  Probabilistic independence can dramatically simplify the task of eliciting,
representing, and computing with probabilities in large domains. A key
technique in achieving these benefits is the idea of graphical modeling. We
survey existing notions of independence for utility functions in a
multi-attribute space, and suggest that these can be used to achieve similar
advantages. Our new results concern conditional additive independence, which we
show always has a perfect representation as separation in an undirected graph
(a Markov network). Conditional additive independencies entail a particular
functional for the utility function that is analogous to a product
decomposition of a probability function, and confers analogous benefits. This
functional form has been utilized in the Bayesian network and influence diagram
literature, but generally without an explanation in terms of independence. The
functional form yields a decomposition of the utility function that can greatly
speed up expected utility calculations, particularly when the utility graph has
a similar topology to the probabilistic network being used.
