Pairwise, t-Wise, and Pareto Optimalities

Abstract

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. OUR PRINCIPAL CONCERN in this inquiry is with the decentralization of the trading process. The analysis departs from the familiar Arrow-Debreu general equilibrium framework to examine the efficiency of economies deprived of the coordinating function of the Walrasian price mechanism. The alternative, presented here, is to permit trade to take place only in small groups-say up to t traders in number. We envision an exchange economy wherein groups form and reform in order to barter-as individuals and as small coalitions. If all such small groups may form, then such a process might eventually converge to an equilibrium from which no reallocation involving t or fewer traders could result in a Pareto preferable allocation. That is, an allocation which is t-wise optimal. The dynamics of pairwise barter trade to achieve a pairwise optimal allocation is thoroughly studied in Feldman [2]. The corresponding analysis for trade in larger groups represents an open research topic, though we certainly expect Feldman's analysis to generalize. It is by no means apparent that such a t-wise optimal allocation would be Pareto optimal. This reflects the difficulty of achieving a reallocation which is preferable for a large group through a sequence of weakly desirable small group trades. Since Pareto optimality is such an essential condition in welfare economics, it is useful to discover under what circumstances the two optimality concepts