Necessary players and values

TL;DR

The paper addresses fair payoff allocation in transferable-utility cooperative games with necessary players, introducing the $Γ$ value and axiomatizing it via a necessary-players property. It builds a chain of intermediate constructions, $G$ and $γ$, and then defines $Γ$ as an invariant adjustment, extending these ideas to coalitional structures through $γ^C$ and $Γ^C$. It provides axiomatic characterizations for $Γ$, $γ$, and $G$, and gives coalitional extensions that characterize the Owen and Banzhaf-Owen values using necessary-player properties, illustrated by elevator-cost sharing scenarios. The work situates these values within the family of ideal values, connects them to equal-division benchmarks, and offers a robust framework for cost allocation and coalition-structure problems where certain players are essential for any outcome.

Abstract

In this paper we introduce the $Γ$ value, a new value for cooperative games with transferable utility. We also provide an axiomatic characterization of the $Γ$ value based on a property concerning the so-called necessary players. A necessary players of a game is one without which the characteristic function is zero. We illustrate the performance of the $Γ$ value in a particular cost allocation problem that arises when the owners of the apartments in a building plan to install an elevator and share its installation cost; in the resulting example we compare the proposals of the $Γ$ value, the equal division value and the Shapley value in two different scenarios. In addition, we propose an extension of the $Γ$ value for cooperative games with transferable utility and with a coalition structure. Finally, we provide axiomatic characterizations of the coalitional $Γ$ value and of the Owen and Banzhaf-Owen values using alternative properties concerning necessary players.

Theorems & Definitions (16)

• Definition 1
• Example 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6
• Theorem 7
• Example 8
• Theorem 9
• Theorem 10