Large Electorates Research Articles

Dislike of general opinion makes for tight elections.

In modern democracies, the outcome of elections and referendums is often remarkably tight. The repetition of these divisive events are the hallmark of a split society; to the physicist, however, it is an astonishing feat for such large collections of diverse individuals. Many sociophysics models reproduce the emergence of collective human behavior with interacting agents, which respond to their environment according to simple rules, modulated by random fluctuations. A paragon of this class is the Ising model which, when interactions are strong, predicts that order can emerge from a chaotic initial state. In contrast with many elections, however, this model favors a strong majority. Here we introduce a new element to this classical theory, which accounts for the influence of opinion polls on the electorate. This brings about a new phase in which two groups divide the opinion equally. These political camps are spatially segregated, and the sharp boundary that separates them makes the system size dependent, even in the limit of a large electorate. Election data show that, since the early 1990s, countries with more than about a million voters often found themselves in this state, whereas elections in smaller countries yielded more consensual results. We suggest that this transition hinges on the electorate's awareness of the general opinion.

Campaign Finance Vouchers Do Not Expand the Diversity of Donors: Evidence from Seattle

Abstract Donating to a campaign is inherently costly, and as a result the composition of campaign donors differs from the composition of the electorate. What happens when the financial barriers to campaign finance participation are removed? This paper analyzes Seattle’s recent campaign finance reform, where all registered voters receive four $25 vouchers to donate to candidates abiding by stricter campaign finance restrictions. Utilizing individual- and census block group-level data combined with administrative donation records, I find that those most mobilized by the availability of vouchers belong to groups already overrepresented within the donor pool. This finding is significant across race, income, past political participation, age, and partisanship. In some cases, the availability of vouchers appears to pull the donor pool further from parity with the larger electorate.

Large elections and interim turnout

We examine the effect of the interim release of turnout information on elections in a model of a large electorate with a common voting cost, private values, and two alternatives. We consider that (i) one group of citizens votes before the rest and that (ii) the individuals of the second group know the first group's turnout—but not the vote tally—before they vote. The alternative with more votes in total is implemented. Our main result is that there is an equilibrium in which the ex-ante probability that each alternative is implemented is one half and, hence, the same as when no information about turnout is released. We also show several equilibrium properties of our political game which both broaden our theoretical understanding of the costly-voting paradigm and provide testable hypotheses about elections. Our insights remain valid for some extensions of our baseline setup.

Examining the limits of the Condorcet Jury Theorem: Tradeoffs in hierarchical information aggregation systems

Condorcet’s Jury Theorem states that the correct outcome is reached in direct majority voting systems with sufficiently large electorates as long as each voter’s independent probability of voting for that outcome is greater than 1/2. Previous research has found that switching to a hierarchical system always leads to an inferior result. Yet, in many situations direct voting is infeasible (e.g., due to high implementation or infrastructure costs), and hierarchical voting may provide a reasonable alternative. This paper examines differences in accuracy rates of hierarchical and direct voting systems for varying group sizes, abstention rates, and voter competences. We derive three main results. First, we prove that indirect two-tier systems differ most from their direct counterparts when group size and number are equal (i.e., when each equals [Formula: see text], where Nd is the total number of voters in the direct system). In multitier systems, we prove that this difference is maximized when group size equals [Formula: see text], where n is the number of hierarchical levels. Second, we show that while direct majority rule always outperforms indirect voting for homogeneous electorates, hierarchical voting gains in accuracy when either the number of groups or the number of individuals within each group increases. Third, we prove that when voter abstention and competency are correlated within groups, hierarchical systems can outperform direct voting. The results have implications beyond voting, including information processing in the brain, collective cognition in animal groups, and information aggregation in machine learning.

Iterative Calculus of Voting under Plurality

We formalize a voting model for plurality elections that combines Iterative Voting and Calculus of Voting. Each iteration, autonomous agents simultaneously maximize the utility they expect from candidates. Agents are aware of neither other individuals’ preferences or choices, nor of the distribution of preferences. They know only of candidates’ latest vote shares and with that calculate expected rewards from each candidate, pondering the probability that voting for each would alter the election. We define the general form of those pivotal probabilities, then we derive efficient exact and approximated calculations. Lastly, we prove formally the model converges with asymptotically large electorates and show via simulations that it nearly always converges even with very few agents.

Wisdom of the crowd? Information aggregation in representative democracy

The Condorcet Jury Theorem and subsequent literature establish the feasibility of information aggregation in a common-value environment with exogenous policy options: a large electorate of imperfectly informed voters almost always selects the correct policy option. Rather than directly voting for policies, citizens in modern representative democracies elect candidates who make strategic policy commitments. We show that intermediation by candidates sometimes improves policy choices and sometimes impedes information aggregation. Somewhat paradoxically, the possibility of information aggregation by voters encourages strategic conformism by candidates. Correlated information or partisan biases among voters can mitigate the political failure we uncover. We also discuss possible institutional solutions.

