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Measuring majority power and veto power of voting rules

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Abstract

We study voting rules with respect to how they allow or limit a majority from dominating minorities: whether a voting rule makes a majority powerful and whether minorities can veto the candidates they do not prefer. For a given voting rule, the minimal share of voters that guarantees a victory to one of the majority’s most preferred candidates is the measure of majority power; and the minimal share of voters that allows the minority to veto each of their least preferred candidates is the measure of veto power. We find tight bounds on such minimal shares for voting rules that are popular in the literature and used in real elections. We order the rules according to majority power and veto power. Instant-runoff voting has both the highest majority power and the highest veto power; plurality rule has the lowest. In general, the greater is the majority power of a voting rule, the greater its veto power. The three exceptions are: voting with proportional veto power, Black’s rule and Borda’s rule, which have relatively weak majority power and strong veto power, thus providing minority protection. Our results can shed light on how voting rules provide different incentives for voter participation and candidate nomination.

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Fig. 1

Notes

  1. The instant-runoff is a special case of the single transferable vote (STV) when we select a single winner.

  2. A version of plurality with runoff—a two-round system—is used for presidential elections in France and Russia. The US presidential election system with primaries also resembles the plurality with runoff rule given the dominant positions of the two major political parties. Instant-runoff voting currently is used in parliamentary elections in Australia and presidential elections in India and Ireland. According to the Center of Voting and Democracy (fairvote.org, 2009), the instant-runoff and plurality with runoff rules have the best prospects for adoption in the United States. In the United Kingdom; a 2011 referendum proposing a switch from plurality rule to instant-runoff voting lost when almost 68% voted No.

  3. For example, the process is used in non-partisan blanket primaries in the United States, which is a version of plurality with runoff.

  4. More specifically, the comparison is made for a particular setting: for each given total number of candidates m and each given number of preferred candidates k we find the minimal size of the qualified mutual majority q(k, m).

  5. Because of this duality, the (q, k)-majority criterion and the (q, l)-veto criterion can together be referred to as the qualified mutual majority criterion.

  6. Under that rule, each group of voters can veto the share of candidates that is approximately the same as the voting share of the group. The rule selects the candidates that have not been vetoed by any group.

  7. Black’s rule selects a Condorcet winner (otherwise known as the pairwise majority winner) if it exists and a Borda winner otherwise.

  8. The literature on strategic voting is prolific; see e.g., Kondratev and Mazalov (2019) and the references therein.

  9. In fact, they might be unable even to do that. In our example presented in Table 1, under instant-runoff, the 43% minority prefers John, but John is deleted first.

  10. The collection (McLean and Urken 1995) contains English translations of original works by Borda, Condorcet, Nanson, Dodgson, and other early researches.

  11. Any social choice rule is a voting rule. Voting rules exist that are not social choice rules; for example, approval voting, preference approval voting (Brams 2009), and majority judgement (Balinski and Laraki 2011).

  12. The “extremely desirable” criteria of universality, non-imposition, anonymity, neutrality and unanimity are satisfied by all voting rules considered in this paper (Fischer et al. 2016; Taylor 2005; Tideman 2006; Zwicker 2016).

  13. The mutual majority criterion is implied by a more general axiom for multi-winner voting called Droop-Proportionality for Solid Coalitions (Woodall 1997).

  14. For a voting rule, the majority criterion is satisfied if and only if the absolute majority winner paradox never occurs. For a voting rule, the majority loser criterion is satisfied if and only if the absolute majority loser paradox never occurs (Felsenthal and Nurmi 2018; Diss et al. 2018).

  15. The (q, k, m)-majority criterion is even more general than the concepts \(q{-}PSC\) (Proportionality for Solid Coalitions) formalized by Aziz and Lee (2017), \(\pi _{PSC}\) (Janson 2018), and the threshold of exclusion (Rae et al. 1971; Lijphart and Gibberd 1977) if they are applied to single-winner elections. The weak mutual majority criterion defined by Kondratev (2018) is a particular case of \(q=k/(k+1)\). Also, q-majority decisiveness proposed by Baharad and Nitzan (2002) is a particular case of \(k=1\). A somewhat similar approach but for q-Condorcet consistency is developed by Baharad and Nitzan (2003), Courtin et al. (2015), and Mahajne and Volij (2019). All of these approaches are based on worst-case analysis.

  16. One likewise can see that the unanimity criterion is equivalent to the (\(1-\varepsilon ,1\))-majority criterion with an infinitely small \(\varepsilon >0\).

