An Ascending Polynomial Running Time Vickrey Auction for Selling Bases of an Integer Polymatroid

In practice, the rules in most open English auctions require participants to raise bids by a sizeable, discrete amount. Furthermore, some bidders are typically more aggressive in seeking to become the “current bidder” during competitive bidding. Most auction theory, however, has assumed bidders can place any tiny “continuous” bid increase, and recommend as optimal the tiniest possible increase. This article examines how incorporating discrete bidding and bidder aggressiveness affect optimal strategies for an important decision for auction sellers, which is setting the lowest acceptable bid at which to sell the property. We investigate two alternative methods sellers often use to enforce this decision. These are setting an irrevocable reserve before the auction, and covert shilling, where the seller or confederates pose as bona fide bidders and raise bona fide bids, unsuspected by bidders. These optimal strategies interest auction participants, especially sellers who must recognize the bidding rules and bidder aggressiveness they will encounter in actual auctions. We also examine how these strategies change with the auction context, such as the number of bidders, and how they differ from corresponding strategies already identified for continuous bidding. Our model examines open English auctions where bidders have independent, private valuations. We find that discrete bidding does affect these strategies, as does the aggressiveness of the bidder with the highest valuation, relative to the average aggressiveness of all other remaining bidders. We identify the seller's optimal discrete reserve, and show that if the highest valuator is relatively more (less) aggressive, this increases (decreases) from the optimal continuous reserve, and also increases (decreases) as the number of bidders increases. With continuous bidding, by contrast, this reserve is invariant to the number of bidders. As this bidder becomes relatively more aggressive, for a given number of bidders, the optimal discrete reserve increases, while as he or she becomes less aggressive, the seller's expected auction utility increases, which increases the set of auctions where discrete bidding generates higher seller welfare than continuous. We propose a covert shilling model that requires shilling sellers, and any confederates and auctioneers, to outwardly act no differently than with reserves, to avoid detection. We identify cases where the seller optimally shills once the bona fide bidding has stopped, and identify the corresponding optimal point to stop shilling and accept the next bona fide bid, if offered. This stopping point does not depend on where bona fide bidding stops, or aggressiveness, or the number of bidders, or on whether shill bids alternate with bona fide bids or are consecutively entered. We also find that the optimal lowest acceptable bid with shilling can be higher (lower) than that with reserves if the highest valuator is sufficiently unaggressive (aggressive). By comparison, in continuous bidding shilling and reserves yield identical lowest acceptable bids. Sometimes the seller using a shilling strategy optimally should not shill at all, and instead accept the bid where bona fide bidding stops. This can occur when that bid, or the number of bidders, is sufficiently high, or when the highest valuator is as, or less, aggressive than other bidders. Optimal shilling can be as practical to implement as reserves, because it does not require sellers to have any information beyond that needed in a reserve auction. If sellers shill optimally, they can never be worse off compared to using a reserve, and can be better off. Shilling can make bidders worse off, but can also make them better off when the seller using a shilling strategy optimally accepts bids below the optimal reserve. In these latter cases, shilling Pareto dominates reserves, ex ante. We provide numerical examples to illustrate these results. We discuss how our results might be affected if shilling is not covert, or bidders' valuations have a common value component rather than being independent, or by the rules used in many discrete bid Internet auctions.

Banking: A Mechanism Design Approach

The authors study banking using the tools of mechanism design, without a priori assumptions about what banks are, who they are, or what they do. Given preferences, technologies, and certain frictions - including limited commitment and imperfect monitoring - they describe the set of incentive feasible allocations and interpret the outcomes in terms of institutions that resemble banks. The bankers in the authors' model endogenously accept deposits, and their liabilities help others in making payments. This activity is essential: if it were ruled out the set of feasible allocations would be inferior. The authors discuss how many and which agents play the role of bankers. For example, they show agents who are more connected to the market are better suited for this role since they have more to lose by reneging on obligations. The authors discuss some banking history and compare it with the predictions of their theory.