Scoring rules, ballot truncation, and the truncation paradox

A voting rule that permits some voters to favor a candidate by revealing only the initial segment of their sincere rankings is said to be vulnerable to the truncation paradox. In this paper, we consider four models for counting truncated ballots, optimistic, pessimistic (the most common), averaged, and round-down. Under the impartial anonymous culture assumption, the choice of model generally has a real impact on truncation-paradox vulnerability, but exceptions exist. When the election is decided by a one-shot scoring rule, the optimistic model is invulnerable to the truncation paradox, but all other models are vulnerable. We identify new voting rules immune to the truncation paradox, such as the Modified Borda Count. To obtain a more complete picture of the impact of processing model, we assess the likelihood of the truncation paradox in three-candidate elections with large electorates, focusing not only on one-shot scoring rules but also scoring rules with one-by-one or below-average elimination. Our assessment confirms that the processing model for truncated ballots may really matter.

Electorate involvement disorder: Universal relationship between the amplitude and electorate size in second round of Brazilian Presidential Election

The voter’s opinion is not expressed exclusively in the vote for one of the candidates, it may be implicit among abstentions or invalid votes, which are blank and null votes. In this study, we investigated the electorate participation in second rounds of Brazilian presidential elections, by consider the list, per ballot box, of the proportions of the three possible “forms of voter participation” in the election result: abstaining from voting, voting blank or null, or voting for one of the candidates, which called here of the electorate involvement. We used the entropy concept of Statistical Mechanics relating these proportions as a measure of the “disorder” of the electoral involvement. We found a generalized logarithmic model, parameter free, for the entropy amplitude per city as a function of the electorate size, showing that the range of the electorate involvement disorder is maintained over all election years and tends to grow more slowly in the larger electorates.

Manipulable outcomes within the class of scoring voting rules

Coalitional manipulation in voting is considered to be any scenario in which a group of voters decide to misrepresent their votes in order to secure an outcome they all prefer to the outcome of the election when they vote honestly. The present paper is devoted to studying coalitional manipulability within the class of scoring voting rules. For any such rule and any number of alternatives, we introduce a new approach allowing us to characterize all the outcomes that are manipulable by a coalition of voters. This then opens the possibility of determining the probability of manipulable outcomes for some well-studied scoring voting rules in the case of small number of alternatives and large electorates, under a well-known assumption on individual preference profiles.

The effect of handicaps on turnout for large electorates with an application to assessment voting

We analyze the effect of handicaps on turnout. A handicap is a difference in the vote tally between alternatives that is added to the vote tally generated by voters. Handicaps are implicit in many existing democratic procedures. Within a costly voting framework with private values we show that turnout incentives diminish considerably across the board if handicaps are large, while low handicaps yield more mixed predictions. The results extend beyond the baseline model—e.g. by including uncertainty and behavioral motivations—and can be applied to the design of Assessment Voting. This is a new voting procedure where (i) some randomly selected citizens vote for one of two alternatives, and the results are published; (ii) the remaining citizens vote or abstain; and (iii) the final outcome is obtained by applying the majority rule to all votes combined. If the size of the first voting group is appropriate, large electorates choose the majority's preferred alternative with high probability, and average participation costs are moderate or low.

Voting for Compromises: Alternative Voting Methods in Polarized Societies

Democratic societies have been increasingly confronted with extreme, knife-edge election outcomes that affect everybody’s lives and contribute to social instability. Even if political compromises based on social conventions as equity or economic arguments as efficiency are available, polarized societies might fail to select them. We demonstrate that part of the problem might be purely technical and, hence, potentially solvable. We study different voting methods in three experiments (total N = 5, 820), including small, medium-sized, and large electorates, and find that currently-used methods (Plurality Voting and Rank-Order systems) can lead voters to overwhelmingly support egoistic options. In contrast, alternative, more nuanced methods (Approval Voting and Borda Count) reduce the support for egoistic options and favor equity and efficiency, avoiding extreme outcomes. Those methods differ in whether they favor equity or efficiency when the latter benefits a majority. Our evidence suggests that targeted changes in the electoral system could favor socially-desirable compromises and increase social stability.

Electoral reforms and the representativeness of turnout

AbstractVoters tend to be richer, more conservative, and more educated than non-voters. While many electoral reforms promise to increase political participation, these policy instruments may have multidimensional and differential effects that can increase or decrease the representativeness of turnout. We develop an approach that allows us to estimate these effects and assess the impact of postal voting on representational inequality in Swiss referendums using individual-level ($N = 79,\; 000$) and aggregate-level data from 1981 to 2009. We find that postal voting mobilizes equally across a wide range of political and sociodemographic groups but more strongly activates high earners, those with medium education levels, and less politically interested individuals. Yet, those who vote are not less politically knowledgeable and the effects on the composition of turnout remain limited. Our results inform research on the consequences of electoral reforms meant to increase political participation in large electorates.

