1993 Reports

Limited Arbitrage, Gains from Trade and Arrow's Theorem

Chichilnisky, Graciela

"The expression 'limited arbitragers' used to describe economies where only bounded,
or limited, opportunities for gains are available to the traders from their initial endowments.
This concept was rigorously defined in [4], [5] and shown to be central
to the problem of resource allocation; it is also linked to the social diversity of the
economy [7]. It turns out that a simple geometric interpretation can be given to limited
arbitrage: here I show that it is equivalent to bounding the gains from trade,
namely the sum of utilities increases which the traders can achieve from their initial
endowments (Proposition 2, Section 1). From this geometry a somewhat unexpected
new link emerges: a close connection with Arrow's impossibility theorem [1]. I establish
that markets have limited arbitrage if and only in they have no Condorcet triples
beyond certain utility levels (Proposition 3, Section 3). This means that on choices of
great importance, irrational or intransitive behavior does not arise. Since Condorcet
triples are the building blocks of Arrow's theorem, limited arbitrage appears to be at
the core of social choice theory. The connection between limited arbitrage and the
concept of no-arbitrage used in financial markets is discussed in Section 2."