  17. In each generalized scoring rule, ties are broken using a sequence of non-generalized scoring rules (Smith 1973; Young 1975).

  18. For completeness of results, we should mention well-studied voting rules that satisfy the mutual majority criterion. They are the Condorcet extensions: Nanson’s (1882), see also McLean and Urken (1995), Baldwin’s (1926), maximal likelihood (Kemeny 1959), ranked pairs (Tideman 1987), Schulze’s (2011), successive elimination (see e.g., Felsenthal and Nurmi 2018) and those tournament solutions that are refinements of the top cycle (Good 1971; Schwartz 1972). Other rules include the single transferable vote (Hare 1859), Coombs’ (1964), Bucklin’s (see e.g., Felsenthal and Nurmi 2018), median voting rule (Bassett and Persky 1999), majoritarian compromise (Sertel and Yılmaz 1999) and q-approval fallback bargaining (Brams and Kilgour 2001). For their formal definitions and properties, we also advise Brandt et al. (2016), Felsenthal and Nurmi (2018), Fischer et al. (2016), Taylor (2005), Tideman (2006), and Zwicker (2016).

  19. When only \(m=2\) candidates compete, simple majority rule is the most natural as it satisfies a number of other important axioms according to May’s Theorem (1952).

  20. This tight bound \(q=(k-1)/k\) for Simpson’s rule coincides with the tight bound of the q-majority equilibrium (Greenberg 1979; Kramer 1977), with the minimal quota, that guarantees the acyclicity of preferences (Craven 1971; Ferejohn and Grether 1974; Usiskin 1964).

  21. For the proof, we use (Saari 2000, Proposition 5).

  22. Using the same arguments as in the proof of Theorem 9, one can check that voting rules implemented by sequential voting by veto (Mueller 1978; Moulin 1982, 1983; Felsenthal and Machover 1992) have the same tight bound on the size of the qualified mutual majority q.

  23. Equivalently, the absolute majority loser paradox never occurs.

  24. Equivalently, q-majority consistency (or q-majority decisiveness) is satisfied.

  25. Previously, the concept of veto power in voting was introduced by Baharad and Nitzan (2005, 2007b) for settings with \(l=1\) and by Moulin (1981, 1982, 1983) for settings with an arbitrary l. Their concepts also are based on the worst-case analysis, but differ from ours in that they involve strategic voting.

  26. The fundamental characterization of the class of generalized and non-generalized scoring rules was introduced by Smith (1973) and Young (1975). They use the criteria of universality, anonymity, neutrality, and consistency (reinforcement) for the generalized scoring rules, and additionally the continuity (Archimedean) criterion for the non-generalized scoring rules. Characterizations of specific scoring rules usually involve the fundamental result presented above; see Chebotarev and Shamis (1998) for a review and Richelson (1978) and Ching (1996) for the case of plurality rule.

  27. The rule constructed in the proof of Theorem 12 does not satisfy the criteria of the Condorcet loser, majority loser, and reversal symmetry. In contrast, the convex median voting rule satisfies the three criteria (Kondratev 2018), but has a higher tight bound on the size of qualified mutual majority according to Theorem 7.

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Acknowledgements

We are grateful to Richard Ericson, Shmuel Nitzan and Dan Felsenthal for their encouraging and valuable feedback. We thank participants of the 7th International Workshop on Computational Social Choice, the 14th Meeting of the Society for Social Choice and Welfare, the 10th Lisbon Meetings in Game Theory and Applications for their helpful comments; we thank Elena Yanovskaya, and other our colleagues from the HSE St. Petersburg Game theory lab for their suggestions and support. We also thank Jennifer Rontganger (WZB Berlin Social Science Center) for help with language editing.

Funding

Support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged. Kondratev is partially supported by the Russian Foundation for Basic Research via the Project No. 18-31-00055. Nesterov is partially supported by the Grant 19-01-00762 of the Russian Foundation for Basic Research.

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  1. National Research University Higher School of Economics, St. Petersburg, Russian Federation

    Aleksei Y. Kondratev & Alexander S. Nesterov

  2. Institute for Regional Economic Studies Russian Academy of Sciences, St. Petersburg, Russian Federation

    Aleksei Y. Kondratev

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Kondratev, A.Y., Nesterov, A.S. Measuring majority power and veto power of voting rules. Public Choice 183, 187–210 (2020). https://doi.org/10.1007/s11127-019-00697-1

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