Markets provide a widely used solution to the problem of allocating resources among the members of the economy. A market equilibrium is individually optimal and clears the markets. The efficiency of competitive market allocations is what makes them desirable.
A different solution to the resource allocation problem is provided by social choice rules. These rules derive a social preference as a function of individual preferences in a fair fashion, for example through voting. An allocation which is optimal according to a social preference, is called a social allocation.
A third solution to the problem of resource allocations is the core: these are allocations from which no coalition would want to deviate. This paper establishes a clear connection between equilibrium theory, social choice theory and game theory by showing that, for a well defined social choice problem, a condition which is necessary and sufficient to solve the social choice problem limited arbitrage - is the same as the condition which is necessary and sufficient to establish the existence of an equilibrium and the core. The connection is strengthened by establishing that a market allocation, which is in the core, can always be realized as a social allocation.
How does limited arbitrage work? Limited arbitrage is a condition on traders endowments and preferences. Introduced in [18] it bounds potential gains from trade, and the scope for mutually advantageous reallocation in the economy. In mathematical terms, limited arbitrage is the non-empty intersection of a family of cones. Somewhat surprisingly, this nonempty intersection is the same as a topological condition: the contractibility of spaces of preferences. This connection arises from a new result: afamily ofconvex sets intersects if and only if every subfamily has a contractible union, proved in Chichilnisky [15, 19]. It has been shown that contractibility is a restriction on the diversity of preferences [44]. Through its connection with contractibillty, therefore, limited arbitrage implies a restriction on the traders diversity [22], [23, 26, 28]. Indeed, the central role of limited arbitrage in resource allocation is explained by its connection with social diversity [22], [23], [26], [28]. Social diversity is crucial for resource allocation.
