Within the framework of separable utility theory, a condition, called reduction invariance, is shown to be equivalent to the 2-parameter family of weighting functions that Prelec (1998) derived from the condition called compound invariance. Reduction invariance, which is a variant on the reduction of compound gambles, is appreciably simpler and more easily testable than compound invariance, and a simpler proof is provided. Both conditions are generalized leading to more general weighting functions that include, as special cases, the families of functions that Prelec called exponential-power and hyperbolic logarithm and that he derived from two other invariance principles. However, of these various families, only Prelec's compound-invariance family includes, as a special case, the power function, which arises from the simplest probabilistic assumption of reduction of compound gambles. Copyright 2001 Academic Press.