## Abstract

Both the Chamberlin–Courant and Monroe rules are voting rules that solve the problem of fully proportional representation: given a set of candidates and a set of voters, they select committees of candidates whose members represent the voters so that the voters’ total dissatisfaction is minimized. These two rules suffer from a common disadvantage, namely being computationally intractable. As both the Chamberlin–Courant and Monroe rules, explicitly or implicitly, partition voters so that the voters in each part share the same representative, they can be seen as clustering algorithms. This suggests studying approximation algorithms for these voting rules by means of cluster analysis, which is the subject of this paper. Using ideas from cluster analysis we develop several approximation algorithms for the Chamberlin–Courant and Monroe rules and experimentally analyze their performance. We find that our algorithms are computationally efficient and, in many cases, are able to provide solutions which are very close to optimal.

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## Notes

Single transferable vote (STV) can also be extended to a fully proportional representation rule. However, the quality bounds for the resulting assignment would be unclear. A common practice of splitting the electorate into electoral districts and using a single-winner voting rule in each district can solve the second problem but not the first.

In their original works, Chamberlin and Courant (1983) and Monroe (1995) discussed the issue of choosing representative bodies (such as parliaments) and, thus, used the term

*misrepresentation*instead of*dissatisfaction*. However, since now their rules are used for many other applications, it is more typical to use the latter term.For clarity, we refer to the voters as females and to the candidates as males.

Dodgson’s rule is a single-winner voting rule based on the Condorcet principle and is quite distinct from the rules studied in this paper. For some more details on the rule, see, e.g., the work of Brandt (2009).

Indeed, one may argue that instead of using the indirect approach, one could simply compute election results by solving an appropriate integer linear program (ILP). The problem with relying on ILP solvers is that we never know how long it would take to find the election result, and the laws need to provide hard guarantees. For example, even for 100 candidates, 1000 voters, and committee of size 10, we found elections where the CPLEX ILP solver needed over 18 h to compute Monroe winning committees (see Sect. 4.6) for details). For elections with millions of voters, we cannot be certain that an ILP solver would compute election results within any reasonable amount of time.

In the dissatisfaction-based model it is known that constant-factor polynomial-time approximation algorithms do not exist unless \({{\mathrm {P}}}= {{\mathrm {NP}}}\) Skowron et al. (2015b).

There is a certain technicality here regarding the Monroe rule. Sometimes it is impossible to guarantee that the local Borda winner is the representative of the voters in a given group because he already is a representative of a different group. We describe how we deal with this issue later.

They also considered three other models. The uniform disc model appears to be the most basic one.

This last condition, if a bit weird at first sight, takes into account the possibility that while the committee is required to contain

*k*candidates, there can be fewer than*k*candidates that represent any voters; e.g., if*V*is unanimous, then \(\varPhi (V)\) is a singleton.Given an assignment function \(\varPhi \), we may compute its approximation ratio as \(\beta = \frac{\varGamma (\varPhi )}{\varGamma (\varPhi _{{{\mathrm {OPT}}}})}\). However, since \(\varGamma (\varPhi _{{{\mathrm {OPT}}}})\) might be equal to 0, we slightly modify this formula and compute the approximation ratio as \(\beta = \frac{1+\varGamma (\varPhi )}{1+\varGamma (\varPhi _{{{\mathrm {OPT}}}})}\).

Further, for the cases that they do not formally cover, it is often possible to extend their proofs easily.

For our experiments we have also included an upper bound of executing at most 30 iterations.

Recall that when computing the new candidate \(c'\), we break ties in favor of the incumbent representative

*c*.In the satisfaction-based model, the algorithm guarantees that the committee it outputs is \((1 - \frac{1}{e})\)-approximate for both the Monroe and Chamberlin–Courant systems. For the case of Monroe rule based on the Borda satisfaction function, Skowron et al. (2015b) prove yet stronger approximation guarantees.

That is, under IAC we completely ignore the identities of the voters. Moreover, a profile with all votes identical is as likely as any particular profile with all votes distinct. In the IC model, the latter is typically much more likely than the former.

For further experiments we do not provide plots of total dissatisfaction, but here we wanted to provide additional evidence showing how our algorithms behave, to justify some of our intuitions.

Since the Chamberlin–Courant always assigns each voter to the highest ranked committee member, the dissatisfaction of the optimal Chamberlin–Courant committee never increases with the committee size.

Recall that under R(1,5)C, we obtain 5 committees by running D1 five times, then each run of D5 gives another 5 committees. Since we make five runs of D5, we obtains 25 committees. Altogether, we have to run the clustering heuristic for 30 committees.

In fact, the Chamberlin–Courant rule was designed so that every elected representative is the Borda winner among the voters whom he represents.

This is a theoretical guarantee. Skowron et al. (2015b) have shown algorithms that ensure that the average positions of the voters’ representatives in every Monroe and Chamberlin–Courant election are very close to the top; see their paper for details of the argument.

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## Acknowledgements

Arkadii Slinko and Piotr Faliszewski gratefully acknowledge the support by Marsden Fund 3706352 of The Royal Society of New Zealand. Nimrod Talmon was supported by a postdoctoral fellowship from I-CORE ALGO.

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A conference version of this paper was presented at the Fifteenth International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2016). Besides more elaborate discussion, the current version of the paper presents results of many more experiments with a more detailed analysis and uses the dissatisfaction-based model instead of the satisfaction-based one.

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Faliszewski, P., Slinko, A., Stahl, K. *et al.* Achieving fully proportional representation by clustering voters.
*J Heuristics* **24**, 725–756 (2018). https://doi.org/10.1007/s10732-018-9376-y

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DOI: https://doi.org/10.1007/s10732-018-9376-y

### Keywords

- Multiwinner elections
- Voting
- Fully proportional representation
- Clustering