Walrasian Pricing in Multi-unit Auctions

2011

In a setup where a divisible good is to be allocated to a set of bidders with budget constraints, we introduce a mechanism in the spirit of the Vickrey auction. In the mechanism we propose, understating budgets or values is weakly dominated. Since the revenue is increasing in budgets and values, all kinds of equilibrium deviations from true valuations turn out to be beneficial to the auctioneer.

SIAM Journal on Computing, 2017

We consider auctions in which greedy algorithms, paired with first-price or critical-price payment rules, are used to resolve multi-parameter combinatorial allocation problems. We study the price of anarchy for social welfare in such auctions. We show for a variety of equilibrium concepts, including Bayes-Nash equilibrium, low-regret bidding sequences, and asynchronous best-response dynamics, the resulting price of anarchy bound is close to the approximation factor of the underlying greedy algorithm.

We study a double auction environment where buyers and sellers have interdependent valuations and multi-unit demand and supply. We propose a new mechanism which satisfies ex post incentive compatibility, individual rationality, feasibility, non-wastefulness, and no budget deficit. Moreover, this mechanism is asymptotically efficient in that the trade outcome in the mechanism converges to the efficient level as in a competitive equilibrium as the numbers of the buyers and sellers become large. Our mechanism is the first double auction mechanism with these properties in the interdependent values setting.

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.

Computing Research Repository, 2009

Motivated by sponsored search auctions with hard budget constraints given by the advertisers, we study multi-unit auctions of a single item. An important example is a sponsored result slot for a keyword, with many units representing its inventory in a month, say. In this single-item multi-unit auction, each bidder has a private value for each unit, and a private budget which is the total amount of money she can spend in the auction. A recent impossibility result [Dobzinski et al., FOCS'08] precludes the existence of a truthful mechanism with Paretooptimal allocations in this important setting. We propose Sort-Cut, a mechanism which does the next best thing from the auctioneer's point of view, that we term semi-truthful. While we are unable to give a complete characterization of equilibria for our mechanism, we prove that some equilibrium of the proposed mechanism optimizes the revenue over all Pareto-optimal mechanisms, and that this equilibrium is the unique one resulting from a natural rational bidding strategy (where every losing bidder bids at least her true value). Perhaps even more significantly, we show that the revenue of every equilibrium of our mechanism differs by at most the budget of one bidder from the optimum revenue (under some mild assumptions).

International Journal of Economic Theory, 2015

We revisit the benchmark model of auctions and consider a more general class of utility functions that allow for income effects. We assume that all individuals have the same utility function but have different incomes. Incomes are private information. We analyze first-price, secondprice, and all-pay auctions and show that non-quasilinearity changes many basic results of the benchmark model. While Vickrey's (1961) result on second-price auctions is very robust, revenue equivalence breaks down even with risk-neutral bidders, high enough incomes and identically and independently distributed types. In most cases, we find that all-pay auctions fetch the highest expected revenue.

Economic Theory, 2010

We construct a model of multi-unit auctions in which I bidders bid for two indivisible units of a common value good. Using a first-order approach, we find that there are equilibria in which bidders bid the same price for both units in the discriminatory auction, but not in the uniform auction. When there are only two bidders, under certain conditions there are linear equilibria for both the discriminatory and the uniform auction formats. In all equilibria, bidders equalize the expected marginal benefit of bidding to the marginal costs of bidding. We show that comparison of the seller's expected revenue across auction formats depends only on the ratio of the precision of private information to the precision of public information.

The B.E. Journal of Theoretical Economics, 2000

We design a Generalized Position Auction for players with private values and private budget constraints. Our mechanism is a careful modification of the Generalized English Auction of Edelman, . By enabling multiple price trajectories that ascent concurrently we are able to retrieve all the desired properties of the Generalized English Auction, that was not originally designed for players with budgets. In particular, the ex-post equilibrium outcome of our auction is Pareto-efficient and envy-free. Moreover, we show that any other position auction that satisfies these properties and does not make positive transfers must obtain in ex-post equilibrium the same outcome of our mechanism, for every tuple of distinct types. This uniqueness result holds even if the players' values are fixed and known to the seller, and only the budgets are private. JEL Classification Numbers: C70, D44, D82