How Many Citizens Have Already Voted? The Role of (Interim) Turnout Rate Polls in Elections

We examine the effect of the interim release of turnout information on elections in a costly voting model of a large electorate with private values and two alternatives. We consider that (i) one group of citizens votes before the rest and that (ii) the individuals of the second group know the first group’s turnout—but not the vote tally—before they vote. The alternative that receives more votes in total is implemented. We show that the ex ante probability that each alternative is implemented is one half and, hence, the same as when no information about turnout is released in the midst of voting. One interpretation of this result is that interim information about turnout may not distort the election’s outcome (from an ex ante perspective). This is consistent with many countries (a) reporting the ongoing turnout rate at several points in time during election day, (b) allowing early voting and making public the number of citizens who did so before the day of election, and/or (c) considering different opening hours for voting stations across different time zones, if there are some. Beyond our main result, we provide several further equilibrium properties of our political game which both broaden our contribution to the theoretical understanding of the costly voting paradigm and provide additional testable hypotheses about elections.

A prospect-theory model of voter turnout

We incorporate prospect-theory preferences in a game-theoretic model to study voter turnout. We show that voter turnout is heavily affected by agents having subjective reference points with respect to the vote or abstain decision and their subjective probability weighting in the decision-making process. Using empirically based parameter values, we show that our model has lower prediction error than other game-theoretic models with standard expected-utility preferences. We also find that our model maintains desirable comparative statics effects and leads to higher turnout predictions in larger electorates.

Truth-revealing voting rules for large populations

We propose a new solution to the problem of strategic voting for large electorates. For any deterministic voting rule, we design a stochastic rule that asymptotically approximates it in the following sense: for a sufficiently large population of voters, the stochastic voting rule (i) incentivizes every voter to reveal her true preferences and (ii) produces the same outcome as the deterministic rule, with arbitrarily high probability. We then apply these results to obtain an implementation in Bayesian Nash equilibrium.

Accountability with Large Electorates

We show that ignorant voters can succeed in establishing high levels of electoral accountability. In our model, an incumbent politician is confronted with a large number of voters who receive fuzzy private signals about her performance. A sampling effect enables the incumbent to form a precise estimate of the median voter's signal, and the resulting level of accountability is as if the incumbent faced a perfectly informed social planner. Public information or ideological preferences can impair the beneficial impact of the sampling effect on accountability; overconfidence of voters can restore the full benefit of the sampling effect.

Wisdom of the Crowd? Information Aggregation and Electoral Incentives

In representative democracy, voters elect candidates who strategically propose policies. In a common value environment with imperfectly informed voters and candidates, we establish that intermediation by candidates can render information aggregation unfeasible even when a large electorate presented with exogenous options would almost always select the correct policy. In fact, the possibility of information aggregation encourages candidates' conformism and stifles the competition among ideas. Neither liberalizing access to candidacy nor introducing additional frictions in voters information guarantees feasible information aggregation. Thus, the political failure we uncover is due to the intermediation by candidates---that is, the nature of representative democracy.

Detecting Election Fraud from Irregularities in Vote-Share Distributions

I develop a novel method to detect election fraud from irregular patterns in the distribution of vote-shares. I build on a widely discussed observation that in some elections where fraud allegations abound, suspiciously many polling stations return coarse vote-shares (e.g., 0.50, 0.60, 0.75) for the ruling party, which seems highly implausible in large electorates. Using analytical results and simulations, I show that sheer frequency of such coarse vote-shares is entirely plausible due to simple numeric laws and does not by itself constitute evidence of fraud. To avoid false positive errors in fraud detection, I propose a resampled kernel density method (RKD) to measure whether the coarse vote-shares occur too frequently to raise a statistically qualified suspicion of fraud. I illustrate the method on election data from Russia and Canada as well as simulated data. A software package is provided for an easy implementation of the method.

Assessment Voting in Large Electorates

We analyze the effect of handicaps on turnout. A handicap is a difference in the vote tally between alternatives that strategic voters take as predetermined when they decide whether to turn out for voting. Handicaps are implicit in many existing democratic procedures. Within a costly voting framework with private values, we show that turnout incentives diminish considerably across the board if handicaps are large, while low handicaps yield more mixed predictions. The results extend beyond the baseline model — e.g. by including uncertainty and behavioral motivations — and can be applied to the optimal design of Assessment Voting. This is a new voting procedure where (i) some randomly-selected citizens vote for one of two alternatives, and the results are published; (ii) the remaining citizens vote or abstain, and (iii) the final outcome is obtained by applying the majority rule to all votes combined. If the size of the first voting group is appropriate, large electorates will choose the majority’s preferred alternative with high probability and average participation costs will be moderate or low.

Detecting Election Fraud from Irregularities in Vote-Share Distributions

I develop a novel method to detect election fraud from irregular patterns in the distribution of vote-shares. I build on a widely discussed observation that in some elections where fraud allegations abound, suspiciously many polling stations return coarse vote-shares (e.g., 0.50, 0.60, 0.75) for the ruling party, which seems highly implausible in large electorates. Using analytical results and simulations, I show that sheer frequency of such coarse vote-shares is entirely plausible due to simple numeric laws and does not by itself constitute evidence of fraud. To avoid false positive errors in fraud detection, I propose a resampled kernel density method (RKD) to measure whether the coarse vote-shares occur too frequently to raise a statistically qualified suspicion of fraud. I illustrate the method on election data from Russia and Canada as well as simulated data. A software package is provided for an easy implementation of the method.