# Multiwinner approval voting: an apportionment approach

## Abstract

To ameliorate ideological or partisan cleavages in councils and legislatures, we propose modifications of approval voting in order to elect multiple winners, who may be either individuals or candidates of a political party. We focus on two divisor methods of apportionment, first proposed by Jefferson and Webster, that fall within a continuum of apportionment methods. Our applications of them depreciate the approval votes of voters who have had one or more approved candidates elected and give approximately proportional representation to political parties. We compare a simple sequential rule for allocating approval votes with a computationally more complex simultaneous (nonsequential) rule that, nonetheless, is feasible for many elections. We find that our Webster apportionments tend to be more representative than ours based on Jefferson—by giving more voters at least one representative of whom they approve. But our Jefferson apportionments, with equally spaced vote thresholds that duplicate those of cumulative voting in two-party elections, are more even-handed. By enabling voters to express support for more than one candidate or party, these apportionment methods will tend to encourage coalitions across party or factional lines, thereby diminishing gridlock and promoting consensus in voting bodies.

## Keywords

Approval voting Multiple winners Apportionment Divisor methods Cumulative voting## Notes

### Acknowledgements

We thank two reviewers, the associate editor, and the editor for valuable comments.

## References

- Aziz, H., Brill, M., Conitzer, V., Elkind, E., Freeman, R., & Walsh, T. (2017). Justified representation in approval-based committee voting.
*Social Choice and Welfare,**48*(2), 461–485.CrossRefGoogle Scholar - Balinski, M. L., & Young, H. P. (2001).
*Fair representation: Meeting the ideal of one-man, one-vote*(2nd ed.). Washington, DC: Brookings Institution.Google Scholar - Blais, A., & Massicotte, L. (2002). Electoral systems. In L. LeDuc, R. S. Niemi, & P. Norris (Eds.),
*Comparing democracies 2: New challenges in the study of elections and voting*(pp. 40–69). London: Sage.Google Scholar - Brams, S. J. (1990). Constrained approval voting: A voting system to elect a governing board.
*Interfaces,**20*(5), 65–79.CrossRefGoogle Scholar - Brams, S. J. (2004).
*Game theory and politics*(2nd ed.). Mineola, NY: Dover.Google Scholar - Brams, S. J. (2008).
*Mathematics and democracy: Designing better and voting and fair-division procedures*. Princeton, NJ: Princeton University Press.Google Scholar - Brams, S. J., & Brill, M. (2018).
*The excess method: A multiwinner approval voting procedure to allocate wasted votes*. New York: New York University.Google Scholar - Brams, S. J., & Fishburn, P.C. (2007).
*Approval voting*(2nd ed.). New York: Springer.Google Scholar - Brams, S. J., & Kilgour, D. M. (2014). Satisfaction approval voting. In R. Fara, D. Leech, & M. Salles (Eds.),
*Voting power and procedures: Essays in honor of Dan Felsenthal and Moshé Machover*(pp. 323–346). Cham: Springer.Google Scholar - Brams, S. J., Kilgour, D. M., & Sanver, M. R. (2007). The minimax procedure for electing committees.
*Public Choice,**132*(33–34), 401–420.CrossRefGoogle Scholar - Brill, M., Freeman, R., Janson, F., & Lackner, M. (2017). Phragmén’s voting methods and justified representation. In
*Proceedings of the 31*^{st}*AAAI conference on artificial intelligence*(*AAAI*-*17*). Palo Alto, CA: AAAI Press, pp. 406–413.Google Scholar - Brill, M., Laslier, J.-F., & Skowron, P. (2018). Multiwinner approval rules as apportionment methods.
*Journal of Theoretical Politics,**30*(3), 358–382.CrossRefGoogle Scholar - Chamberlin, J. R., & Courant, P. H. (1983). Representative deliberations and representative decisions: Proportional representation and the Borda rule.
*American Political Science Review,**77*(3), 718–733.CrossRefGoogle Scholar - Cox, G. W. (1997).
*Making votes count: Strategic coordination in the world’s electoral systems*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Edelman, P. H. (2006a). Getting the math right: Why California has too many seats in the House of Representatives.
*Vanderbilt Law Review,**59*(2), 296–346.Google Scholar - Edelman, P. H. (2006b). Minimum total deviation apportionments. In B. Simeone & F. Pukelsheim (Eds.),
*Mathematics and democracy: Recent advances in voting systems and social choice*(pp. 55–64). Berlin: Springer.CrossRefGoogle Scholar - Elkind, E., Faliszewski, P., Skowron, P., & Slinko, A. (2017). Properties of multiwinner voting rules.
*Social Choice and Welfare,**48*(3), 599–632.CrossRefGoogle Scholar - Faliszewski, P., Skowron, P., Slinko, A., & Talmon, N. (2017). Multiwinner voting: A new challenge for social choice theory. In U. Endress (Ed.),
*Trends in computational social choice*(pp. 27–47). Amsterdam: ILLC University of Amsterdam.Google Scholar - Kilgour, D. M. (2010). Approval balloting for multi-winner elections. In J.-F. Laslier & M. R. Sanver (Eds.),
*Handbook on approval voting*(pp. 105–124). Berlin: Springer.CrossRefGoogle Scholar - Kilgour, D. M. (2018). Multi-winner voting.
*Estudios de Economia Applicada,**36*(1), 167–180.Google Scholar - Kilgour, D. M., Brams, S. J., & Sanver, M. R. (2006). How to elect a representative committee using approval balloting. In B. Simeone & F. Pukelsheim (Eds.),
*Mathematics and democracy: Recent advances in voting systems and collective choice*(pp. 893–895). Berlin: Springer.Google Scholar - Kilgour, D. M., & Marshall, E. (2012). “Approval balloting for fixed-size committees. In D. S. Felsenthal & M. Machover (Eds.),
*Electoral systems: Studies in social welfare*(pp. 305–326). Berlin: Springer.CrossRefGoogle Scholar - Laslier, J.-F., & Sanver, M. R. (Eds.). (2010).
*Handbook on approval voting*. Berlin: Springer.Google Scholar - Liptak, A. (2018). Supreme Court avoids an answer on partisan gerrymandering.
*New York Times*(June 18).Google Scholar - Monroe, B. L. (1995). Fully proportional representation.
*American Political Science Review,**89*(4), 925–940.CrossRefGoogle Scholar - Potthoff, R. F. (2014). An underrated 1911 relic can modify divisor methods to prevent quota violation in proportional representation and U.S. House apportionment.
*Representation,**50*(2), 193–215.CrossRefGoogle Scholar - Potthoff, R. F., & Brams, S. J. (1998). Proportional representation: Broadening the options.
*Journal of Theoretical Politics,**10*(2), 147–178.CrossRefGoogle Scholar - Pukelsheim, F. (2014).
*Proportional representation: Apportionment methods and their applications*. Cham: Springer.CrossRefGoogle Scholar - Sánchez-Fernández, L., Fernández Garcia, N., & Fisteus, J. A. (2016). Fully open extensions of the D’Hondt method. Preprint. https://arxiv.org/pdf/1609.05370v1.pdf.
- Sivarajan, S. N. (2018). A generalization of the minisum and minimax voting methods Preprint. https://arxiv.org/pdf/1611.01364v2.pdf.
- Subiza, B., & Peris, J. E. (2014). A consensual committee using approval balloting. Preprint. https://web.ua.es/es/dmcte/documentos/qmetwp1405.pdf.
- Toplak, J. (2008). Equal voting weight of all: Finally “one person, one vote” from Hawaii to Maine?
*Temple Law Review,**81*(1), 123–175.Google Scholar