A Double-Sided Multiunit Combinatorial Auction for Substitutes: Theory and Algorithms

Combinatorial exchanges have existed for a long time in securities markets. In these auctions buyers and sellers can place orders on combinations, or bundles of different securities. These orders are conjunctive: they are matched only if the full bundle is available. On business-to-business (B2B) exchanges, buyers have the choice to receive the same product with different attributes; for instance the same product can be produced by different sellers. A buyer indicates his preference by submitting a disjunctive order, where he specifies how much of the product he wants, and how much he values each attribute. Only the goods with the best attributes and prices will be matched. This article considers a doubled-sided multi-unit combinatorial auction for substitutes, that is, a uniform price auction where buyers and sellers place both types of orders, conjunctive and disjunctive. We prove the existence of a linear price which is both competitive and surplus-maximizing when goods are perfectly divisible, and nearly so otherwise. We describe an algorithm to clear the market, which is particularly efficient when the number of traders is large.Combinatorial auction, economic equilibrium

On the Concentration of Allocations and Comparisons of Auctions in Large Economies

We analyze competitive pressures in a sequence of auctions with a growing number of bidders, in a model that includes private and common valuations as special cases. We show that the key determinant of bidders' surplus (and implicitly auction revenue) is how the goods are distributed. In any setting and sequence of auctions where the allocation of good(s) is concentrated among a shrinking proportion of the population, the winning bidders enjoy no surplus in the limit. If instead the good(s) are allocated in a dispersed manner so that a non- vanishing proportion of the bidders obtain objects, then in any of a wide class of auctions bidders enjoy a surplus that is bounded away from zero. Moreover, under dispersed allocations, the format of the auction matters. If bidders have constant marginal utilities for objects up to some limit, then uniform price auctions lead to higher revenue than discriminatory auctions. If agents have decreasing marginal utilities for objects, then uniform price auctions are asymptotically efficient, while discriminatory auctions are asymptotically {\sl in}efficient. Finally, we show that in some cases where dispersed allocations are efficient, revenue may increase by bundling goods at the expense of efficiency.Auction, Competition, Mechanism, Asymptotic Efficiency, Revenue Equivalence

Distributed Channel Assignment in Cognitive Radio Networks: Stable Matching and Walrasian Equilibrium

We consider a set of secondary transmitter-receiver pairs in a cognitive radio setting. Based on channel sensing and access performances, we consider the problem of assigning channels orthogonally to secondary users through distributed coordination and cooperation algorithms. Two economic models are applied for this purpose: matching markets and competitive markets. In the matching market model, secondary users and channels build two agent sets. We implement a stable matching algorithm in which each secondary user, based on his achievable rate, proposes to the coordinator to be matched with desirable channels. The coordinator accepts or rejects the proposals based on the channel preferences which depend on interference from the secondary user. The coordination algorithm is of low complexity and can adapt to network dynamics. In the competitive market model, channels are associated with prices and secondary users are endowed with monetary budget. Each secondary user, based on his utility function and current channel prices, demands a set of channels. A Walrasian equilibrium maximizes the sum utility and equates the channel demand to their supply. We prove the existence of Walrasian equilibrium and propose a cooperative mechanism to reach it. The performance and complexity of the proposed solutions are illustrated by numerical simulations.Comment: submitted to IEEE Transactions on Wireless Communicaitons, 13 pages, 10 figures, 4 table

Monetary Exchange with Multilateral Matching

This paper analyzes monetary exchange in a search model allowing for multilateral matches to be formed, according to a standard urn-ball process. We consider three physical environments: indivisible goods and money, divisible goods and indivisible money, and divisible goods and money. We compare the results with Kiyotaki and Wright (1993), Trejos and Wright (1995), and Lagos and Wright (2005) respectively. We …nd that the multilateral matching setting generates very simple and intuitive equilibrium allocations that are similar to those in the other papers, but which have important di¤erences. In particular, sur- plus maximization can be achieved in this setting, in equilibrium, with a positive money supply. Moreover, with ‡exible prices and directed search, the …rst best allocation can be attained through price posting or through auctions with lotteries, but not through auctions without lotteries. Finally, analysis of the case of divisible goods and money can be performed without the assumption of large families (as in Shi (1997)) or the day and night structure of Lagos and Wright (2005)Matching, Money, Directed Search

