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Register a new userThe Pseudo-Dimension of Near-Optimal Auctions
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Added by
Jamie Morgenstern
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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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.
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The question of the minimum menu-size for approximate (i.e., up-to-$varepsilon$) Bayesian revenue maximization when selling two goods to an additive risk-neutral quasilinear buyer was introduced by Hart and Nisan (2013), who give an upper bound of $O(frac{1}{varepsilon^4})$ for this problem. Using the optimal-transport duality framework of Daskalakis et al. (2013, 2015), we derive the first lower bound for this problem - of $Omega(frac{1}{sqrt[4]{varepsilon}})$, even when the values for the two goods are drawn i.i.d. from nice distributions, establishing how to reason about approximately optimal mechanisms via this duality framework. This bound implies, for any fixed number of goods, a tight bound of $Theta(logfrac{1}{varepsilon})$ on the minimum deterministic communication complexity guaranteed to suffice for running some approximately revenue-maximizing mechanism, thereby completely resolving this problem. As a secondary result, we show that under standard economic assumptions on distributions, the above upper bound of Hart and Nisan (2013) can be strengthened to $O(frac{1}{varepsilon^2})$.
We identify the first static credible mechanism for multi-item additive auctions that achieves a constant factor of the optimal revenue. This is one instance of a more general framework for designing two-part tariff auctions, adapting the duality framework of Cai et al [CDW16]. Given a (not necessarily incentive compatible) auction format $A$ satisfying certain technical conditions, our framework augments the auction with a personalized entry fee for each bidder, which must be paid before the auction can be accessed. These entry fees depend only on the prior distribution of bidder types, and in particular are independent of realized bids. Our framework can be used with many common auction formats, such as simultaneous first-price, simultaneous second-price, and simultaneous all-pay auctions. If all-pay auctions are used, we prove that the resulting mechanism is credible in the sense that the auctioneer cannot benefit by deviating from the stated mechanism after observing agent bids. If second-price auctions are used, we obtain a truthful $O(1)$-approximate mechanism with fixed entry fees that are amenable to tuning via online learning techniques. Our results for first price and all-pay are the first revenue guarantees of non-truthful mechanisms in multi-dimensional environments; an open question in the literature [RST17].
We study single-good auctions in a setting where each player knows his own valuation only within a constant multiplicative factor delta{} in (0,1), and the mechanism designer knows delta. The classical notions of implementation in dominant strategies and implementation in undominated strategies are naturally extended to this setting, but their power is vastly different. On the negative side, we prove that no dominant-strategy mechanism can guarantee social welfare that is significantly better than that achievable by assigning the good to a random player. On the positive side, we provide tight upper and lower bounds for the fraction of the maximum social welfare achievable in undominated strategies, whether deterministically or probabilistically.
We study two standard multi-unit auction formats for allocating multiple units of a single good to multi-demand bidders. The first one is the Discriminatory Auction, which charges every winner his winning bids. The second is the Uniform Price Auction, which determines a uniform price to be paid per unit. Variants of both formats find applications ranging from the allocation of state bonds to investors, to online sales over the internet, facilitated by popular online brokers. For these formats, we consider two bidding interfaces: (i) standard bidding, which is most prevalent in the scientific literature, and (ii) uniform bidding, which is more popular in practice. In this work, we evaluate the economic inefficiency of both multi-unit auction formats for both bidding interfaces, by means of upper and lower bounds on the Price of Anarchy for pure Nash equilibria and mixed Bayes-Nash equilibria. Our developments improve significantly upon bounds that have been obtained recently in [Markakis, Telelis, ToCS 2014] and [Syrgkanis, Tardos, STOC 2013] for submodular valuation functions. Moreover, we consider for the first time bidders with subadditive valuation functions for these auction formats. Our results signify that these auctions are nearly efficient, which provides further justification for their use in practice.
The market economy deals with many interacting agents such as buyers and sellers who are autonomous intelligent agents pursuing their own interests. One such multi-agent system (MAS) that plays an important role in auctions is the combinatorial auctioning system (CAS). We use this framework to define our concept of fairness in terms of what we call as basic fairness and extended fairness. The assumptions of quasilinear preferences and dominant strategies are taken into consideration while explaining fairness. We give an algorithm to ensure fairness in a CAS using a Generalized Vickrey Auction (GVA). We use an algorithm of Sandholm to achieve optimality. Basic and extended fairness are then analyzed according to the dominant strategy solution concept.
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