V-set and M ₁ bargaining set are either redundant or useless. Seven trials each of six 5-person games with a condorcet winner (core), however, suggest that usual static solution concepts may be inadequate for treating games with any interesting degree of strategic complexity.     

If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs rule

We consider four factors relevant to picking a voting rule to be used to select a single candidate from among a set of choices: (1) avoidance of Condorcet losers, (2) choice of Condorcet winners, (3) resistance to manipulability via strategic voting, (4) simplicity. However, we do not try to evaluate all voting rules that might be used to select a single alternative. Rather, our focus is restricted to a comparison between a rule which, under the name ‘instant runoff,’ has recently been pushed by electoral reformers in the US to replace plurality-based elections, and which has been advocated for use in plural societies as a means of mitigating ethnic conflict; and another similar rule, the ‘Coombs rule.’ In both rules, voters are required to rank order candidates. Using the instant runoff, the candidate with the fewest first place votes is eliminated; while under the Coombs rule, the candidate with the most last place votes is eliminated. The instant runoff is familiar to electoral system specialists under the name ‘alternative vote’ (i.e., the single transferable vote restricted to choice of a single candidate). The Coombs rule has gone virtually unmentioned in the electoral systems literature (see, however, Chamberlin et al., 1984). Rather than considering the properties of these two rules in the abstract, we evaluate them in the politically realistic situations where voters are posited to have (at least on balance) single-peaked preferences over alternatives. Evaluating the two rules under this assumption, we argue that the Coombs rule is directly comparable in that Coombs is always as good as AV with respect to two of our four criteria and it is clearly superior to AV with respect to one of the four criteria, namely criterion (2), and is potentially inferior only with respect to criterion (3). Key to this argument are two new propositions. The first new result shows that, under the posited assumption, for four alternatives or fewer, AV is always as likely or more likely to select the Condorcet winner than plurality. The second new result shows that, under the same assumptions, the Coombs rule will always select the Condorcet winner regardless of the number of alternatives.     

Approval voting,Condorcet's principle,and runoff elections

Approval voting allows each voter to vote for as many candidates as he wishes in an election but not cast more than one vote for each candidate of whom he approves. If there is a strict Condorcet candidate — a candidate who defeats all others in pairwise contests — approval voting is shown to be the only nonranked voting system that is always able to elect the strict Condorcet candidate when voters use sincere admissible strategies. Moreover, if a strict Condorcet candidate must be elected under ordinary plurality voting when voters use admissible strategies, then he must also be elected under approval voting when voters use admissible strategies, but the converse does not hold. The widely used plurality runoff method can also elect a strict Condorcet candidate when voters use admissible strategies on the first ballot, but some of these may have to be insincere to get the strict Condorcet candidate onto the runoff ballot. Furthermore, there is no case in which the strict Condorcet candidate is invariably elected under the plurality runoff method when voters use admissible first-ballot strategies. Thus, approval voting is superior to the plurality runoff method with respect to the Condorcet principle in its ability to elect the strict Condorcet candidate by sincere voting and in its ability to guarantee the election of the strict Condorcet candidate when voters use admissible strategies. In addition, approval voting is more efficient since it requires only one election and is probably less subject to strategic manipulation.     

Condorcet Polling

For a single-winner multi-candidate election, it is broadly accepted that the Condorcet candidate (if one exists) should win. Voting systems do not always elect the Condorcet winner. Public opinion polls are not generally designed to try to identify a Condorcet candidate. They could easily be constructed to do so, however. The resulting process may be called Condorcet polling, for which various designs are presented herein. Information from Condorcet polling may enable some voters, under a plurality or runoff system, to bring about an outcome they prefer by voting strategically for the Condorcet candidate when they would not otherwise do so.     

Minor candidates as kingmakers

We consider a sequential entry model with three candidates who cannot commit to any policy announcement during the campaign. The study focuses on how a minor candidate, who wins only when unopposed, influences the electoral outcome. We show that unless the Condorcet winner (i.e., the winner in every pairwise vote) coincides with the grand winner (i.e., the winner of the three-candidate competition), the minor candidate is a kingmaker in the sense that his preferred rival wins regardless of the order of the entry decisions. To influence the outcome, the minor candidate could either (i) enter strategically without any chance to win, or (ii) enter if and only if the Condorcet winner already has entered.     

