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One person, one weight: when is weighted voting democratic?

Abstract

In a classical “jury theorem” setting, the collective performance of a group of independent decision-makers is maximized by a voting rule that assigns weight to individuals compatibly with skills. The primary concern is that such weighted voting interferes with majoritarianism, since excessive power may be granted to a competent minority. In this paper, we address a surprisingly undertheorized issue of much significance to collective decision-making: the overlap of optimal weighted voting and the democratic, ubiquitous simple majority rule which is typically adopted in public decision-making. Running Monte Carlo simulations on the distribution of skills in large groups, our main findings are rather counterintuitive. In terms of procedure, the optimal allocation of weights is almost always democratic or “semi-democratic”, in that it satisfies or draws close to “one person, one weight”. In terms of outcome, the chosen alternative under optimal weighted voting is almost always the one that would have been selected by the simple majority rule, which satisfies “one person, one vote”. We thereby submit that the decision rules supported by the proceduralist and epistemic approaches to collective decision-making, effectively coincide more often than one would expect.

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Notes

  1. For discussions on proceduralist vis-à-vis epistemic justifications see, e.g., Estlund (1993), Cohen (1986), List and Goodin (2001). See also Dietrich (2006) and references therein.

  2. A notable exception is Goodin and Spiekermann (2018), who use Monte Carlo simulations to estimate group competence in a closely related jury theorem setting.

  3. The term “epistocracy” has been originally introduced by Estlund (2003).

  4. The truncated normal distribution is defined as follows. Let \(X\) be a normal random variable with mean \(\mu\) and standard deviation \(\sigma\), and let \(f_{X}\) be its probability density function. Let \(\left( {a,b} \right)\) be a real interval. Let \(Y\) be the random variable “\(X\) conditional on \(a < X < b\)”. Note that the probability density function of \(Y\) equals \(f_{X} /\Pr (a < X < b)\) in the interval \(\left( {a,b} \right)\) and \(0\) elsewhere. Then, \(Y\) has a truncated normal distribution with parameters \(\mu , \sigma , a, b\).

  5. The underlying explanation is formal. It reflects the fact that, when \(n = 5\), all possible systems of weights (rules) can be partitioned to just seven equivalence classes of rules. Suppose that we ask the same question for \(n = 3\). In this case, there are just two systems of weights that describe efficient rules where the weight of a more competent individual is larger than or equal to that of an equally or less competent individual: the expert rule and the simple majority rule. In this case every system of weights is equivalent to (1,0,0) or to (1,1,1) that define, respectively, the expert rule and simple majority rule. Equivalence means that the winning coalitions of the two systems are identical.

  6. A weight system can always be represented in a normalized form where the positive weights are integers, and the smallest positive weight is 1. If the weight of an individual is equal to 0, then he-she is considered inessential.

  7. Clearly, the “expert rule” is a special case of epistocracy, as \(n - 1\) decision-makers (all but the expert herself) are inessential.

  8. Individual competence is assumed to be state-of-nature independent, namely, voter \(i\)’s probability of being correct is \(p_{i}\) regardless of which alternative is the correct one. Moreover, we assume that each voter’s decision does not depend on decisions made by other voters, that is, decisions are not correlated.

  9. The choice of the particular value of the largest standard deviation was based on the understanding that the results are robust to an increase of this value.

  10. The results for the beta distribution of skills are presented in Table OA1 and Figure OA1 in the Online Appendix. It turns out that these results are qualitatively similar to the truncated normal distribution when the standard deviation is low or medium, or when the mean is medium. In particular, the optimal rule is semi-democratic for large groups. However, when the standard deviation is high and the mean is either high or low, there is a substantial difference: even for large groups, the optimal rule is epistocratic or semi-epistocratic. This can be explained by the fact that, in these ranges, the beta distribution puts most of the probability mass near the extreme ends of the range, namely, near 0.5 and near 1. In other words, most voters are either highly skillful or extremely incompetent. In such a population, it makes sense that the optimal rule bestows substantial decision-making power on the formers. We believe that the truncated normal distribution, which puts most of the probability mass near the mean, better reflects the actual distribution of skills in society.