We provide a reduction from revenue maximization to welfare maximization in multidimensional Bayesian auctions with arbitrary - possibly combinatorial - feasibility constraints and independent bidders with arbitrary - possibly combinatorial-demand constraints, appropriately extending Myerson's single-dimensional result [21] to this setting. We also show that every feasible Bayesian auction - including in particular the revenue-optimal one - can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidder's type ti is transformed into a virtual type fi(ti), via a bidder-specific function. Then, the allocation maximizing virtual welfare is chosen. Using this characterization, we show how to find and run the revenue-optimal auction given only black-box access to an implementation of the VCG allocation rule. We generalize this result to arbitrarily correlated bidders, introducing the notion of a second-order VCG allocation rule. Our results are computationally efficient for all multidimensional settings where the bidders are additive, or can be efficiently mapped to be additive, albeit the feasibility and demand constraints may still remain arbitrary combinatorial. In this case, our mechanisms run in time polynomial in the number of items and the total number of bidder types, but not type profiles. This is polynomial in the number of items, the number of bidders, and the cardinality of the support of each bidder's value distribution. For generic correlated distributions, this is the natural description complexity of the problem. The runtime can be further improved to polynomial in only the number of items and the number of bidders in itemsymmetric settings by making use of techniques from [15].

The objective of this paper is to investigate whether the affiliation effect is of only marginal importance or it is of the same order of magnitude as the winner's curse and competition effects. This paper is an application of Pinkse and Tan in which they show that bids can be decreasing in the number of bidders in private value auctions provided that the bidders' private values are affiliated. They argue that the affiliation effect is also present in common value auctions. If the affiliation effect is substantial then a regression of bids on the number of bidders will not help in distinguishing between the common and private value paradigms. We use the Offshore Continental Shelf auction data-set to estimate these effects and find that the affiliation effect is smaller than the other two effects in terms of size. Therefore, at least in this application it appears unlikely that the affiliation effect would often offset the competition effect.

We show that every feasible, Bayesian, multi-item multi-bidder mechanism for independent, additive bidders can be implemented as a mechanism that: (a) allocates every item independently of the other items; (b) for the allocation of each item it uses a strict ordering of all bidders' types; and allocates the item using a distribution over hierarchical mechanisms that iron this ordering into a non-strict ordering, and give the item uniformly at random to the bidders whose reported types dominate all other reported types according to the non-strict ordering. Combined with cyclic-monotonicity our results provide a characterization of feasible, Bayesian Incentive Compatible mechanisms in this setting. Our characterization is enabled by a new, constructive proof of Border's theorem [Border 1991], and a new generalization of this theorem to independent (but not necessarily identically distributed) bidders, improving upon the results of [Border 2007, Che-Kim-Mierendorf 2011]. For a single item and independent bidders, we show that every feasible reduced form auction can be implemented as a distribution over hierarchical mechanisms that are consistent with the same strict ordering of all bidders' types, which every mechanism in the support of the distribution irons to a non-strict ordering. We also give a polynomial-time algorithm for determining feasibility of a reduced form auction, or providing a separation hyperplane from the set of feasible reduced forms. To complete the picture, we provide polynomial-time algorithms to find and exactly sample from a distribution over hierarchical mechanisms consistent with a given feasible reduced form. All these results generalize to multi-item reduced form auctions for independent, additive bidders. Finally, for multiple items, additive bidders with hard demand constraints, and arbitrary value correlation across items or bidders, we give a proper generalization of Border's Theorem, and characterize feasible reduced form auctions as multi-commodity flows in related multi-commodity flow instances. We also show that our generalization holds for a broader class of feasibility constraints, including the intersection of any two matroids.