The architecture of the Electoral College,the House size effect,and the referendum paradox

Using data from U.S. presidential elections, we show how seemingly insignificant changes to what we call the “architecture” of the Electoral College can cause different candidates to be elected President, even when no one changes how they vote. We consider varying the size of the House of Representatives, the method of apportionment, the number of “Senate” electoral votes cast by each state, and the lower bound on the number of “House” electoral votes cast by each state. We consider, in particular, elections with a “referendum paradox”. In these elections, the electoral vote winner is not the popular vote winner. Our work extends Neubauer and Zeitlin (2003) who analyzed the case of the 2000 election. We give an explanation for the effects that we observe in the data.     

Weak Condorcet winner(s) revisited

A strong Condorcet winner (SCW) is an alternative, x, that a majority of voters rank higher than z, for every other alternative, z. A weak Condorcet winner (WCW) is an alternative, y, that no majority of voters rank below any other alternative, z, but is not a SCW. There has been some confusion in the voting/social choice literature as to whether particular voting rules that are SCW-consistent are also WCW-consistent. The purpose of this paper is to revisit this issue, clear up the confusion that has developed, and determine whether three additional SCW-consistent voting rules—that as far as we know have not been investigated to date regarding their possible WCW consistency—are indeed WCW-consistent.     

Group Components of U.S. Presidential Voting Across the States

Examinations of the sociodemographic group foundations of presidential voting and electoral coalitions rely on national samples (e.g., the American National Election Studies). However, recent developments in the state party and electoral change literature suggest variation across the United States in the group bases of political coalitions and in the process of electoral adjustment. Moreover, the strategic implications of the electoral college suggest a focus on state electorates. We estimate multivariate, group-based logit models of presidential vote choice using 1988 CBS/ New York Times and 1992 Voter Research and Surveys exit poll data from each of the largest states. Our results reveal noteworthy variation in the nature of group influences on presidential voting, in the composition of presidential electoral coalitions, and in cleavage structures across the states. This mapping exercise suggests limitations in theoretical and empirical accounts of presidential voting, political cleavage, and electoral change that do not accommodate the geopolitical diversity of the United States.     

Collective decision-making of voters with heterogeneous levels of rationality

This paper studies the collective decision-making processes of voters who have heterogeneous levels of rationality. Specifically, we consider a voting body consisting of both rational and sincere voters. Rational voters vote strategically, correctly using both their private information and the information implicit in other voters’ actions to make decisions; sincere voters vote according to their private information alone. We first characterize the conditions under which the presence of sincere voters increases, reduces, or does not alter the probabilities of making correct collective decisions. We also discuss how the probabilities change when the incidence of sincere voters in the population varies. We then characterize the necessary and sufficient condition under which informational efficiency can be achieved when sincere voters coexist with rational voters. We find that when sincere voters are present, supermajority rules with high consensus levels are not as desirable as they are in rational voting models, as informational efficiency fails under such voting rules.

     

A note on Nanson's rule

E.J. Nanson, in 1883, proposed a voting rule designed mainly to select the Condorcet winner. Unfortunately, much of the collective choice literature has misstated Nanson's rule. This note shows that the misstated versions of the rule and Nanson's actual rule can yield different outcomes, and it provides a counter example, which shows that Nanson's rule, although it satisfies the strong Condorcet condition, does not satisfy the weak Condorcet condition.     

Dead Heat: The 2006 Public Choice Society Election

In 2006, the Public Choice Society chose a new president using approval voting. There were five candidates, and the election was extremely close. We indicate the sources of support of the different candidates, based in part on spectral analysis, by voters who cast between one and five votes. Using preference information that was also gathered, we show that two candidates different from the approval voting winner, including the apparent Condorcet winner, might have won under different voting systems. Because most voters did not indicate their complete preference rankings, however, these differences are hardly robust, especially since the outcome was essentially a dead heat.     