  11. All the claims involving numerical values can be confirmed by observing Table 1 in the Appendix.

  12. The above results are relatively robust to the alternative assumption of beta distribution of skills.

  13. This result should not be surprising given Condorcet’s (1785) asymptotic result under the SMR.

  14. Under the alternative beta distribution of skills, the following findings are obtained. Under the impartial culture assumption, the expected optimality likelihood of democracy or semi-democracy, in 3, 5, 7, 9 and 11-member groups, is at least 0.61 and the expected overlap between the alternatives selected by the majority of weights and votes is at least 0.93. In large groups, \(n \ge 21\), the expected optimality likelihood of democracy or semi-democracy is 0.92 and the expected overlap between the alternatives supported by the majority of weights and votes is at least 0.95.

  15. The optimality likelihood of semi-epistocracy is presented in column 6 in Table 1 in the Appendix, and in Fig. 1.

  16. We are indebted to an anonymous referee for pointing out this idea.

  17. This value is the average of the sums of the entries in columns 4 and 6 in Table 1, corresponding to 3,5,7,9 and 11-member groups.

  18. Notice that Table 1, unlike Fig. 1, is a reduced form of the simulation findings because, for each of the 6 group sizes, it presents 9 average optimality probabilities that correspond to the nine possible combinations of the parameter ranges (each parameter is classified into one of three ranges: lower, intermediate and upper range). In the simulation, see Fig. 1, for each of the 6 group sizes, we have computed 81 average optimality probabilities that correspond to the 9 possible mean values and the 9 possible values of the standard deviation.

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Acknowledgements

The authors are very much indebted to the AE and two anonymous referees whose insightful comments originated substantial improvements.

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Appendix A

Appendix A

Table 1 The optimality likelihoods of the four rule categories (democracy, semi-democracy, epistocracy and semi-epistocracy), as well as the optimality likelihood of the expert rule (a special case of epistocracy), and the probability of overlap between the alternatives selected by the majority of weights and votes (overlap B-M-W-V) under the truncated normal distribution of skills

Table 1 (column 4) reports the probability that the optimal weighted voting rule satisfies condition (1). When the standard deviation of skills is sufficiently small, condition (1) is satisfied for any value of the mean and the SMR is the optimal rule. The Condorcet Jury Theorem is obtained in the special case of competence-homogeneity, that is, any combination of (\(\mu\), \(\sigma\)) where \(\sigma = 0\) and \(\mu > 0.5\). For \(n \le 11,\) the likelihood of SMR optimality is relatively high, when the mean of the skill distribution is in its middle range and the standard deviation is in its lower range (column 4).

For the values of the standard deviation in the lower range, the optimality likelihood of democracy is maximal when the mean competence is 0.75. In the low range of the standard deviation and medium range of skill, the probability that the SMR is optimal is above 0.5, for \(n < 11\). As is evident from Table 1 and from the findings of the simulation when run for group-sizes larger than 11, the probability that the SMR is optimal is decreasing with \(n\) and with \(\mu\). For sufficiently large decision-making groups and sufficiently large standard deviation, it approaches 0. All the above observations regarding the optimality likelihood of SMR remain valid under the beta distribution of skills (see column 4 in Table OA1 in the Online Appendix).

As mentioned, when \(n = 3\) or \(5\), there is no difference between the probability of optimality of epistocracy and of the expert rule. For larger decision-making bodies, since there are several sets that render the majority inessential, the probability of epistocracy being optimal is clearly higher than that of the expert rule (columns 7 and 8). The question is whether the probability of epistocracy being optimal is substantial, given that the number of potentially optimal outcomes that allow it is relatively small. In small groups, \(5 \le n \le 11\), epistocracy is unlikely to be optimal. In large groups, the probability that it is optimal is small – converges to 0 when \(n\) is sufficiently large, regardless of the range of the standard deviation.

The estimated degree of overlap between the alternatives selected by the majority of weights and the majority of voters is presented in column 9 of Table 1. In other words, this column presents the probability that the simple majority rule, which prevails in organizations and institutions, and the optimal weighted majority rule (democratic rule, semi-democratic rule, semi-epistemic rule or epistemic rule) select the same alternative.Footnote 18

Fig. 3
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Optimality likelihood of democracy (d), semi-democracy (sd), epistocracy (e) and semi-epistocracy (se) as a function of the mean competence and standard deviation for the normal distribution of skills and \(n = 3, 5, 7, 9, 11, 21\)

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Baharad, R., Nitzan, S. & Segal-Halevi, E. One person, one weight: when is weighted voting democratic?. Soc Choice Welf 59, 467–493 (2022). https://doi.org/10.1007/s00355-022-01393-8

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