On minimum sum representations for weighted voting games


A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the “yea” voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243–260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation.

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  1. 1.

    Other aliases are weighted (majority) games or threshold functions.

  2. 2.

    We remark that the counts for weighted voting games with 6≤n≤8 voters are wrongly stated in de Keijzer (2009). The causative buglet is fixed by now (personal communication).

  3. 3.

    We would like to mention the (unpublished) diploma thesis (Tautenhahn 2008) containing the enumeration for 9 voters (without using the results from this paper; compare footnote 8).

  4. 4.

    For a more extensive introduction we refer to Taylor and Zwicker (1999).

  5. 5.

    Another representation, which takes the possible symmetry of voters into account, is described e.g. in Carreras and Freixas (1996). There are several applications where this representation is more convenient, see e.g. Kurz and Tautenhahn (2012).

  6. 6.

    Here one may also read the coalitions as integers written in their binary expansion and use the ordinary ordering ≤ of integers.

  7. 7.

    The numbers of weighted voting games and complete simple games coincide for n≤5 voters but their ratio converges to zero with increasing n, see also Table 2. An asymptotic upper bound for weighted voting games is given in de Keijzer et al. (2010) and an asymptotic lower bound for complete simple games, there called regular Boolean functions, is given in Peled and Simeone (1985).

  8. 8.

    We remark that it is also possible to do the enumeration for 9 voters without the presented ideas, as demonstrated in Tautenhahn (2008). Using some heuristics to find suitable weights on the one hand and to find dual multipliers of the inequalities on the other hand to prove the non-existence of weights, roughly 4 months of computation time were necessary.

  9. 9.

    Due to the definition of a weighted voting game the corresponding linear program has rational solutions, which can be scaled to be integers.

  10. 10.

    We have verified these results using our approach outlined below.

  11. 11.

    As a check of the correctness of our computer calculations we have verified that we have found the dual games and all examples from the list in Freixas and Molinero (2010).


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The author thanks the anonymous referees for carefully reading a preliminary version of this article and giving useful comments to improve the presentation.

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Correspondence to Sascha Kurz.

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Kurz, S. On minimum sum representations for weighted voting games. Ann Oper Res 196, 361–369 (2012). https://doi.org/10.1007/s10479-012-1108-3

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  • Simple games
  • Weighted voting games
  • Minimum realizations
  • Realizations with minimum sum