Abstract
In this paper we introduce a procedure based on sampling to estimate the Owen value of a cooperative game. It is an adaptation of an analogous procedure for the estimation of the Shapley value, and it is specially useful when dealing with games having large sets of players. We provide some results in order to choose a sample size guaranteeing a bound for the absolute error with a given probability, and illustrate our procedure with an example taken from the game theoretical literature.
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- 1.
Alternatively, a TU-game can map each coalition to the cost that its members support when cooperating. In this case, the game is said to be a cost game and is denoted by (N, c).
- 2.
Hoeffding’s inequality: Let \(\sum _{j=1}^rX_j\) be the sum of r independent random variables such that \(a_j\le X_j\le b_j\) for all \(j\in \{1,\dots ,r\}\). Then \(P(|\sum _{j=1}^rX_j-E(\sum _{j=1}^rX_j)|\ge t)\le 2 \exp (\frac{-2t^2}{\sum _{j=1}^r(b_j-a_j)^2})\).
- 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.
Popoviciu’s inequality on variances: Let M and m be an upper and a lower bound on the values of a bounded random variable X with variance \(\text{ Var }(X)\). Then, \(\text{ Var }(X)\le {\frac{1}{4}}(M-m)^{2}\).
- 5.
The peseta was the official currency in Spain in 1993. One peseta is about 0.006 euros.
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Owen G (1977) Values of games with a priori unions. In: Henn R, Moeschlin O (eds) Mathematical economics and game theory. Springer, Berlin
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Acknowledgements
This work has been supported by MINECO grants MTM2014-53395-C3-1-P, MTM2014-53395-C3-2-P, MTM2014-53395-C3-3-P, and by the Xunta de Galicia through the ERDF (Grupos de Referencia Competitiva ED431C-2016-015 and ED431C-2016-040, and Centro Singular de Investigación de Galicia ED431G/01). Authors acknowledge J. Costa and P. Saavedra-Nieves for their comments on an earlier version of this paper.
Ignacio García-Jurado and M. Gloria Fiestras-Janeiro would like to sincerely thank Pedro Gil for his affectionate help and welcome at various moments in their careers. In particular, Pedro Gil was the organizer of the first conference that they attended, held in Gijón, Spain, in 1985.
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Saavedra-Nieves, A., García-Jurado, I., Fiestras-Janeiro, M.G. (2018). Estimation of the Owen Value Based on Sampling. In: Gil, E., Gil, E., Gil, J., Gil, M. (eds) The Mathematics of the Uncertain. Studies in Systems, Decision and Control, vol 142. Springer, Cham. https://doi.org/10.1007/978-3-319-73848-2_33
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DOI: https://doi.org/10.1007/978-3-319-73848-2_33
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