How Competitive Are Competitive Payoff Vectors? Money and Commodity Bounds in Market Games

How many commodities are required to represent a totally balanced game with transferable utility? The answer depends on whether the representing economy reproduces only coalition values or also makes the assumed utility transfers feasible. Hart’s (n − 1)-commodity bound concerns a reduced-form market representing coalition values by aggregate surplus. Its complete quasi-linear representation contains an additional transferable numéraire, which is necessary for payoff feasibility as well as competitive exchange. Counting this commodity yields a sharp worst-case bound of n, already without an equilibrium requirement. We then establish a lower bound for Walrasian representation: every concave pure-exchange representation of the n-player unanimity game that realizes even one core payoff vector at nonzero equilibrium prices requires at least n commodities. The proof expresses Hart’s commodity lower bound through the rank of net trades and combines it with individual budget balance. Building on Shapley and Shubik’s classical direct-market construction, we make explicit its budget-feasible realization of the core. A self-contained proof distinguishes trading-profit maximization with monetary settlement from utility maximization under physical-commodity budgets; the two interpretations yield the same set of payoff vectors, but need not use the same allocation. A coalition-value criterion extends the lower bound beyond unanimity and yields a quantitative welfare bound for low-dimensional representing economies. The analysis separates commodity accounting, transferable-payoff feasibility, and Walrasian implementation.