Bang-Bang Correlation Sensitivity: A Local-to-Global Dichotomy for Correlation Preference

Binary choice under risk can depend on how the alternative outcomes are jointly distributed. We show that in a finite outcome space, continuity, completeness, and betweenness impose a local-to-global restriction on this dependence: indifference between two alternatives with one common marginal distribution, however their outcomes are jointly distributed, implies correlation insensitivity for all lotteries supported on the same outcomes. In the correlation-sensitive case, these three axioms alone characterize correlation weighted utility, which combines statewise comparisons with a bilinear function of the marginal lotteries. In the correlation-insensitive case, an additional projective independence axiom yields Fishburn's (1982) skew-symmetric bilinear representation. Strong independence reduces correlation weighted utility to Lanzani's (2022) correlation expected utility representation. We extend the correlation-sensitive characterization to finitely supported lotteries on general metrizable outcome spaces. The resulting framework distinguishes correlation effects from departures from classical independence, and yields testable restrictions linking choices between identically distributed lotteries to correlated Allais behavior.