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

The price system is often said to economize on information. This paper asks how much information it saves. I develop a formal framework for measuring the informational burden that agents would have to bear if they had to reason individually over feasible coalitional alternatives. Taking the convergence theorem of Debreu and Scarf (1963) as the benchmark, I ask how much coalition-feasibility information must be held, and how it must be distributed, for individual acceptance decisions to recover the core. Agents' information is represented as finite sets of meaning-bearing sentences in a formal language, and reasoning is modeled as proof in Gentzen's sequent calculus. The main result characterizes the minimal informational structure under which unanimous acceptability coincides with the core for all transferable-utility games on a fixed player set: every coalition must be known to at least one of its members. Removing information about even a single coalition from all of its members suffices to break the equivalence for some game. Applying this result to the $k$-fold replica economy, I identify and count the load-bearing coalitions whose feasibility information must be distributed across the population. The associated average per-agent informational burden grows as $\Theta(4^k/k^{3/2})$ -- exponentially in the size of the economy. This is, precisely and formally, what the price system saves.