A characterization of the existence of succinct linear representation of subset-valuations

Decisions that involve bundling or unbundling a large number of objects, such as deciding on the bundle structure or optimizing bundle prices, are based on underlying valuation function over the set of all possible bundles. Given that the number of possible bundles (i.e., subsets of the given set of objects) is exponential in the number of objects, it is important for the decision-maker to be able to represent this valuation function succinctly. Identifying all structural sources of synergy in subset valuations might point to simple and concise representation of the valuation function. We characterize additive and multiplicative representations of synergies in subset valuations and subset utility, which in turn points to necessary and sufficient conditions for a succinct representation of subset valuations to exist.