Weighted voting: Vote of representative weighted by number of election votes

I was wondering if there is any literature on voting systems where voters elect representatives, and in the representatives' assembly the vote of each representative is weighted by the number of votes she was elected with.

E.g.: A special but easy example would be the US supreme court. The 'voters' here are the US senators, their 'representatives' are the supreme court judges. Instead of each judge having an equal vote their vote could have a weight equal to the number of votes they were confirmed with. This would make a difference, as consensus candidates seem to be confirmed by more votes - at least in the current system. Changing the supreme court's voting mechanism could of course affect confirmations, but this may be a way to incentivize consensus.

In case of general elections and parliamentary representatives such a weighted voting system would incentivize representatives to try to represent all the voters in their district, not just the majority. (Assuming representatives care about their voting power, but one could always tie their salary to the number of votes they were elected by.)

The idea is simple, so it has probably been raised before. Please refer me to some literature.

You ask an excellent question, as it has been partly but not fully discussed in the scientific literature.

Your exact proposal (weighting representatives by the amount of votes they got in the election) has been discussed on a blog post (in french) by Jean-François Laslier see here.

In practice, representatives are often elected on a geographical basis, and their weight is determined by the local area they represent (typically: its population). There is a whole literature on that subject. Balinski and Young [BY] focus on how to design (integer) voting weights that are as close as possible to proportional to the populations. A consequent literature has questioned the desirability of the idea of proportionality, a representative paper in this area is Barbera and Jackson [BJ].

Besides, your proposal seems to be one feature (among other) of the liquid democracy system advocated by democracy activists. In that system, each one can delegate or vote on each decision, and delegates' weight equals the number of received delegations. The system has been experimented by the German Pirate party. It has been studied recently in computer science. One issue is that some influent members may get too many votes [GKMP].

[BY]: Balinski and Young. Fair Representation. 1982

[BJ]: Barbera and Jackson. On the Weights of Nations: Assigning Voting Weights in a Heterogeneous Union. 2006

[GKMP]: Gölz, P., Kahng, A., Mackenzie, S., & Procaccia, A. D.. The Fluid Mechanics of Liquid Democracy. 2018

The following recent article might be relevant to your question:

Weighted representative democracy, Marcus Pivato and Arnold Soh, Journal of Mathematical Economics 88, pp.52-63, 2020.

Abstract: We propose a new system of democratic representation. Any voter can choose any legislator as her representative; thus, different legislators can represent different numbers of voters. Decisions in the legislature are made by weighted majority voting, where the weight of each legislator is proportional to the population of voters she represents. We show that, if the size of the electorate is very large, then with very high probability, the decisions obtained in the legislature agree with those which would have been reached by a popular referendum decided by simple majority vote.

Here is a link to an open-access version on SSRN: https://ssrn.com/abstract=3468210

Here is a link to the official publication: https://www.sciencedirect.com/science/article/pii/S030440682030032X

Here are some other recent articles on similar topics:

• Abramowitz, B., Mattei, N., 2019. Flexible representative democracy: an introduction with binary issues. In: Proceedings of the 28th International Joint Conference on Artificial Intelligence. AAAI Press, pp. 3–10.

• Meir, R., Sandomirskiy, F. and Tennenholtz, M., 2020. Representative Committees of Peers. https://arxiv.org/abs/2006.07837

Finally, here are some older publications which proposed similar voting systems:

• Alger, D., 2006. Voting by proxy. Public Choice 126 (1-2), 1–26.

• Cohensius, G., Mannor, S., Meir, R., Meirom, E., Orda, A., 2017. Proxy voting for better outcomes. In: Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems. International Foundation for Autonomous Agents and Multiagent Systems, pp. 858–866.

• Green-Armytage, J., 2015. Direct voting and proxy voting. Constitutional Political Economy 26 (2), 190– 220.

• Miller, J. C., 1969. A program for direct and proxy voting in the legislative process. Public Choice 7 (1), 107–113.

• Tullock, Gordon, 1967. Toward a mathematics of politics. University of Michigan Press, Ann Arbor.

• Tullock, Gordon, 1992. Computerizing politics. Mathematical and Computer Modelling 16 (8-9), 59–65.

Generally most voting reform advocates try to tweak the voting method to select the candidate that maximizes utility for all voters rather than weighting who gets elected post vote because that would get difficult to keep track of and there could be a lot of confounding variables. With your SCOTUS example; (Ignoring the fact that it used to be very common for 98-0 approvals) what if a senator dies in a state like MA where the governor has to hold a special rather than appoint? Would we just give up on appointing justices unless the same party help POTUS and the Senate? Or to broaden that to a general election, what if it rained really bad on election day? Senators elected during midterms (turnout 35%-40%) would be worthless compared to ones elected in presidential years (turnout 60%). If we did it by raw votes That would really upset small states and any way it was done would definitely violate the constitution. But if we didn't do it for the senate and just did it for SCOTUS isn't that just perpetuating the EXTREME bias for rural states that chamber already has to another branch?

Some much better voting methods: For Multi Winner elections there are several methods to make sure that the electorate is appropriately represented. IMO MMP is the best in use but I think RRV would be better. And if you are stuck with single winner the best way to maximize honest voting and get the best result is STAR. But if you are just having an up or down vote like for SCOTUS there is no need to get fancy.

• Sorry but this does not really address my question, which is a reference request. E.g. in your answer you write that utility maximization is usually the goal, but then don't really say anything about that. I am pretty sure one could build a framework to examine if this voting rule does well there. Sep 6, 2018 at 6:38