Utility of gambling under p(olynomial)-additive joint receipt and segregation or duplex decomposition

This article continues our exploration of the utility of gambling, but here under the assumption of a non-additive but p(olynomial)-additive representation of joint receipt. That assumption causes the utility of gambling term to have a multiplicative impact, rather than just an additive one, on what amounts to ordinary weighted utility. Assuming the two rational recursions known as branching and upper gamble decomposition, we investigate separately the rational property of segregation and the non-rational one of duplex decomposition. Under segregation, we show that each pair of disjoint events either exhibits weight complementarity or both support no utility of gambling, a property called UofG-singular. We develop representations for both cases. The former representation is a simple weighted utility with each term multiplied by a function of the event underlying that branch. The latter representation is an ordinary rank-dependent utility with Choquet weights but with no utility of gambling. Under duplex decomposition, we show that weights have the intuitively unacceptable property that there is essentially no dependence upon the events, making duplex decomposition, in this context, of little behavioral interest.