A characterization of Pareto surfaces

Billera, Louis J.; Bixby, Robert E.

doi:10.1090/S0002-9939-1973-0325163-1

A characterization of Pareto surfaces
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by Louis J. Billera and Robert E. Bixby PDF

Proc. Amer. Math. Soc. 41 (1973), 261-267 Request permission

Abstract:

Given $n$ concave continuous functions ${u_i}$ defined over the unit $m$-cube ${I^m}$, the corresponding attainable set $V$ and Pareto surface $P$ are defined. In the economic interpretation, $V$ corresponds to the set of attainable utility outcomes realized through trading, and $P$ the set of such outcomes for which no trader can attain more without another getting less. Sets of the form of $V$ and $P$ are characterized among all subsets of ${R^n}$. The notion of complexity (the smallest $m$ for which a given $V$ can be realized) is briefly discussed, as is the idea of a “market game".

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