Abstract
Taking advantage of recent progress in simulation techniques, this study replicates and extends previous research on social heterogeneity and choice failure under Condorcet and Borda. The simulation results can be summarized in three points: First, under a uniform distribution of preference profiles (Impartial Culture, IC), Borda is less likely to fail in selecting a group winner than Condorcet. With empirical preference profiles, obtained from the Inglehart (The silent revolution: changing values and political styles among western publics, Princeton University Press: Princeton, 1977) item battery presented in the GGSS (German General Social Survey (ALLBUS) 1980–2012. Study-No. 4578, GESIS—Leibniz-Institut für Sozialwissenschaften, Cologne, 2014), however, Condorcet is less likely to fail in selecting a group winner. Second, under IC, the probability that Condorcet and Borda provide identical group winners strongly decreases with group size, whereas with the GGSS sample, the probability of identical winners slowly increases with group size. Third, while unimodality has hardly any effect on choice failure, higher levels of bimodality are associated with a strong decrease in the probability of choice failure under both methods. In sum, these results corroborate conclusions from previous simulations that Riker’s (Liberalism against populism: a confrontation between the theory of democracy and the theory of social choice, Waveland, Illinois, 1982) general dismissal of majoritarian democracy as inaccurate remains incorrect. However, our results also indicate that making a well informed choice for either Condorcet or Borda becomes more important with higher degrees of preference polarization.
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- 1.
- 2.
Different methods for value assignment exist. For example, some assign values according to the number of alternatives which are ranked below an alternative, so that high values indicate a high ranking of the alternative.
- 3.
See Reilly (2002) for an empirical exploration of the use of Borda count for elections in the Pacific Island states of Nauru and Kiribati.
- 4.
A simulation with \( n = 3 \) and \( m = 3 \) and 1,000,000 samples takes approximately 40s and circa 80s for \( m = 4 \) and \( n = 99 \).
- 5.
Beyond that, the Inglehart (1977) item battery captures respondents’ attitudes towards policies more generally. To explore this subtle meaning of the Inglehart item battery in further depth, we conducted an auxiliary analysis in which we regress a dummy for “materialists” and another dummy for “post-materialists” on self-reported political interest and ideological left-right placement. Respondents with a “mixed” value orientation serve as the reference category. The regression models control for gender, age and education. The models were estimated with OLS, robust standard errors and the use of sample weights. The estimation results indicate that higher political interest and leftist ideology is positively associated with “post-materialists”, while lower political interest and right leaning ideological views are positively associated with “materialists”. Generally speaking, these correlation patterns indicate that the Inglehart item battery covers a meaningful political preference order.
- 6.
The dashed line in Fig. 1 indicates the uniform distribution of the IC assumption.
- 7.
We also estimated models with other items removed from the data to check for the robustness of our findings. The results are all comparable to the simulations presented here.
- 8.
With odd n and \( m = 4 \), it is not possible that all four alternatives are tied.
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Acknowledgements
We are grateful for the comments and suggestions provided by the audiences of the 2016 Annual Meeting of the Study Group “Dynamics of Collective Decisions” at the Hanse Wissenschaftskolleg (HWK) and the 2016 Annual Meeting of the DVPW Working Group “Handlungs- und Entscheidungstheorie” (AK HET). Furthermore, this paper benefited from comments of the two reviewers.
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Appendix
Distributions used for bimodality simulations
Distributions used for unimodality simulations
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Jankowski, M., Tepe, M. (2017). Social Heterogeneity and Choice Failure Under Condorcet and Borda. In: Linhart, E., Debus, M., Kittel, B. (eds) Jahrbuch für Handlungs- und Entscheidungstheorie. Jahrbuch für Handlungs- und Entscheidungstheorie. Springer VS, Wiesbaden. https://doi.org/10.1007/978-3-658-16714-1_6
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