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Quid Pro Quo allocations in Production-Inventory games

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Document pages: 25 pages

Abstract: The concept of Owen point, introduced in Guardiola et al. (2009), is anappealing solution concept that for Production-Inventory games (PI-games)always belongs to their core. The Owen point allows all the players in the gameto operate at minimum cost but it does not take into account the cost reductioninduced by essential players over their followers (fans). Thus, it may be seenas an altruistic allocation for essential players what can be criticized. Theaim this paper is two-fold: to study the structure and complexity of the coreof PI-games and to introduce new core allocations for PI-games improving theweaknesses of the Owen point. Regarding the first goal, we advance further onthe analysis of PI-games and we analyze its core structure and algorithmiccomplexity. Specifically, we prove that the number of extreme points of thecore of PI-games is exponential on the number of players. On the other hand, wepropose and characterize a new core-allocation, the Omega point, whichcompensates the essential players for their role on reducing the costs of theirfans. Moreover, we define another solution concept, the Quid Pro Quo set(QPQ-set) of allocations, which is based on the Owen and Omega points. Amongall the allocations in this set, we emphasize what we call the Solomonic QPQallocation and we provide some necessary conditions for the coincidence of thatallocation with the Shapley value and the Nucleolus.

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