We efficiently solve optimal multi-dimensional mechanism design problem for independent additive bidders with arbitrary demands when either number of bidders is held constant or number of items is held constant. In first setting, we need that each bidder's values for items are sampled from a possibly correlated, item-symmetric distribution, allowing different distributions for each bidder. In second setting, we allow values of each bidder for items to be arbitrarily correlated, but assume that distribution of bidder types is bidder-symmetric. These symmetric distributions include i.i.d. distributions, as well as many natural correlated distributions. E.g., an item-symmetric distribution can be obtained by taking an arbitrary distribution, and forgetting names of items; this could arise when different members of a bidder population have various sorts of correlations among items, but items are the with respect to a random bidder from population.For all ∈>0, we obtain a computationally efficient additive ∈-approximation, when value distributions are bounded, or a multiplicative (1-∈)-approximation when value distributions are unbounded, but satisfy Monotone Hazard Rate condition, covering a widely studied class of distributions in Economics. Our running time is polynomial in max{#items,#bidders}, and not size of support of joint distribution of all bidders' values for all items, which is typically exponential in both number of items and number of bidders. Our mechanisms are randomized, explicitly price bundles, and in some cases can also accommodate budget constraints.Our results are enabled by several new tools and structural properties of Bayesian mechanisms, which we expect to find applications beyond settings we consider here; indeed, there has already been follow-up research [Cai et al. 2012; Cai and Huang 2012] making use of our tools in both symmetric and non-symmetric settings. In particular, we provide a symmetrization technique that turns any truthful mechanism into one that has same revenue and respects all symmetries in underlying value distributions. We also prove that item-symmetric mechanisms satisfy a natural strong-monotonicity property which, unlike cyclic-monotonicity, can be harnessed algorithmically. Finally, we provide a technique that turns any given ∈-BIC mechansism (i.e. one where incentive constraints are violated by ∈) into a truly-BIC mechanism at cost of O(√∈) revenue.

On Information and Competition in Private Value Auctions

Information disclosure in optimal auctions

We present approximation and online algorithms for a number of problems of pricing items for sale so as to maximize seller's revenue in an unlimited supply setting. Our first result is an O(k)-approximation algorithm for pricing items to single-minded bidders who each want at most k items. This improves over recent independent work of Briest and Krysta [5] who achieve an O(k2) bound. For the case k = 2, where we obtain a 4-approximation, this can be viewed as the following graph vertex pricing problem: given a (multi) graph G with valuations we on the edges, find prices pi ≥ 0 for the vertices to maximize Σ (pi + pj). {e=(i,j):we ≥ pi + pj.We also improve the approximation of Guruswami et al. [11] from O(log m + log n) to O(log n), where m is the number of bidders and n is the number of items, for the "highway problem" in which all desired subsets are intervals on a line.Our approximation algorithms can be fed into the generic reduction of Balcan et al. [2] to yield an incentive-compatible auction with nearly the same performance guarantees so long as the number of bidders is sufficiently large. In addition, we show how our algorithms can be combined with results of Blum and Hartline [3], Blum et al. [4], and Kalai and Vempala [13] to achieve good performance in the online setting, where customers arrive one at a time and each must be presented a set of item prices based only on knowledge of the customers seen so far.

$ \newcommand{\val}{w} \newcommand{\expect}[1]{{\mathbf E}\left[ #1 \right]} \newcommand{\setsize}[1]{\left| #1\right|} \newcommand{\nitems}{n} \newcommand{\ncust}{m} $ We present approximation and online algorithms for problems of pricing a collection of items for sale so as to maximize the seller's revenue in an unlimited supply setting. Our first result is an $O(k)$-approximation algorithm for pricing items to single-minded bidders who each want at most $k$ items. This improves over work of Briest and Krysta (2006) who achieve an $O(k^2)$ bound. For the case $k=2$, where we obtain a $4$-approximation, this can be viewed as the following graph vertex pricing problem: given a (multi) graph $G$ with valuations $\val_{ij}$ on the edges, find prices $p_i \geq 0$ for the vertices to maximize $$\sum\limits_{\{(i,j):\val_{ij} \geq p_i+p_j\}} \left(p_i + p_j \right)\,.$$ We also improve the approximation of Guruswami et al. (2005) for the “highway problem” in which all desired subsets are intervals on a line, from $O(\log {\ncust} + \log {\nitems})$ to $O(\log {\nitems})$, where $\ncust$ is the number of bidders and $\nitems$ is the number of items. Our approximation algorithms can be fed into the generic reduction of Balcan et al. (2005) to yield an incentive-compatible auction with nearly the same performance guarantees so long as the number of bidders is sufficiently large. In addition, we show how our algorithms can be combined with results of Blum and Hartline (2005) and Kalai and Vempala (2003) to achieve good performance in the online setting, where customers arrive one at a time and each must be presented a set of item prices based only on knowledge of the customers seen so far.

