Abstract
In federal unions, how many weights should be given to each state in a two-tier voting system, given that the majority rule is used at each level? We apply a majority criterion to evaluate these voting rules: An apportionment of the seats among the states is majority efficient if it minimizes the probability of electing the candidate who receives less than 50% of the votes in a two candidate competition over the whole union. Depending on the assumptions that we use to describe the electoral process, either the proportional or the square root rule can emerge as an optimal solution.
Marc Feix passed away on July 4th 2005 at the age of 78. Though he never saw the last version of this paper, he greatly contributed to an early version of this piece of work.
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Notes
- 1.
- 2.
For other examples in US, United Kingdom and France, see Feix et al. (2004).
- 3.
The reader can find very nice introductions to these concepts in a series of papers (Gelman et al., 2002, 2004).
- 4.
If the number of weights is even, ties may occur with majority rule. A way to avoid such situation is to assume that the number of weights is odd, or to flip a fair coin to take a decision in case of a draw, or to ask for a new election until a clear decision is obtained, etc.
- 5.
Or, equivalently, \(a_i\) representatives are elected to seat at the federal level (European assembly).
- 6.
In a weighted majority game, each player is endowed with a positive weights \(a_i\). A coalition is winning if the sum of the weights of its members exceeds some predefined quota q.
- 7.
A formal version of Penrose’s statement has been proposed recently (Lindner and Machover 2004). Using simulations, the validity of the approximation for numerous partition of the population among states has been tested; It has been shown that it is valid with a probability close to one for the non normalized Banzhaf index and a quota of 50% (Chang et al. 2006). The proportionality between the weight and the Banzhaf index is even better for some super majority rules (Feix et al. 2007; Słomiczyǹski and Życzkowski 2007). To give an example, the proportionality between the mandantes and the Banzhaf power is almost perfectly met for the enlarged European Union if we attribute to each state a number of weight in proportion to the square root of its population and use a quota of 61.5% (Feix et al. 2007). Recent works with applications to the European Union are perfect examples of this tradition (Felsenthal and Machover 1998, 2001, 2004).
- 8.
We are indebted to John Roemer and Hannu Nurmi, who mentioned this reference to us ten years ago.
- 9.
Recall that different weights can lead to the same set of winning coalitions (Taylor and Zwicker 1999).
- 10.
The only “missing” game is defined by \(\tilde{a} = (3,3,3,0,0)\).
- 11.
Some simulations, not presented in this paper, have shown that a second local minima, above the first one, may exist for higher values of \(\alpha \) and \(\beta \), before reaching the dictatorial case.
- 12.
Ideally, we should have drawn a new set of elections for each value of \(\beta \), as we did in the previous section. This would have allowed us to define a clear statistical test about the optimality of \(\beta =1\) compared to other values of \(\beta \). However, one immediately realizes that drawing a new batch of data for each \(\beta \) would have enormously increased the computation time.
- 13.
Again, we cannot propose a proper test as we use the same batch of elections to compute the \(P(N,\tilde{n}, \alpha , IC)\) for different values of \(\alpha \).
- 14.
To compute the values for an even number of states, we flip a fair coin in case of a tie in terms of weights. This explains the discontinuities we observe for \(\alpha = \beta = 0\) between odd and even values of N.
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Acknowledgements
Preliminary versions of this work have been presented since 2004, and we apologize for the long time we took to write down a final version. During all these years, we benefited from many comments and remarks, especially during the Voting Power and Procedure meetings, that were organized on a regular basis by Rudy Fara, Dan Felsenthal, Dennis Leech, Moshe Machover and Maurice Salles. We thank all the participants of these meetings for the stimulating discussions that we had over the past years. We are grateful to Stefan Napel, who kept asking us when the final version of the paper would be published. Special thanks are also due to Hannu Nurmi and John Roemer, who rediscovered a forgotten article due to Kenneth May. We also benefited from the discussions we had Guillermo Owen, who helped us to understand the differences between the different probabilistic approaches to the power indices. Special thanks are due to William Gehrlein for his comments on the last version of the manuscript. This paper was part of the SOLITER project selected in the “Gouverner, administrer” program by the French Agence Nationale pour la Recherche (ANR) and has benefited from the ANR-08-GOUV-054 grant as well as the ANR project ComSoc (ANR-09-BLAN-0305).
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Feix, M., Lepelley, D., Merlin, V., Rouet, JL., Vidu, L. (2021). Majority Efficient Representation of the Citizens in a Federal Union. 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_8
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