Assume that binary rank-dependent expected utility (subjective expected uti
lity is a special case) holds for gains and losses separately and that a co
mmutative binary operation of joint receipt on consequences and gambles is
linked to binary gambles via the rational property of segregation. This imp
lies that utility U of joint receipt is either itself additive or is an exp
onential transformation of an additive representation V. For joint receipt
of mixed gains and losses two hypotheses are discussed: either U or V is ad
ditive over mixed joint receipts. Both hypotheses are linked back to gamble
s in two different ways: a generalization of segregation and an empirically
sustained but nonrational property called duplex decomposition. The additi
ve U model yields bilinear expressions like rank-dependent expected utility
. Data favor the latter. The additive V models yield nonbilinear representa
tions of mixed gambles.