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Public Choice

Erschienen in:

27.07.2019

Measuring majority power and veto power of voting rules

verfasst von: Aleksei Y. Kondratev, Alexander S. Nesterov

Erschienen in: Public Choice | Ausgabe 1-2/2020

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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.

Vorheriger Artikel Extensions of the Simpson voting rule to the committee selection setting

Nächster Artikel Paul Dragos Aligica, Peter J. Boettke, and Vlad Tarko: Public governance and the classical-liberal perspective: political economy foundations

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

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.

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

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).

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.

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.

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

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

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.

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

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).

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).

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

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).

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.

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

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

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).

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).

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).

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

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.

Equivalently, the absolute majority loser paradox never occurs.

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

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.

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.

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.

Aleskerov, F., & Kurbanov, E. (1999). Degree of manipulability of social choice procedures. In A. Alkan, C. D. Aliprantis, & N. C. Yannelis (Eds.), Current trends in economics (pp. 13–27). Berlin: Springer.

Aleskerov, F., Karabekyan, D., Sanver, M. R., & Yakuba, V. (2012). On the manipulability of voting rules: The case of 4 and 5 alternatives. Mathematical Social Sciences, 64(1), 67–73.

Aziz, H., & Lee, B. (2017). The expanding approvals rule: Improving proportional representation and monotonicity. arXiv preprint arXiv:170807580v2.

Baharad, E., & Nitzan, S. (2002). Ameliorating majority decisiveness through expression of preference intensity. American Political Science Review, 96(4), 745–754.

Baharad, E., & Nitzan, S. (2003). The Borda rule, Condorcet consistency and Condorcet stability. Economic Theory, 22(3), 685–688.

Baharad, E., & Nitzan, S. (2005). The inverse plurality rule—An axiomatization. Social Choice and Welfare, 25(1), 173–178.

Baharad, E., & Nitzan, S. (2007a). The costs of implementing the majority principle: The golden voting rule. Economic Theory, 31(1), 69–84.

Baharad, E., & Nitzan, S. (2007b). Scoring rules: An alternative parameterization. Economic Theory, 30(1), 187–190.

Baldwin, J. M. (1926). The technique of the Nanson preferential majority system of election. Proceedings of the Royal Society of Victoria, 39, 42–52.

Balinski, M., & Laraki, R. (2011). Majority judgment: Measuring, ranking, and electing. Cambridge: MIT Press.

Bartholdi, J., Tovey, C. A., & Trick, M. A. (1989). Voting schemes for which it can be difficult to tell who won the election. Social Choice and Welfare, 6(2), 157–165.

Bassett, G. W., & Persky, J. (1999). Robust voting. Public Choice, 99(3–4), 299–310.

Black, D. (1958). The theory of committees and elections. Cambridge: University Press.

Brams, S. J. (2009). Mathematics and democracy: Designing better voting and fair-division procedures. Princeton: Princeton University Press.

Brams, S. J., & Kilgour, D. M. (2001). Fallback bargaining. Group Decision and Negotiation, 10(4), 287–316.

Brandt, F., Brill, M., & Harrenstein, P. (2016). Tournament solutions. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. D. Procaccia (Eds.), Handbook of computational social choice (pp. 57–84). New York: Cambridge University Press.

Cabannes, Y. (2004). Participatory budgeting: A significant contribution to participatory democracy. Environment and Urbanization, 16(1), 27–46.

Campbell, D. E., & Kelly, J. S. (2003). A strategy-proofness characterization of majority rule. Economic Theory, 22(3), 557–568.

Caragiannis, I., Hemaspaandra, E., & Hemaspaandra, L. A. (2016). Dodgson’s rule and Young’s rule. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. D. Procaccia (Eds.), Handbook of Computational Social Choice (pp. 103–126). New York: Cambridge University Press.

Casella, A. (2005). Storable votes. Games and Economic Behavior, 51(2), 391–419.

Chebotarev, P. Y., & Shamis, E. (1998). Characterizations of scoring methods for preference aggregation. Annals of Operations Research, 80, 299–332.

Ching, S. (1996). A simple characterization of plurality rule. Journal of Economic Theory, 71(1), 298–302.

Coombs, C. H. (1964). A theory of data. New York: Wiley.

Courtin, S., Martin, M., & Tchantcho, B. (2015). Positional rules and q-Condorcet consistency. Review of Economic Design, 19(3), 229–245.

Craven, J. (1971). Majority voting and social choice. The Review of Economic Studies, 38(2), 265–267.

