Pricing Equilibria and Graphical Valuations

ACM Transactions on Economics and Computation, Vol. 1, No. 1, Article 1, January 2016

26 Pages Posted: 3 Oct 2015 Last revised: 28 Oct 2017

See all articles by Ozan Candogan

Ozan Candogan

University of Chicago - Booth School of Business

Asuman E. Ozdaglar

Massachusetts Institute of Technology (MIT) - Department of Electrical Engineering and Computer Science

Pablo Parrilo

Massachusetts Institute of Technology (MIT) - Electrical Engineering and Computer Science

Date Written: September 27, 2015

Abstract

We study 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. On the other hand, a pricing equilibrium exists when the seller uses an agent-specific graphical pricing rule that involves prices for each item and markups/discounts for pairs of items. We study the existence of pricing equilibria with simpler pricing rules which either (i) require anonymity (so that prices are identical for all agents) while allowing for pairwise markups/discounts, or (ii) involve offering prices only for items. We show that a pricing equilibrium with the la er pricing rule exists if and only if a Walrasian equilibrium exists, whereas the former 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.

Suggested Citation

Candogan, Ozan and Ozdaglar, Asuman E. and Parrilo, Pablo, Pricing Equilibria and Graphical Valuations (September 27, 2015). ACM Transactions on Economics and Computation, Vol. 1, No. 1, Article 1, January 2016, Available at SSRN: https://ssrn.com/abstract=2668330 or http://dx.doi.org/10.2139/ssrn.2668330

Ozan Candogan (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S. Woodlawn Avenue
Chicago, IL 60637
United States

HOME PAGE: http://faculty.chicagobooth.edu/ozan.candogan/

Asuman E. Ozdaglar

Massachusetts Institute of Technology (MIT) - Department of Electrical Engineering and Computer Science ( email )

50 Memorial Drive
Cambridge, MA 02139-4307
United States
617-324-0058 (Phone)

Pablo Parrilo

Massachusetts Institute of Technology (MIT) - Electrical Engineering and Computer Science ( email )

77 Massachusetts Avenue
Cambridge, MA 02139-4307
United States

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