Multilinear Expected Utility
Published Online:1 Nov 1980https://doi.org/10.1287/moor.5.4.502
Abstract
For each i in {1, …, n} let pi be a convex set of finitely additive probability measures defined on an atomic Boolean algebra of subsets of Xi, and let P = P1 × ⋯ × Pn and X = X1 × ⋯ × Xn. Under specified structural assumptions, axioms are stated for a preference relation ≻ on P which are necessary and sufficient for the existence of a real valued utility function u on X for which ∫Xu(x1, …, xn) dpn(xn), …, dp1(x1) is finite for all (p1, …, pn) in P and for which p ≻ q iff
$$ \int_{X}u(x)dp_{n}(x_{n}) \ldots dp_{1}(x_{1})>\int_{X}u(x)dq_{n}(x_{n})\ldots dq_{1}(x_{1}),$$
for all p and q in P. A simpler set of axioms yields the same results when each algebra is a Borel algebra and all measures are countably additive. The axioms allow the Pi to contain nonsimple measures without necessarily implying that u is bounded. The relevance of the formulation to n-person games and other situations is noted.
