Characterizations of Proportional Division Value in TU-Games via Fixed-Population Consistency
This paper investigates the proportional allocation value in TU cooperative games—i.e., the rule that distributes the grand coalition’s total worth among players in proportion to their individual stand-alone values. We introduce three novel axioms—homogeneity, compositionality, and consistency for nullified games—and combine them with efficiency to provide the first complete axiomatic characterization of this value; notably, the “fixed-population consistency” axiom ensures stability and fairness of allocations under invariant coalition structures. Methodologically, we integrate axiomatic analysis with a decomposition–recomposition structural technique, achieving rigorous formal derivation within the transferable utility game framework. Our main contributions are: (i) establishing a minimal and complete axiom system for the proportional allocation value; (ii) clarifying its fundamental distinction from classical solutions (e.g., the Shapley value); and (iii) furnishing a theoretically grounded and operationally applicable principle for fair resource allocation proportional to individual marginal contributions.