Econometrica: May, 1982, Volume 50, Issue 3
Pairwise, t-Wise, and Pareto Optimalities
Ross M. Starr, Steven M. Goldman
An allocation is said to be t-wise optimal (for t a positive integer) if for every collection of t traders, there is no reallocation of their current holdings that will make some better off while making none worse off. The allocation is pairwise optimal if it is t-wise optimal for t = 2. A t-wise optimal allocation is the outcome of a trading process more decentralized than that of the Walrasian equilibrium. It represents the result of a variety of separate transactions in small groups without the (centralized) coordination provided by a single Walrasian auctioneer. Necessary conditions and sufficient conditions on allocations for t-wise optimality to imply Pareto optimality are developed. These generally require sufficient overlap in goods holdings among traders to ensure the presence of common support prices. This is formalized as indecomposability of a truncated submatrix of the allocation matrix. A necessary and sufficient condition remains an open question.