Abstract
A large part of the social choice literature studies voting paradoxes in which seemingly mild properties are violated by common voting rules. In this chapter, we investigate the likelihood of the Condorcet Loser Paradox (CLP) and the Agenda Contraction Paradox (ACP) using Ehrhart theory, computer simulations, and empirical data. We present the first analytical results for the CLP on four alternatives and show that our experimental results, which go well beyond four alternatives, are in almost perfect congruence with the analytical results. It turns out that the CLP—which is often cited as a major flaw of some Condorcet extensions such as Dodgson’s rule, Young’s rule, and MaxiMin—is of no practical relevance. The ACP, on the other hand, frequently occurs under various distributional assumptions about the voters’ preferences. The extent to which it is real threat, however, strongly depends on the voting rule, the underlying distribution of preferences, and, somewhat surprisingly, the parity of the number of voters.
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Notes
- 1.
The Kendall-tau distance counts the number of pairwise disagreements.
- 2.
In a related study, Brandt and Seedig (2016) have found that the number of dimensions does not seem to have a large impact on the results as long as it is at least two.
- 3.
These mixed equilibria are also known as maximal lotteries in probabilistic social choice.
- 4.
For most preference models other than IAC this approach does not work. While for specific combinations of (simple) distributions and voting rules there are some highly tailor-made computations in the literature (cf. Sect. 2), these have to be redesigned for each individual setting.
- 5.
In theory, the analysis can be adapted to also cover more complex rules (e.g., Dodgson’s and Young’s rule, which involve solving an integer linear program). It is unclear, however, how one would translate their definitions to linear inequalities.
- 6.
These upper bounds turn out to be relatively independent from the underlying preference distribution (among the models we considered, cf. Sect. 5.3).
- 7.
We tested 314 preference profiles with strict orders from the PrefLib library as well as the roughly 11 million 4-alternative elections which Mattei et al. (2012) derived from the Netflix Prize data. While about 54,000 of those elections were susceptible to the CLP, it never occurred under the rules we considered in this chapter. In contrast, under plurality it already occurred in twelve out of the 314 PrefLib-instances.
- 8.
- 9.
Fishburn (1977) actually analyzes violations of “Smith’s Condorcet principle”, which are weaker than the CLP.
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Acknowledgements
This material is based on work supported by the Deutsche Forschungsgemeinschaft under grant BR 2312/9-1. The authors thank Nicholas Mattei for providing the Netflix Prize data and Christof Söger for his guidance regarding Normaliz. Preliminary results of this chapter were presented at the 15th International Conference on Autonomous Agents and Multiagent Systems (Singapore, May 2016) and the 6th International Workshop on Computational Social Choice (Toulouse, June 2016).
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Brandt, F., Geist, C., Strobel, M. (2021). Analyzing the Practical Relevance of the Condorcet Loser Paradox and the Agenda Contraction Paradox. In: Diss, M., Merlin, V. (eds) Evaluating Voting Systems with Probability Models. Studies in Choice and Welfare. Springer, Cham. https://doi.org/10.1007/978-3-030-48598-6_5
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