TITLE:
Control Values of Connected Components in Graph Games
AUTHORS:
Letian Chen, Guang Zhang
KEYWORDS:
Cooperative Games, Graph Structure, Single-Layer Restriction, Allocation Rule
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.15 No.11,
November
5,
2025
ABSTRACT: This paper studies an allocation rule for graph games with a control structure. In such graph games, a virtual player
C
is introduced as the control structure; only sub-coalitions that combine with
C
can obtain the coalition payoff. Based on this model, we propose three novel axioms-null player property, control fairness, and linearity of loss-and introduce the control value as the allocation rule. The control value first assigns a Shapley payoff to every player, and the Shapley payoff of the virtual player
C
is then evenly redistributed among all players. We prove that the control value is the unique allocation rule satisfying efficiency, representative null player property, control fairness, and linearity of loss.