There is a widespread belief that in auctions with private values, the seller's revenue is increasing in the number of bidders. We show that there exists an auction setting with symmetric and affiliated private values in which the seller's expected revenue in a first price auction is decreasing in the number of bidders who participate in the auction.

On-line auctions are one of the most popular and successful types of electronic commerce nowadays. I present a theoretical framework to address the kind of ascending auction implemented at eBay, and I check its predictions through two experiments. The results of the experiments did not confirm the theory, but I believe this is due to the bidders' lack of previous experience. I also present an algorithm that automatically gathers data from auctions of stamps at eBay. These data is then used on an exploratory analysis that tries to shed some light on the determinants of final prices, minimum bids and the number of bidders. There were no surprises in this analysis, but I would highlight the measurable effect that seller's reputation has on final prices and the number of bidders. Furthermore, it was possible to observe the occurrence of late bidding. At last, I propose a simple method that determines to which paradigm the on-line auction of stamps at eBay belongs: private value or common value. The result was that the auction of stamps at eBay belongs to the common value paradigm.

This paper studies the nonparametric identification of the first-price auction model with risk averse bidders within the private value paradigm. First, we show that the benchmark model is nonindentified from observed bids. We also derive the restrictions imposed by the model on observables and show that these restrictions are weak. Second, we establish the nonparametric identification of the bidders' utility function under exclusion restrictions. Our primary exclusion restriction takes the form of an exogenous bidders' participation, leading to a latent distribution of private values that is independent of the number of bidders. The key idea is to exploit the property that the bid distribution varies with the number of bidders while the private value distribution does not. We then extend these results to endogenous bidders' participation when the exclusion restriction takes the form of instruments that do not affect the bidders' private value distribution. Though derived for a benchmark model, our results extend to more general cases such as a binding reserve price, affiliated private values, and asymmetric bidders. Last, possible estimation methods are proposed.

Combinatorial auctions CA are a well-studied area in algorithmic mechanism design. However, contrary to the standard model, empirical studies suggest that a bidder's valuation often does not depend solely on the goods assigned to him. For instance, in adwords auctions an advertiser might not want his ads to be displayed next to his competitors' ads. In this paper, we propose and analyze several natural graph-theoretic models that incorporate such negative externalities, in which bidders form a directed conflict graph with maximum out-degree $$\varDelta $$. We design algorithms and truthful mechanisms for social welfare maximization that attain approximation ratios depending on $$\varDelta $$. For CA, our results are twofold: 1i¾?A lottery that eliminates conflicts by discarding bidders/items independent of the bids. It allows to apply any truthful $$\alpha $$-approximation mechanism for conflict-free valuations and yields an $${\mathcal O}\alpha \varDelta $$-approximation mechanism. 2i¾?For fractionally sub-additive valuations, we design a rounding algorithm via a novel combination of a semi-definite program and a linear program, resulting in a cone program; the approximation ratio is $${\mathcal O}\varDelta \log \log \varDelta /\log \varDelta $$. The ratios are almost optimal given existing hardness results. For adwords auctions, we present several algorithms for the most relevant scenario when the number of items is small. In particular, we design a truthful mechanism with approximation ratio $$o\varDelta $$ when the number of items is only logarithmic in the number of bidders.