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We identified unique set theoretic properties for connected coalitions in cycle-free networks. This leads to a new "balanced fairness" axiom, which, with component efficiency, uniquely defines the Shapley value for cooperative games.

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Area of Science:

  • Cooperative game theory
  • Network structures
  • Graph theory

Background:

  • Understanding coalition formation in networks is crucial for cooperative game theory.
  • Existing frameworks often lack specific characterizations for cycle-free network structures.
  • The Shapley value is a prominent solution concept in cooperative games.

Purpose of the Study:

  • To identify set theoretic properties characterizing connected coalitions in cycle-free undirected graphs within accessible union stable network structures.
  • To define a novel fairness axiom for cooperative games on these specific network structures.
  • To characterize the Shapley value using this new axiom and component efficiency.

Main Methods:

  • Investigating set theoretic properties of coalitions.
  • Analyzing unique representations of feasible coalitions as unions of supports.
  • Defining and applying a new fairness axiom ('balanced fairness').

Main Results:

  • Connected coalitions in cycle-free networks exhibit specific set theoretic properties.
  • A unique representation for non-unitary feasible coalitions as unions of non-unitary supports was found.
  • A new 'balanced fairness' axiom, combined with component efficiency, uniquely characterizes the Shapley value.

Conclusions:

  • The study provides a precise characterization of coalitions in cycle-free networks.
  • The 'balanced fairness' axiom offers a new perspective on solution concepts in cooperative game theory.
  • The Shapley value is uniquely determined by component efficiency and balanced fairness on these networks.