Limited Arbitage is Necessary and Sufficient for the Existence of a Competitive Equilibrium and the Core, and Limits Voting Cycles

Abstract

A close connection between the theory of competitive markets, game theory, and Arrow's impossibility theorem is given by limited arbitrage, a condition originally defined on the preferences and endowments of the traders of an Arrow-Debreu economy in Chichilnisky (l99I, l993b, l994, 1995). This condition limits the traders' diversity, and is shown here to bound their mutually beneficial gains from trade. Limited arbitrage is necessary and sufficient for the existence of an Arrow Debreu equilibrium (see also Chichilnisky. 1991, l993b, 1994, I995), and for the existence of the core (see also Chichilnisky, 1993d); it is also necessary and sufficient for the elimination of Condorcet cycles on allocations involving large utility values. Condorcet cycles are the basis for the cyclical behavior of voting rules and a building block of Arrow's impossibility theorem, so that limited arbitrage appears to be at the core of social choice theory.