On complexity of single-minded auction (original) (raw)

The communication complexity of private value single-item auctions

Operations Research Letters, 2006

In this paper we present a new auction, the bisection auction, that can be used for the sale of a single indivisible object. We discuss the issue concerning the information revelation requirement of this auction and the associated amount of data that needs to be transmitted. We show that in the truthtelling equilibrium the bisection auction is economical in its demand for information on the valuations of the players. It requires the players to transmit less information bits to the auctioneer than the Vickrey and English auctions. In particular, we prove that for integer valuations uniformly distributed on the interval [0, L) the bisection auction of n players requires in expectation transmission of at most 2n+log L information bits by the players. Compared with the corresponding number in the Vickrey auction which is n log L, and in the English auction which is on average at least (1/3)nL, the bisection auction turns out to be the best performer. JEL Codes. C72, D44.

On the complexity of combinatorial auctions

Proceedings of the 8th ACM conference on Electronic commerce - EC '07, 2007

The winner determination problem in combinatorial auctions is the problem of determining the allocation of the items among the bidders that maximizes the sum of the accepted bid prices. While this problem is in general NPhard, it is known to be feasible in polynomial time on those instances whose associated item graphs have bounded treewidth (called structured item graphs). Formally, an item graph is a graph whose nodes are in one-to-one correspondence with items, and edges are such that for any bid, the items occurring in it induce a connected subgraph. Note that many item graphs might be associated with a given combinatorial auction, depending on the edges selected for guaranteeing the connectedness. In fact, the tractability of determining whether a structured item graph of a fixed treewidth exists (and if so, computing one) was left as a crucial open problem. In this paper, we solve this problem by proving that the existence of a structured item graph is computationally intractable, even for treewidth 3. Motivated by this bad news, we investigate different kinds of structural requirements that can be used to isolate tractable classes of combinatorial auctions. We show that the notion of hypertree decomposition, a recently introduced measure of hypergraph cyclicity, turns out to be most useful here. Indeed, we show that the winner determination problem is solvable in polynomial time on instances whose bidder interactions can be represented with (dual) hypergraphs having bounded hypertree width. Even more surprisingly, we show that the class of tractable instances identified by means of our approach properly contains the class of instances having a structured item graph.

The communication complexity of the private value single item bisection auction

2004

Towards a characterization of truthful combinatorial auctions

Foundations of Computer …, 2003

This paper analyzes implementable social choice functions (in dominant strategies) over restricted domains of preferences, the leading example being combinatorial auctions. Our work generalizes the characterization of who showed that truthful mechanisms over unrestricted domains with at least 3 possible outcomes must be "affine maximizers". We show that truthful mechanisms for combinatorial auctions (and related restricted domains) must be "almost affine maximizers" if they also satisfy an additional requirement of "independence of irrelevant alternatives". This requirement is without loss of generality for unrestricted domains as well as for auctions between two players where all goods must be allocated. This implies unconditional results for these cases, including a new proof of Roberts' theorem. The computational implications of this characterization are severe, as reasonable "almost affine maximizers" are shown to be as computationally hard as exact optimization.

Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions

Proceedings of the 9th ACM conference on Electronic commerce - EC '08, 2008

We provide tight information-theoretic lower bounds for the welfare maximization problem in combinatorial auctions. In this problem the goal is to partition m items between k bidders in a way that maximizes the sum of bidders' values for their allocated items. Bidders have complex preferences over items expressed by valuation functions that assign values to all subsets of items.

combinatorial auctions without money

Algorithmic Mechanism Design attempts to marry computation and incentives, mainly by leveraging monetary transfers between designer and selfish agents involved. This is principally because in absence of money, very little can be done to enforce truthfulness. However, in certain applications, money is unavailable, morally unacceptable or might simply be at odds with the objective of the mechanism. For example, in Combinatorial Auctions (CAs), the paradigmatic problem of the area, we aim at solutions of maximum social welfare, but still charge the society to ensure truthfulness. We focus on the design of incentive-compatible CAs without money in the general setting of k-minded bidders. We trade monetary transfers with the observation that the mechanism can detect certain lies of the bidders: i.e., we study truthful CAs with verification and without money. In this setting, we characterize the class of truthful mechanisms and give a host of upper and lower bounds on the approximation ratio obtained by either deterministic or randomized truthful mechanisms. Our results provide an almost complete picture of truthfully approximating CAs in this general setting with multi-dimensional bidders.

Combinatorial auctions with budgets

2010

We consider budget constrained combinatorial auctions where bidder i has a private value v i for each of the items in some set S i , agent i also has a budget constraint b i. The value to agent i of a set of items R is |R ∩ S i | • v i. Such auctions capture adword auctions, where advertisers offer a bid for those adwords that (hopefully) reach their target audience, and advertisers also have budgets. It is known that even if all items are identical and all budgets are public it is not possible to be truthful and efficient. Our main result is a novel auction that runs in polynomial time, is incentive compatible, and ensures Pareto-optimality. The auction is incentive compatible with respect to the private valuations, v i , whereas the budgets, b i , and the sets of interest, S i , are assumed to be public knowledge. This extends the result of Dobzinski et al. [3, 4] for auctions of multiple identical items and public budgets to single-valued combinatorial auctions with public budgets.

Dynamic Auction: a Tractable Auction Procedure

2010

Dynamic auctions are trading mechanisms for discovering market-clearing prices and efficient allocations based on price adjustment processes. This paper studies the computational issues of dynamic auctions for selling multiple indivisible items. Although the decision problem of efficient allocations in a dynamic auction in general is intractable, it can be solved in polynomial time if the economy under consideration satisfies the condition of Gross Substitutes and Complements, which is known as the most general condition that guarantees the existence of Walrasian equilibrium. We propose a polynomial algorithm that can be used to find efficient allocations and introduce a double-direction auction procedure to discover a Walrasian equilibrium in polynomial time.

Optimal competitive auctions

Proceedings of the forty-sixth annual ACM symposium on Theory of computing, 2014

We study the design of truthful auctions for selling identical items in unlimited supply (e.g., digital goods) to n unit demand buyers. This classic problem stands out from profit-maximizing auction design literature as it requires no probabilistic assumptions on buyers' valuations and employs the framework of competitive analysis. Our objective is to optimize the worst-case performance of an auction, measured by the ratio between a given benchmark and revenue generated by the auction. We establish a sufficient and necessary condition that characterizes competitive ratios for all monotone benchmarks. The characterization identifies the worst-case distribution of instances and reveals intrinsic relations between competitive ratios and benchmarks in the competitive analysis. With the characterization at hand, we show optimal competitive auctions for two natural benchmarks. The most well-studied benchmark F (2) (•) measures the envy-free optimal revenue where at least two buyers win. Goldberg et al. [13] showed a sequence of lower bounds on the competitive ratio for each number of buyers n. They conjectured that all these bounds are tight. We show that optimal competitive auctions match these bounds. Thus, we confirm the conjecture and settle a central open problem in the design of digital goods auctions. As one more application we examine another economically meaningful benchmark, which measures the optimal revenue across all limited-supply Vickrey auctions. We identify the optimal competitive ratios to be (n n−1) n−1 − 1 for each number of buyers n, that is e − 1 as n approaches infinity.

On complexity of single-minded auction (original) (raw)

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