International Journal of Game Theory

, Volume 37, Issue 1, pp 139–153 | Cite as

Dynamic realization games in newsvendor inventory centralization

Original Paper

Abstract

Consider a set N of n (> 1) stores with single-item and single-period nondeterministic demands like in a classic newsvendor setting with holding and penalty costs only. Assume a risk-pooling single-warehouse centralized inventory ordering option. Allocation of costs in the centralized inventory ordering corresponds to modelling it as a cooperative cost game whose players are the stores. It has been shown that when holding and penalty costs are identical for all subsets of stores, the game based on optimal expected costs has a non empty core (Hartman et al. 2000, Games Econ Behav 31:26–49; Muller et al. 2002, Games Econ Behav 38:118–126). In this paper we examine a related inventory centralization game based on demand realizations that has, in general, an empty core even with identical penalty and holding costs (Hartman and Dror 2005, IIE Trans Scheduling Logistics 37:93–107). We propose a repeated cost allocation scheme for dynamic realization games based on allocation processes introduced by Lehrer (2002a, Int J Game Theor 31:341–351). We prove that the cost subsequences of the dynamic realization game process, based on Lehrer’s rules, converge almost surely to either a least square value or the core of the expected game. We extend the above results to more general dynamic cost games and relax the independence hypothesis of the sequence of players’ demands at different stages.

Keywords

Dynamic realization games Newsvedor centralization game Cooperative game Allocation process Core Least square value 

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References

  1. Anupindi R, Bassok Y (1999) Centralization of stocks: retailers vs. manufacturer. Manage Sci 45:178–191Google Scholar
  2. Blackwell D (1956) An analog of the MinMax Theorem for vector payoffs. Pacific J Math 6:1–8Google Scholar
  3. Burer S, Dror M (2007) Convex optimization of centralized inventory operation (submitted)Google Scholar
  4. Chen M-S, Lin C-T (1989) Effects of centralization on expected costs in a multi-location newsboy problem. J Oper Res Soc 40:597–602CrossRefGoogle Scholar
  5. Eppen GD (1979) Effects of centralization on expected costs in a multi-location newsboy problem. Manage Sci 25:498–501Google Scholar
  6. Feller W (1966) An introduction to probability. Theory and its applications, vol. II. Wiley, New YorkGoogle Scholar
  7. Fernandez FR, Puerto J, Zafra MJ (2002) Cores of stochastic cooperative games. Int Game Theory Rev 4(3):265–280CrossRefGoogle Scholar
  8. Granot D (1977) Cooperative games in stochastic function form. Manage Sci 23:621–630CrossRefGoogle Scholar
  9. Hartman BC, Dror M (1996) Cost allocation in continuous review inventory models. Naval Res Logistics J 43:549–561CrossRefGoogle Scholar
  10. Hartman BC, Dror M (2003) Optimizing centralized inventory operations in a cooperative game theory setting. IIE Trans Oper Eng 35:243–257CrossRefGoogle Scholar
  11. Hartman BC, Dror M (2005) Allocation of gains from inventory centralization in newsvendor environments. IIE Trans Scheduling Logistics 37:93–107Google Scholar
  12. Hartman BC, Dror M, Shaked M (2000) Cores of inventory centralization games. Games Econ Behav 31:26–49CrossRefGoogle Scholar
  13. Lehrer E (2002a) Allocation process in cooperative games. Int J Game Theory 31:341–351CrossRefGoogle Scholar
  14. Lehrer E (2002b) Approachability in infinite dimensional spaces. Int J Game Theory 31:253–268CrossRefGoogle Scholar
  15. Muller A, Scarsini M, Shaked M (2002) The newsvendor game has a nonempty core. Games Econ Behav 38:118–126CrossRefGoogle Scholar
  16. Naurus JA, Anderson JC (1996) Rethinking distribution. Harvard Bus Rev 74(4):113–120Google Scholar
  17. Parlar M (1988) Game theoretic analysis of the substitutable product inventory problem with random demands. Naval Research Logistics 35:397–409CrossRefGoogle Scholar
  18. Ruiz LM, Valenciano F, Zarzuelo JM (1998) The family of least square values for transferable utility games. Games Econ Behav 24:109–130CrossRefGoogle Scholar
  19. Slikker M, Fransoo J, Wouters M (2005) Cooperation between multiple news-vendors with transshipment. Eur J Oper Res 167:370–380CrossRefGoogle Scholar
  20. Suijs J (2000) Cooperative decision-making under risk. Kluwer, BostonGoogle Scholar
  21. Timmer J (2006) The compromise value for cooperative games with random payoffs. Math Methods Oper Res 64:95–106CrossRefGoogle Scholar
  22. Timmer J, Borm P, Tijs S (2003) On three Shapley-like solutions for cooperative games with random payoffs. Int J Game Theory 32:595–613Google Scholar
  23. Timmer J, Borm P, Tijs S (2005) Convexity in stochastic cooperative situations. Int Game Theory Rev 7:25–42CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Management Information Systems, Eller College of ManagementUniversity of ArizonaTucsonUSA
  2. 2.Operations Research CenterUniversidad Miguel HernándezElcheSpain
  3. 3.Facultad de MatemáticasUniversidad de SevillaSevillaSpain

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