Dasgupta, P., & Maskin, E. (2008). On the robustness of majority rule. Journal of the European Economic Association, 6(5), 949–973.

de Borda, J.C. (1781). Mémoire sur les Élections au Scrutin. Histoire de l’Académie Royale des Sciences, Paris.

de Condorcet, M. (1785). Essai sur l’Application de L’Analyse á la Probabilité des Décisions Rendues á la Pluralité des Voix. L’imprimerie royale, Paris.

Diss, M., Kamwa, E., & Tlidi, A. (2018). A note on the likelihood of the absolute majority paradoxes. Economics Bulletin, 38(4), 1727–1734.

Dodgson, C.L. (1876). A method of taking votes on more than two issues. Pamphlet printed by the Clarendon Press.

Fahrenberger, T., & Gersbach, H. (2010). Minority voting and long-term decisions. Games and Economic Behavior, 69(2), 329–345.

FairVoteorg (2009). The center for voting and democracy. single-winner voting method comparison chart. http://archive3.fairvote.org/reforms/instant-runoff-voting/irv-and-the-status-quo/irv-versus-alternative-reforms/single-winner-voting-method-comparison-chart/.

Feld, S. L., & Grofman, B. (1992). Who’s afraid of the big bad cycle? Evidence from 36 elections. Journal of Theoretical Politics, 4(2), 231–237.

Felsenthal, D. S., & Machover, M. (1992). Sequential voting by veto: Making the Mueller-Moulin algorithm more versatile. Theory and Decision, 33(3), 223–240.

Felsenthal, D. S., & Nurmi, H. (2018). Voting procedures for electing a single candidate: Proving their (In) vulnerability to various voting paradoxes. Berlin: Springer.

Ferejohn, J. A., & Grether, D. M. (1974). On a class of rational social decision procedures. Journal of Economic Theory, 8(4), 471–482.

Fischer, F., Hudry, O., & Niedermeier, R. (2016). Weighted tournament solutions. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. D. Procaccia (Eds.), Handbook of computational social choice (pp. 85–102). New York: Cambridge University Press.

Fishburn, P. C. (1974). Paradoxes of voting. American Political Science Review, 68(2), 537–546.

Gehrlein, W. V., & Lepelley, D. (2017). Elections, voting rules and paradoxical outcomes. Berlin: Springer.

Good, I. J. (1971). A note on Condorcet sets. Public Choice, 10(1), 97–101.

Green-Armytage, J., Tideman, T. N., & Cosman, R. (2016). Statistical evaluation of voting rules. Social Choice and Welfare, 46(1), 183–212.

Greenberg, J. (1979). Consistent majority rules over compact sets of alternatives. Econometrica, 47(3), 627–636.

Grofman, B., Feld, S. L., & Fraenkel, J. (2017). Finding the threshold of exclusion for all single seat and multi-seat scoring rules: Illustrated by results for the Borda and Dowdall rules. Mathematical Social Sciences, 85, 52–56.

Hare, T. (1859). Treatise on the election of representatives, parliamentary and municipal. London: Longman, Green, Reader, and Dyer.

Hemaspaandra, E., Hemaspaandra, L. A., & Rothe, J. (1997). Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. Journal of the ACM (JACM), 44(6), 806–825.

Janson, S. (2018). Thresholds quantifying proportionality criteria for election methods. arXiv preprint arXiv:181006377.

Kemeny, J. G. (1959). Mathematics without numbers. Daedalus, 88(4), 577–591.

Kondratev, A. Y. (2018). Positional voting methods satisfying the criteria of weak mutual majority and Condorcet loser. Automation and Remote Control, 79(8), 1489–1514.

Kondratev, A. Y., & Mazalov, V. V. (2019). Tournament solutions based on cooperative game theory. International Journal of Game Theory,. https://doi.org/10.1007/s00182-019-00681-5.CrossRef

Kramer, G. H. (1977). A dynamical model of political equilibrium. Journal of Economic Theory, 16(2), 310–334.

Lackner, M., & Skowron, P. (2018). A quantitative analysis of multi-winner rules. arXiv preprint arXiv:180101527v2.

Lepelley, D. (1992). Une caractérisation du vote à la majorité simple. RAIRO-Operations Research, 26(4), 361–365.

Lijphart, A., & Gibberd, R. W. (1977). Thresholds and payoffs in list systems of proportional representation. European Journal of Political Research, 5(3), 219–244.

Mahajne, M., & Volij, O. (2019). Condorcet winners and social acceptability. Social Choice and Welfare,. https://doi.org/10.1007/s00355-019-01204-7.CrossRef

May, K. O. (1952). A set of independent necessary and sufficient conditions for simple majority decision. Econometrica, 20(4), 680–684.