A precise restatement and extension of Black's theorem on voting orders

Duncan Black originally suggested that ‘the later any motion enters the voting, the greater its chance of adoption.’ We formalize this reasoning as a theorem, which we prove. We then specify the implications of this theorem for agenda control. If the social preference is known and there is no majority winner, one is best off choosing a specific voting order, which may or may not have one's most preferred alternative last. If the social preference is unknown, the optimal agenda is one in which voting is in reverse order of one's preferences.     

The structure of the Banks set

We consider a new solution set for majority voting tournaments recently proposed by Banks (1985), and we examine its internal structure. In particular, we demonstrate that, in the absence of a Condorcet winner, there is always a cycle including precisely the points in the Banks set. We introduce the concept of “external stability” in order to facilitate analysis.     

The problem of indeterminacy in approval,multiple, and truncated voting systems

It is well known that a plurality election need not reflect the true sentiments of the electorate. Some of the proposed reform procedures, such as approval and cumulative voting, share the characteristics that there are several ways to tally each voter's preferences. Voting systems that permit truncated ballots share this feature. It is shown that the election results for any such procedure can be highly indeterminate; all possible election results can occur with the same choice of sincere voters. This conclusion of indeterminacy holds even when measures of voters' sentiments, such as the existence of a Condorcet winner or even much stronger measures, indicate there is considerable agreement among the voters. Then, multiple systems are compared with all standard tallying procedures. For instance, a corollary asserts it is probable for the plurality voting method to elect the Condorcet winner while approval voting has an indeterminate outcome.The work of both authors was supported, in part, by NSF Grant IST 8415348. This work was partially stimulated by comments concerning the assertions about approval voting and other systems that are discussed in (Saari, 1987). In particular, we are pleased to thank S. Brams, Carl Simon, and the other participants for their questions after one of us presented (Saari, 1987) at the July, 1984, Workshop on Mathematical Models in Political Science held at the University of Minnesota in Minneapolis.     

The impact of compulsory voting on inequality and the quality of the vote

Abstract

Democratic elections imply that the electorate holds incumbents accountable for past performance, and that voters select the party that is closest to their own political preferences. Previous research shows that both elements require political sophistication. A number of countries throughout the world have a system of compulsory voting, and this legal obligation boosts levels of voter turnout. Under such rules, citizens with low levels of sophistication in particular are thought to turn out to vote in higher numbers. Is it the case that the quality of the vote is reduced when these less sophisticated voters are compelled to vote? This article investigates this claim by examining the effect of compulsory voting on accountability and proximity voting. The results show that compulsory voting reduces stratification based on knowledge and level of education, and proximity voting, but it does not have an effect on economic accountability. The article concludes with some suggestions on how systems of compulsory voting might mitigate the strength of political sophistication in determining the quality of the vote decision process.     

Dealing in Discipline: Party Switching and Legislative Voting in the Italian Chamber of Deputies, 1988–2000

Compared to U.S. political parties, parties in Italy (and Europe generally) are quite cohesive. Rarely do members of parliament vote against their copartisans in legislative divisions. Yet in Italy in recent years, legislators switch parties with seeming abandon. Between 1996 and spring 2000, one out of four deputies in the Chamber of Deputies switched parties at least once, compared to only 20 switches in the U.S. Congress from 1947 to 1997 ( Nokken 2000 ). We examine the relationship between switching and observed party unity in Italy by focusing on individual legislators' switching decisions and voting behavior. Overall, switchers move out of highly disciplined parties, suggesting that they switch partly in order to escape strong discipline.     

Direct democracy and regional integration: Citizens' perceptions of treaty implications and the Irish reversal on Lisbon

Analyses of voting in European Union referendums typically distinguish between ‘second‐order’ effects and the impact of substantive ‘issues’. In order to explain change in referendum outcome, two types of substantive issues are distinguished in this article. Focusing on Irish voting in the Lisbon Treaty referendums and using data from post‐referendum surveys, it is found that perceptions of treaty implications outperform underlying attitudes to EU integration in predicting vote choice at both referendums, and perceptions of treaty implications are strong predictors of vote change between the referendums. The findings have broadly positive implications for normative assessments of the usefulness of direct democracy as a tool for legitimising regional integration advance.