A note on non-essential players in multi-choice cooperative games

A Characterization of the Multi-Choice Shapley Value With Partially Consistent Property

“…In the proof of Theorem 1 in [6], the first formula in page 428 in [6], we see a reformulation of the H&R Shapley value as follows. To make this article-self contained, we give a simple proof as follows.…”

“…It is symmetric among players and asymmetric among actions, or say choices. The multi-choice Shapley value is monotonic, dummy free of dummy player, dummy free of dummy action [4], transferable utility invariant and independent of non-essential player [6], redundant free [7]. Moreover, in this article, we will prove that it has consistent property that allows a player to reduce some part of his choices instead of all of his choices.…”

Tolerance design optimization on cost–quality trade-off using the Shapley value method

“…The final optimization scheme, which is identified from the Pareto-optimal set, is a kind of payoff allocation among players making everyone's profit not worse than that from one's own efforts. It is a fair and stable trade-off that has adequately considered the interdependent and the interactional relations among all the players' payoffs and can be considered as one kind of importance evaluation of each player based on their contribution to the whole coalition [10,17,18]. The computational procedure is demonstrated through a design example of a vehicle front lamp assembly.…”

The Potential and Consistency Property for Multi-Choice Shapley Value

“…The H&R Shapley value is monotone, transferable utility invariant, dummy free and independent of non-essential players, please see [2] and [5] for details. In 1991, when Hsiao and Raghavan presented [3] in the 2rd International Conference on Game Theory at Stony-Brook, Shapley suggested that we should study the consistent property of the H&R Shapley value.…”