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Sampling methods to estimate the Banzhaf–Owen value

  • S.I. : SING 14
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Abstract

This paper addresses two sampling methods to estimate the Banzhaf–Owen value for general cooperative games. The first approach is based on simple random sampling without replacement of those coalitions that are compatible with the system of unions. Additionally, using the interpretation of the Banzhaf–Owen value as a mean of means, we propose an alternative estimation procedure based on two-stage sampling that reduces the required computation time. Both approaches are analysed by establishing the theoretical statistical properties and bounds of the incurred error and their performance is compared with other sampling methodologies in the literature. Finally, we evaluate these tools on estimating the power of the members of the Board of Governors of the International Monetary Fund (IMF) in 2002 and 2016, and we compare the power of some countries in both compositions.

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Notes

  1. In the case of associating the cost of cooperation, the game is known as a cost game and is denoted by (N, c).

  2. Popoviciu’s inequality on variances: Let M and m be an upper and lower bound on the values of a bounded random variable X with variance \(\text{ Var }(X)\). Then, \(\text{ Var }(X)\le {\frac{1}{4}}(M-m)^{2}\).

  3. (N, v) is a convex game when for every \(i\in N\) and every \(K,T\subseteq N{\setminus }\{ i\}\) with \(K\subset T\), it holds that \(v(K\cup \{ i\})-v(K)\le v(T\cup \{ i\})-v(T)\). (N, v) is a concave game when \((N,-v)\) is convex.

  4. We compute the exact Banzhaf–Owen value for the weighted majority game associated with the IMF using the polynom package of R software to multiply polynoms.

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Acknowledgements

This work was presented at the 14th European Meeting on Game Theory (SING 14). Authors acknowledge the financial support of MTM2017-87197-C3-2-P project, from Ministry of Economy, Industry and Competitiveness (MINECO), State Research Agency (AEI), and Regional Development Fund (FEDER) and UE; of ERDF (Grupos de Referencia Competitiva) ED431C 2016-040 project, from Xunta de Galicia and of CPI+D11-08 (IO1) project, from Universidade de Vigo. The authors would like to thank the anonymous reviewers for their insightful comments and suggestions.

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  1. Departamento de Estatística e Investigación Operativa, Universidade de Vigo, Vigo, Spain

    A. Saavedra-Nieves & M. G. Fiestras-Janeiro

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  1. A. Saavedra-Nieves
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  2. M. G. Fiestras-Janeiro
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Correspondence to A. Saavedra-Nieves.

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Saavedra-Nieves, A., Fiestras-Janeiro, M.G. Sampling methods to estimate the Banzhaf–Owen value. Ann Oper Res 301, 199–223 (2021). https://doi.org/10.1007/s10479-020-03614-8

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