1977

Motivated by sponsored search auctions with hard budget constraints given by the adver-tisers, we study multi-unit auctions of a single item. An important example is a sponsored result slot for a keyword, with many units representing its inventory in a month, say. In this single-item multi-unit auction, each bidder has a private value for each unit, and a private bud-get which is the total amount of money she can spend in the auction. A recent impossibility result [Dobzinski et al., FOCS’08] precludes the existence of a truthful mechanism with Pareto-optimal allocations in this important setting. We propose Sort-Cut, a mechanism which does the next best thing from the auctioneer’s point of view, that we term semi-truthful. While we are unable to give a complete characterization of equilibria for our mechanism, we prove that some equilibrium of the proposed mechanism optimizes the revenue over all Pareto-optimal mech-anisms, and that this equilibrium is the unique one resulting from ...

Journal of Economic Behavior & Organization, 1993

Recently, Bulow and Roberts have indicated that there is a close correspondence between the optimal auction mechanism introduced by Myerson and the problem of a monopolist practising third-degree price discrimination in several markets. They find differences, however, in the choice variables and in the constraints of the two problems. A reinterpretation of the pricediscrimination problem leads to the demonstration that an optimal auction is an optimal way to practise price discimination.

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.

2011

In several e-commerce applications, non-truthful auctions have been preferred over truthful weakly dominant strategy ones partly because of their simplicity and scalability. Although non-truthful auctions can have weaker incentive constraints than truthful ones, the question of how much more revenue they can generate than truthful auctions is not well understood. We study this question for natural and broad classes of non-truthful mechanisms, including quasi-proportional sharing and weakly monotonic auctions. Quasi-proportional sharing mechanisms allocate to each bidder i an amount of resource proportional to a monotonic and concave function f (b i) where b i is the bid of bidder i and ask for a payment of b i. Weakly monotonic auctions refer to a more general class of auctions which satisfy some natural continuity and monotonicity conditions. We prove that although weakly monotonic auctions are much broader and require weaker incentive constraints than dominant strategy auctions, they are not more powerful with respect to the revenue in the setting of selling a single item. Furthermore, we show that quasi-proportional sharing with multiple bidders cannot guarantee a revenue that is larger than the second highest valuation, asymptotically as the number of bidders grows large. However, in a more general single-parameter setting modeled by a downwardclosed set system, a version of the proportional sharing mechanism can obtain a constant factor of the optimal social welfare of the game where the highest valuation is replaced by the second highest valuation, which is essentially the best revenue benchmark in the prior-free framework. This is in sharp contrast to weakly dominant strategy mechanisms that cannot achieve better than log n approximation for this benchmark.

2011

In many auction settings, there is favoritism: the seller's welfare depends positively on the utility of a subset of potential bidders. However, laws or regulations may not allow the seller to discriminate among bidders. We …nd the optimal nondiscriminatory auction in a private value, single-unit model under favoritism. At the optimal auction there is a reserve price, or an entry fee, which is decreasing in the proportion of preferred bidders and in the intensity of the preference. Otherwise, the highest-valuation bidder wins. We show that, at least under some conditions, imposing a no-discrimination constraint raises expected seller revenue.

2015

We study the classic setting of envy-free pricing, in which a single seller chooses prices for its many items, with the goal of maximizing revenue once the items are allocated. Despite the large body of work addressing such settings, most versions of this problem have resisted good approximation factors for maximizing revenue; this is true even for the classic unit-demand case. In this paper we study envy-free pricing with unit-demand buyers, but unlike previous work we focus on large markets: ones in which the demand of each buyer is infinitesimally small compared to the size of the overall market. We assume that the buyer valuations for the items they desire have a nice (although reasonable) structure, i.e., that the aggregate buyer demand has a monotone hazard rate and that the values of every buyer type come from the same support.

Econometrica, 2002

An analogue of multi-unit auction is provided when bidders have interdependent values and one-dimensional private information. The analogue is strategically equivalent to a collection of two-bidder single-unit second-price auctions and it possesses an e¢cient ex-post equilibrium.