Balancedness and Core of Stochastic Cooperative Games with Proportional Distributions

This paper studies the balancedness and core properties of stochastic cooperative games under proportional distribution mechanisms. Different from decomposed payoff structures in the existing literature, where individual payoffs are separated into deterministic and stochastic components, we consider a restricted allocation environment in which each player’s payoff is a fixed proportional share of the realized stochastic coalition payoff. This proportional allocation rule is relevant for institutional settings where exposure shares are determined ex ante and cannot be adjusted after uncertainty is realized. Under homogeneous preference structures, we employ expectation–variance and quantile-based evaluation criteria to transform random coalition outcomes into comparable deterministic indices. We show that, under the proportional distribution mechanism, the equivalence between balancedness and core non-emptiness is not unconditional. Rather, it can be recovered only under additional structural assumptions, especially homogeneous preferences and positivity of evaluated coalition values. These results clarify the boundary under which fixed proportional sharing can support stable cooperation in stochastic cooperative games.