McLean, I., & Urken, A. B. (1995). Classics of social choice. Ann Arbor: The University of Michigan Press.

Merrill, S. (1984). A comparison of efficiency of multicandidate electoral systems. American Journal of Political Science, 28(1), 23–48.

Miller, N. R. (2019). Reflections on Arrow’s theorem and voting rules. Public Choice, 179(1–2), 113–124.

Moulin, H. (1981). The proportional veto principle. The Review of Economic Studies, 48(3), 407–416.

Moulin, H. (1982). Voting with proportional veto power. Econometrica, 50(1), 145–162.

Moulin, H. (1983). The strategy of social choice. Amsterdam: North-Holland Publishing Company.

Mueller, D. C. (1978). Voting by veto. Journal of Public Economics, 10(1), 57–75.

Nanson, E. J. (1882). Methods of election. Transactions and Proceedings of the Royal Society of Victoria, 19, 197–240.

Nitzan, S. (2009). Collective preference and choice. Cambridge: Cambridge University Press.

Plassmann, F., & Tideman, T. N. (2014). How frequently do different voting rules encounter voting paradoxes in three-candidate elections? Social Choice and Welfare, 42(1), 31–75.

Rae, D., Hanby, V., & Loosemore, J. (1971). Thresholds of representation and thresholds of exclusion: An analytic note on electoral systems. Comparative Political Studies, 3(4), 479–488.

Richelson, J. T. (1978). A characterization result for the plurality rule. Journal of Economic Theory, 19(2), 548–550.

Saari, D. G. (2000). Mathematical structure of voting paradoxes. Economic Theory, 15(1), 1–53.

Sanver, M. R. (2002). Scoring rules cannot respect majority in choice and elimination simultaneously. Mathematical Social Sciences, 43(2), 151–155.

Schulze, M. (2011). A new monotonic, clone-independent, reversal symmetric, and Condorcet-consistent single-winner election method. Social Choice and Welfare, 36(2), 267–303.

Schwartz, T. (1972). Rationality and the myth of the maximum. Nous, 6(2), 97–117.

Sertel, M. R., & Yılmaz, B. (1999). The majoritarian compromise is majoritarian-optimal and subgame-perfect implementable. Social Choice and Welfare, 16(4), 615–627.

Simpson, P. B. (1969). On defining areas of voter choice: Professor Tullock on stable voting. The Quarterly Journal of Economics, 83(3), 478–490.

Skowron, P. (2018). Proportionality degree of multiwinner rules. arXiv preprint arXiv:181008799v1.

Smith, J. H. (1973). Aggregation of preferences with variable electorate. Econometrica, 41(6), 1027–1041.

Stein, W. E., Mizzi, P. J., & Pfaffenberger, R. C. (1994). A stochastic dominance analysis of ranked voting systems with scoring. European Journal of Operational Research, 74(1), 78–85.

Taylor, A. D. (2005). Social choice and the mathematics of manipulation. New York: Cambridge University Press.

Tideman, N. (1995). The single transferable vote. Journal of Economic Perspectives, 9(1), 27–38.

Tideman, N. (2006). Collective decisions and voting: The potential for public choice. Aldershot: Ashgate Publishing Ltd.

Tideman, T. N. (1987). Independence of clones as a criterion for voting rules. Social Choice and Welfare, 4(3), 185–206.

Tideman, T. N., & Plassmann, F. (2014). Which voting rule is most likely to choose the “best” candidate? Public Choice, 158(3–4), 331–357.

Usiskin, Z. (1964). Max-min probabilities in the voting paradox. The Annals of Mathematical Statistics, 35(2), 857–862.

Woeginger, G. J. (2003). A note on scoring rules that respect majority in choice and elimination. Mathematical Social Sciences, 46(3), 347–354.

Woodall, D. R. (1997). Monotonicity of single-seat preferential election rules. Discrete Applied Mathematics, 77(1), 81–98.

Young, H. P. (1975). Social choice scoring functions. SIAM Journal on Applied Mathematics, 28(4), 824–838.

Young, H. P. (1977). Extending Condorcet’s rule. Journal of Economic Theory, 16(2), 335–353.

Zwicker, W. S. (2016). Introduction to the theory of voting. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. D. Procaccia (Eds.), Handbook of computational social choice (pp. 23–56). New York: Cambridge University Press.

Titel: Measuring majority power and veto power of voting rules
verfasst von: Aleksei Y. Kondratev
Alexander S. Nesterov
Publikationsdatum: 27.07.2019
Verlag: Springer US
Erschienen in: Public Choice / Ausgabe 1-2/2020
Print ISSN: 0048-5829
Elektronische ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-019-00697-1

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