In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders can have arbitrary demand and budget constraints. Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with discrete correlated type space for each bidder. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles. We believe this approach will be useful in other Bayesian mechanism design contexts.
Finally, several researchers have considered the behavior specific types of auctions, for instance, auctions that are sequential by item and second price within each item @cite_21 @cite_4 , and ascending price auctions @cite_25 @cite_26 . The goal here is to analyze the improvement in revenue (or social welfare) by optimal sequencing, or to study incentive compatibility of commonly used ascending price mechanisms. However, analyzing the performance of sequential or ascending price auctions is difficult in general, and there is little known in terms of optimal mechanisms (or even approximately optimal mechanisms) in these settings.
@cite_21 @cite_4 @cite_25 @cite_26
1594189438 1965611011 2952244728 2189241046
We study sequential auctions for private value objects and unit-demand bidders using second-price sealed-bid rules. We analyze this scenario from the seller's perspective and consider several approaches to increasing the total revenue. We derive the equilibrium bidding strategies for each individual auction.We then study the problem of selecting an optimal agenda, i.e., a revenue-maximizing ordering of the auctions. We describe an efficient algorithm that finds an optimal agenda in the important special case when the revenue of each auction is guaranteed to be strictly positive. We also show that the seller can increase his revenue by canceling one or more auctions, even if the number of bidders exceeds the number of objects for sale, and analyze the bidders' behavior and the seller's profit for different cancellation rules. We performed controlled experiments of human participants in a continuous sequence of ad auctions, similar to those used by Internet companies. The goal of the research was to understand users' strategies in making bids. We studied the behavior under two auction types: (1) the Generalized Second-Price (GSP) auction and (2) the Vickrey--Clarke--Groves (VCG) payment rule, and manipulated also the participants' knowledge conditions: (1) explicitly given valuations and (2) payoff information from which valuations could be deduced. We found several interesting behaviors, among them are: No convergence to equilibrium was detected; moreover the frequency with which participants modified their bids increased with time. We can detect explicit "better-response" behavior rather than just mixed bidding. While bidders in GSP auctions do strategically shade their bids, they tend to bid higher than theoretically predicted by the standard VCG-like equilibrium of GSP. Bidders who are not explicitly given their valuations but can only deduce them from their gains behave a little less "precisely" than those with such explicit knowledge, but mostly during an initial learning phase. VCG and GSP yield approximately the same (high) social welfare, but GSP tends to give higher revenue. We consider prior-free auctions for revenue and welfare maximization when agents have a common budget. The abstract environments we consider are ones where there is a downward-closed and symmetric feasibility constraint on the probabilities of service of the agents. These environments include position auctions where slots with decreasing click-through rates are auctioned to advertisers. We generalize and characterize the envy-free benchmark from Hartline and Yan (2011) to settings with budgets and characterize the optimal envy-free outcomes for both welfare and revenue. We give prior-free mechanisms that approximate these benchmarks. A building block in our mechanism is a clinching auction for position auction environments. This auction is a generalization of the multi-unit clinching auction of (2008) and a special case of the polyhedral clinching auction of (2012). For welfare maximization, we show that this clinching auction is a good approximation to the envy-free optimal welfare for position auction environments. For profit maximization, we generalize the random sampling profit extraction auction from (2002) for digital goods to give a 10.0-approximation to the envy-free optimal revenue in symmetric, downward-closed environments. The profit maximization question is of interest even without budgets and our mechanism is a 7.5-approximation which improving on the 30.4 bound of Ha and Hartline (2012). This paper develops a general approach, rooted in statistical learning theory, to learning an approximately revenue-maximizing auction from data. We introduce t-level auctions to interpolate between simple auctions, such as welfare maximization with reserve prices, and optimal auctions, thereby balancing the competing demands of expressivity and simplicity. We prove that such auctions have small representation error, in the sense that for every product distribution F over bidders’ valuations, there exists a t-level auction with small t and expected revenue close to optimal. We show that the set of t-level auctions has modest pseudo-dimension (for polynomial t) and therefore leads to small learning error. One consequence of our results is that, in arbitrary single-parameter settings, one can learn a mechanism with expected revenue arbitrarily close to optimal from a polynomial number of samples.