Information Sufficiency and the Core

This paper introduces a formal framework for analyzing information sufficiency in cooperative decision-making. Unlike models of incomplete information, we consider settings where agents may lack conceptual access to some cooperative opportunities. Using Gentzen-style sequent calculus, we define a syntactic criterion, called $\textsf{C}_i$-acceptability, to determine when a proposed distribution is justifiable given limited structural information. We show that, under a mild assumption that each coalition's potential is known to at least one member, the set of unanimously accepted payoffs coincides with the core. This reinterprets the core not as a predictive solution, but as the boundary of what can be justified under minimal information. Moreover, our results offer a new perspective on Debreu-Scarf's theorem: while the core converges to the competitive equilibrium in replicated markets, this convergence obscures a key asymmetry. Competitive equilibrium requires only local information--preferences and prices--regardless of market size, whereas reaching the core entails increasing informational demands that may ultimately exceed any agent's cognitive capacity.