Welfare and Revenue Guarantees for Competitive Bundling Equilibrium

We study equilibria of markets with $m$ heterogeneous indivisible goods and $n$ consumers with combinatorial preferences. It is well known that a competitive equilibrium is not guaranteed to exist when valuations are not gross substitutes. Given the widespread use of bundling in real-life markets, we study its role as a stabilizing and coordinating device by considering the notion of \emph{competitive bundling equilibrium}: a competitive equilibrium over the market induced by partitioning the goods for sale into fixed bundles. Compared to other equilibrium concepts involving bundles, this notion has the advantage of simulatneous succinctness ($O(m)$ prices) and market clearance. Our first set of results concern welfare guarantees. We show that in markets where consumers care only about the number of goods they receive (known as multi-unit or homogeneous markets), even in the presence of complementarities, there always exists a competitive bundling equilibrium that guarantees a logarithmic fraction of the optimal welfare, and this guarantee is tight. We also establish non-trivial welfare guarantees for general markets, two-consumer markets, and markets where the consumer valuations are additive up to a fixed budget (budget-additive). Our second set of results concern revenue guarantees. Motivated by the fact that the revenue extracted in a standard competitive equilibrium may be zero (even with simple unit-demand consumers), we show that for natural subclasses of gross substitutes valuations, there always exists a competitive bundling equilibrium that extracts a logarithmic fraction of the optimal welfare, and this guarantee is tight. The notion of competitive bundling equilibrium can thus be useful even in markets which possess a standard competitive equilibrium

A Double-Track Auction for Substitutes and Complements

We propose a new t^atonnement process called a double-track auction for efficiently allocating multiple heterogeneous indivisible items in two distinct sets S1 and S2 to many buyers who view items in the same set as substitutes but items across the two sets as complements. The auctioneer initially announces sufficiently low prices for items in one set, say S1, but sufficiently high prices for items in the other set S2. In each round, the buyers respond by reporting their demands at the current prices and the auctioneer adjusts prices upwards for items in S1 but downwards for items in S2 based on buyers' reported demands until the market is clear. Unlike any existing auction, this auction is a blend of a multi-item ascending auction and a multi-item descending auction. We prove that the auction finds an efficient allocation and its market-clearing prices in finitely many rounds. Based on the auction we also establish a dynamic, efficient and strategy-proof mechanism.Market design, dynamic auction, t^atonnement process, gross substitutes and complements, Walrasian equilibrium, incentives.

Monetary Exchange with Multilateral Matching

This paper analyzes monetary exchange in a search model allowing for multilateral matches to be formed, according to a standard urn-ballprocess. We consider three physical environments: indivisible goods and money, divisible goods and indivisible money, and divisible goods and money. We compare the results with Kiyotaki and Wright (1993), Trejos and Wright (1995), and Lagos and Wright (2005) respectively. We find that the multilateral matching setting generates very simple and intuitive equilibrium allocations that are similar to those in the other papers, but which have important differences. In particular, surplus maximization can be achieved in this setting, in equilibrium, with a positive money supply. Moreover, with flexible prices and directed search, the first best allocation can be attained through price posting or through auctions with lotteries, but not through auctions without lotteries. Finally, analysis of the case of divisible goods and money can be performed without the assumption of large families (as in Shi (1997)) or the day and night structure of Lagos and Wright (2005).monetary exchange; directed search; ex post bidding; multilateral matching

Monopolistic Provision of Excludable Public Goods under Private Information

This paper characterizes the optimal contract designed by a profit-maximizing monopolist, who can provide an indivisible and excludable public good to a group of n potential consumers, whose valuations are private information. The analysis takes distribution costs and congestion effects into account. The second-best allocation rule, which is welfare-maximizing under the constraint of non-negative profits, is characterized. Properties of the optimal mechanism in the case of many potential consumers are analyzed and it is shown that in this case the monopolist can use simple posted-price contracts. Finally, implications for public intervention are discussed.