Pricing Equilibria and Graphical Valuations

Abstract

We characterize pricing equilibria for graphical valuations, which is a class of valuations that admit a compact representation. These valuations are associated with a value graph, whose nodes correspond to items, and edges encode (pairwise) complementarities/substitutabilities between items. It is known that for graphical valuations a Walrasian equilibrium (a pricing equilibrium that relies on anonymous item prices) does not exist in general. Our first contribution is to establish the existence of a pricing equilibrium when the seller uses an agent-specific graphical pricing rule that involves prices for each item and markups/discounts for pairs of items. We then characterize the existence of pricing equilibria with simpler pricing rules which either (i) involve offering prices only for items, or (ii) require anonymity (so that prices are identical for all agents) while allowing for pairwise markups/discounts. We show that a pricing equilibrium with the former pricing rule exists if and only if a Walrasian equilibrium exists, whereas the latter pricing rule may guarantee the existence of a pricing equilibrium even for graphical valuations that do not admit a Walrasian equilibrium. Interestingly, by exploiting a novel connection between the existence of a pricing equilibrium and the partitioning polytope associated with the underlying graph, we also establish that for simple (series-parallel) value graphs a pricing equilibrium with anonymous graphical pricing rule exists if and only if a Walrasian equilibrium exists. These equivalence results imply that simpler pricing rules (i) and (ii) do not guarantee the existence of a pricing equilibrium for all graphical valuations. We extend our results to additively decomposable valuations, which generalize graphical valuations by allowing the valuations to be additive over bundles of cardinality greater than two. We establish that for this class a pricing equilibrium with a pricing rule that similarly decomposes over the same bundles always exists. Our results suggest that for valuations that admit a simple structure (e.g., additively decomposable over bundles of small cardinality), it is always possible to obtain a pricing equilibrium using a simple pricing rule with a similar structure.