Estimation of bankruptcy rules with a priori unions for establishing new systems of quotas

  1. Alejandro Saavedra-Nieves
  2. Paula Saavedra-Nieves
Zeitschrift:
BEIO, Boletín de Estadística e Investigación Operativa

ISSN: 1889-3805

Datum der Publikation: 2023

Ausgabe: 39

Nummer: 1

Art: Artikel

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Andere Publikationen in: BEIO, Boletín de Estadística e Investigación Operativa

Zusammenfassung

This paper addresses a sampling procedure for estimating extensions of the random arrival rule to those bankruptcy situations where there exist a priori unions. It is based on simple random sampling with replacement and it adapts an estimation method of the Owen value for transferable utility games with a priori unions, especially useful when the set of involved agents is sufficiently large. We analyse the theoretical statistical properties of the resulting estimator as well as we provide some bounds for the incurred error. Its performance is evaluated on two well-studied examples in literature where this allocation rule can be exactly obtained. Finally, we apply this sampling method to provide a new quota system for the milk market in Galicia (Spain) for checking the role of different territorial structures when they are taken as a priori unions. The resulting quotas estimator is also compared with two classical rules in bankruptcy literature.

Bibliographische Referenzen

  • Aumann, R. J. and M. Maschler (1985). ✭✭Game theoretic analysis of a bankruptcy problem from the Talmud✮✮. In: Journal of Economic Theory 36.2, pp. 195–213.
  • Borm, P., L. Carpente, B. Casas-Méndez, and R. Hendrickx (2005). ✭✭The constrained equal awards rule for bankruptcy problems with a priori unions✮✮. In: Annals of Operations Research 137.1, pp. 211–227.
  • Castro, J., D. Gómez, and J. Tejada (2009). ✭✭Polynomial calculation of the Shapley value based on sampling✮✮. In: Computers & Operations Research 36.5, pp. 1726–1730.
  • Curiel, I. J., M. Maschler, and S. H. Tijs (1987). ✭✭Bankruptcy games✮✮. In: Zeitschrift f¨ur Operations Research 31.5, A143–A159.
  • Fernández-García, F. R. and J. Puerto-Albandoz (2006). ✭✭Teoria de Juegos Multiobjetivo✮✮. In: Imagraf Impresores SA, Sevilla.
  • Frutos, M. A. de (1999). ✭✭Coalitional manipulations in a bankruptcy problem✮✮. In: Review of Economic Design 4.3, pp. 255–272.
  • González-D´ıaz, J., I. Garc´ıa-Jurado, and M. G. Fiestras-Janeiro (2010). An Introductory Course on Mathematical Game Theory. Vol. 115. Providence: American Mathematical Society.
  • Hoeffding, W. (1963). ✭✭Probability Inequalities for Sums of Bounded Random Variables✮✮. In: Journal of the American Statistical Association 58.301, pp. 13–30.
  • Maleki, S. (2015). ✭✭Addressing the computational issues of the Shapley value with applications in the smart grid✮✮. PhD thesis. University of Southampton.
  • Moulin, H. (2002). ✭✭Axiomatic cost and surplus sharing✮✮. In: Handbook of Social Choice and Welfare 1, pp. 289–357.
  • O’Neill, B. (1982). ✭✭A problem of rights arbitration from the Talmud✮✮. In: Mathematical Social Sciences 2.4, pp. 345–371.
  • Owen, G. (1975). ✭✭Multilinear extensions and the Banzhaf value✮✮. In: Naval Research Logistics (NRL) 22.4, pp. 741–750.
  • Owen, G. (1977). ✭✭Values of games with a priori unions✮✮. In: Mathematical Economics and Game Theory. Springer, pp. 76–88.
  • Popoviciu, T. (1935). ✭✭Sur les ´equations alg´ebriques ayant toutes leurs racines r´eelles✮✮. In: Mathematica 9, pp. 129–145.
  • Pulido, M., J. Sánchez-Soriano, and N. Llorca (2002). ✭✭Game theory techniques for university management: an extended bankruptcy model✮✮. In: Annals of Operations Research 109.1-4, pp. 129– 142.
  • R Core Team (2023). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria. url: https://www.R-project.org/.
  • Saavedra-Nieves, A., I. García-Jurado, and M. G. Fiestras-Janeiro (2018). ✭✭Estimation of the Owen Value Based on Sampling✮✮. In: The Mathematics of the Uncertain. Studies in Systems, Decision and Control. Springer, pp. 347–356.
  • Saavedra-Nieves, A. and P. Saavedra-Nieves (2020). ✭✭On systems of quotas from bankruptcy perspective: the sampling estimation of the random arrival rule✮✮. In: European Journal of Operational Research 285.2, pp. 655–669. issn: 0377-2217.
  • Schmeidler, D. (1969). ✭✭The nucleolus of a characteristic function game✮✮. In: SIAM Journal on Applied Mathematics 17.6, pp. 1163–1170.
  • Shapley, L. S. (1953). ✭✭A value for n-person games✮✮. In: Contributions to the Theory of Games 2.28, pp. 307–317.
  • Thomson, W. (2015). ✭✭Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: an update✮✮. In: Mathematical Social Sciences 74, pp. 41–59.
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