We consider a basic resource allocation game, where the players' strategy spaces are subsets of @math and cost utility functions are parameterized by some common vector @math and, otherwise, only depend on the own strategy choice. A strategy of a player can be interpreted as a vector of resource consumption and a joint strategy profile naturally leads to an aggregate consumption vector. We assume that resources can be priced, that is, the game is augmented by a price vector @math and players have quasi-linear overall costs utilities meaning that in addition to the original costs utilities, a player needs to pay the corresponding price per consumed unit. We investigate the following question: for which aggregated consumption vectors @math can we find prices @math that induce an equilibrium realizing the targeted consumption profile? For answering this question, we develop a duality-based framework and derive a characterization of the existence of such @math and @math . We show that our characterization can help to unify parts of three largely independent streams in the literature -- tolls in transportation systems, Walrasian market equilibria and congestion control in communication networks. Besides reproving existing results we establish novel existence results by drawing connections to polyhedral combinatorics and discrete convexity.
For multi-unit items, several recent papers studied the existence of Walrasian equilibria. @cite_45 investigated the existence of Walrasian equilibria in multi-unit auctions and identified general conditions on the demand sets and valuations related to discrete convexity. The conditions of Milgrom and Struluvici @cite_4 and Ausubel @cite_76 appear as special cases of those in @cite_45 . Baldwin and Klemperer @cite_20 explored a connection with tropical geometry and gave necessary and sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result is also closely related to the work of Danilov, Koshevoy, and Murota @cite_45 , see also Sun and Yang @cite_71 . Tran and Yu @cite_57 gave a new proof of the sufficiency condition of @cite_20 using a unimodularity theorem in integer programming. For a comparison of the above works especially with respect to the role of discrete convexity, we refer to the excellent survey of Shioura and Tamura @cite_54 . @cite_74 @cite_64 showed that valuations classes (beyond GS valuations) based on graphical structures also imply the existence of Walrasian equilibria. Their proof uses integrality of optimal solutions of an associated linear min-cost flow formulation.
@cite_64 @cite_4 @cite_54 @cite_57 @cite_45 @cite_71 @cite_74 @cite_76 @cite_20
2966800818 2304647284 2159078604 2626895635
In a recent and ongoing work, Baldwin and Klemperer explored a connection between tropical geometry and economics. They gave a sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result, which we call the Unimodularity Theorem, can also be traced back to the work of Danilov, Koshevoy, and Murota in discrete convex analysis. We give a new proof of the Unimodularity Theorem via the classical unimodularity theorem in integer programming. We give a unified treatment of these results via tropical geometry and formulate a new sufficient condition for competitive equilibrium when there are only two types of product. Generalizations of our theorem in higher dimensions are equivalent to various forms of the Oda conjecture in algebraic geometry. We study pricing equilibria for graphical valuations, whichare 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 latter 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. We propose a new Walrasian tatonnement process called a double-track procedure for efficiently allocating multiple heterogeneous indivisible items in two distinct sets to many buyers who view items in the same set as substitutes but items across the two sets as complements. In each round of the process, a Walrasian auctioneer first announces the current prices for all items, buyers respond by reporting their demands at these prices, and then the auctioneer adjusts simultaneously the prices of items in one set upward but those of items in the other set downward. It is shown that this procedure converges globally to a Walrasian equilibrium in finitely many rounds. Copyright 2009 The Econometric Society. Our first result shows membership in PPAD for the problem of computing approximate equilibria for an Arrow-Debreu exchange market for piecewise-linear concave (PLC) utility functions. As a corollary we also obtain membership in PPAD for Leontief utility functions. This settles an open question of Vazirani and Yannakakis (2011). Next we show FIXP-hardness of computing equilibria in Arrow-Debreu exchange markets under Leontief utility functions, and Arrow-Debreu markets under linear utility functions and Leontief production sets, thereby settling these open questions of Vazirani and Yannakakis (2011). As corollaries, we obtain FIXP-hardness for PLC utilities and for Arrow-Debreu markets under linear utility functions and polyhedral production sets. In all cases, as required under FIXP, the set of instances mapped onto will admit equilibria, i.e., will be "yes" instances. If all instances are under consideration, then in all cases we prove that the problem of deciding if a given instance admits an equilibrium is ETR-complete, where ETR is the class Existential Theory of Reals. As a consequence of the results stated above, and the fact that membership in FIXP has been established for PLC utilities, the entire computational difficulty of Arrow-Debreu markets under PLC utility functions lies in the Leontief utility subcase. This is perhaps the most unexpected aspect of our result, since Leontief utilities are meant for the case that goods are perfect complements, whereas PLC utilities are very general, capturing not only the cases when goods are complements and substitutes, but also arbitrary combinations of these and much more. Finally, we give a polynomial time algorithm for finding an equilibrium in Arrow-Debreu exchange markets under Leontief utility functions provided the number of agents is a constant. This settles part of an open problem of Devanur and Kannan (2008).