Kybernetika 57 no. 5, 840-855, 2021

Division schemes under uncertainty of claims

Xianghui Li, Yang Li and Wei ZhengDOI: 10.14736/kyb-2021-5-0840

Abstract:

In some economic or social division problems, we may encounter uncertainty of claims, that is, a certain amount of estate has to be divided among some claimants who have individual claims on the estate, and the corresponding claim of each claimant can vary within a closed interval or fuzzy interval. In this paper, we classify the division problems under uncertainty of claims into three subclasses and present several division schemes from the perspective of axiomatizations, which are consistent with the classical bankruptcy rules in particular cases. When claims of claimants have fuzzy interval uncertainty, we settle such type of division problems by turning them into division problems under interval uncertainty.

Keywords:

interval, division scheme, bankruptcy, fuzzy

Classification:

91A12, 03B52

References:

  1. R. J. Aumann and M. Maschler: Game theoretic analysis of a bankruptcy problem from the Talmud. J. Econom. Theory 36 (1982), 195-213.   DOI:10.1016/0021-9045(82)90041-7
  2. Cooperative interval games: A survey. Cent. Europ. J. Oper. Res. 18 (2010), 397-411.   DOI:10.1007/s10100-009-0116-0
  3. R. Branzei, D. Dimitrov, S. Pickl and S. Tijs: How to cope with division problems under interval uncertainty of claims? Int. J. Uncertain. Fuzz. 12 (2004), {191-200.   DOI:10.1142/S021848850400276X
  4. I. J. Curiel, M. Maschler and S. H. Tijs: Bankruptcy games. Z. Oper. Res. 31 (1987), {A143-A159}.   DOI:10.1007/BF02109593
  5. T. Driessen: Cooperative Games, Solutions and Applications. Kluwer Academic Publishers, 1988.   CrossRef
  6. I. Elishakoff: Resolution of two millennia-old Talmudic mathematical conundrums. BeOr HaTorah 21 (2012), 61-76.   CrossRef
  7. I. Elishakoff and A. Bégin-Drolet: Talmudic bankruptcy problem: special and general solutions. Scientiae Mathematicae Japonicae 69 (2009), 387-403.   CrossRef
  8. H. Habis and P. J. J. Herings: Stochastic bankruptcy games. Int. J. Game Theory} 42 (2013), {973-988}.   DOI:10.1007/s00182-012-0350-x
  9. L. Mallozzi, V. Scalzo and S. Tijs: Fuzzy interval cooperative games. Fuzzy Set Syst. 165 (2011), 1, 98-105.   DOI:10.1016/j.fss.2010.06.005
  10. J. D. Moreno-Ternero and A. Villar: The Talmud rule and the securement of agents' awards. Math. Soc. Sci. 47 (2004), {245-257}.   DOI:10.1016/s0165-4896(03)00087-8
  11. B. O'Neill: A problem of rights arbitration from the Talmud. Math. Soc. Sci. 2 (1982), 345-371.   CrossRef
  12. M. Pulido, J. Sánchez-Soriano and N. Llorca: Game theory techniques for university management: an extended bankruptcy model. Ann. Oper. Res. 109 (2002), {129-142}.   DOI:10.1023/A:1016395917734
  13. D. Schmeidler: The nucleolus of a characeristic function. SIAM J. Appl. Math. 17 (1969), 1163-1170.   DOI:10.1137/0117107
  14. W. J. Zhao and J. C. Liu: Interval-valued fuzzy cooperative games based on the least square excess and its application to the profit allocation of the road freight coalition. Symmetry 10 (2018), 709.   DOI:10.3390/sym10120709
  15. S. Tijs: Bounds for the core of a game and the t-value. In O. Moeschlin, \& D. Pallaschke (Eds.), Game Theory Math. Econom. (1981), pp. 123-132. North-Holland Publishing Company.   CrossRef
  16. R. R. Yager and V. Kreinovich: Fair division under interval uncertainty. Int. J. Uncert. Fuzz. 8 (2000), 611-618.   CrossRef
  17. X. Yu and Q. Zhang: Core for game with fuzzy generalized triangular payoff value. Int. J. Uncert. Fuzz. 27 (2019), 789-813.   DOI:10.1142/S0218488519500351