Measuring Information Burden: From Coalition-Based Reasoning to the Price System

The price system is often said to economize on information, but economics has lacked a formal measure of how much information it saves. This paper develops such a measure. We construct a proof-theoretic framework for decentralized cooperative decision-making in which an agent's information is a finite set of sentences in a formal language and her reasoning is an explicit proof in Gentzen's sequent calculus. Within this framework, we define $C_i$-acceptability, a criterion under which agent $i$ accepts a proposed allocation whenever she can prove, from the information she holds, that no coalition containing her can feasibly improve upon it. Theorem 1 characterizes the minimal informational structure under which unanimous acceptability coincides exactly with the core: every coalition must be known to at least one of its members, and this coverage condition is tight. The core is thus reinterpreted as the boundary of what can be unanimously justified under decentralized reasoning. We apply the framework to the Debreu-Scarf replica economy. There, sentence-level measurement becomes intractable, but Theorem 1 permits aggregation to the coalitional level: the relevant burden is the number of load-bearing coalitions whose feasibility sets must be known somewhere in the population. For the $k$-fold replica, we identify these coalitions exactly and show that the average per-agent informational burden grows as $\Theta (4^k/k^{3/2})$ and therefore without bound. The paper thereby gives a precise answer, in this setting, to the Hayekian question of how much information the price system saves relative to direct decentralized reasoning.