Alon Eden

The Invisible Hand of Dynamic Market Pricing

Walrasian prices, if they exist, have the property that one can assign every buyer some bundle in her demand set, such that the resulting assignment will maximize social welfare. Unfortunately, this assumes carefully breaking ties amongst different bundles in the buyer demand set. Presumably, the shopkeeper cleverly convinces the buyer to break ties in a manner consistent with maximizing social welfare. Lacking such a shopkeeper, if buyers arrive sequentially and simply choose some arbitrary bundle in their demand set, the social welfare may be arbitrarily bad. In the context of matching markets, we show how to compute dynamic prices, based upon the current inventory, that guarantee that social welfare is maximized. Such prices are set without knowing the identity of the next buyer to arrive. We also show that this is impossible in general (e.g., for coverage valuations), but consider other scenarios where this can be done. We further extend our results to Bayesian and bounded rationality models.

The Competition Complexity of Auctions: A Bulow-Klemperer Result for Multi-Dimensional Bidders

A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to n bidders whose values are drawn i.i.d. from a regular distribution, the simple welfare-maximizing VCG mechanism (in this case, a second price-auction) with one additional bidder extracts at least as much revenue in expectation as the optimal mechanism. The beauty of this theorem stems from the fact that VCG is a prior-independent mechanism, where the seller possesses no information about the distribution, and yet, by recruiting one additional bidder it performs better than any prior-dependent mechanism tailored exactly to the distribution at hand (without the additional bidder). In this work, we establish the first full Bulow-Klemperer results in multi-dimensional environments, proving that by recruiting additional bidders, the revenue of the VCG mechanism surpasses that of the optimal (possibly randomized, Bayesian incentive compatible) mechanism. For a given environment with i.i.d. bidders, we term the number of additional bidders needed to achieve this guarantee the environments competition complexity. Using the recent duality-based framework of Cai et al. [2016] for reasoning about optimal revenue, we show that the competition complexity of n bidders with additive valuations over m independent, regular items is at most n+2m-2 and at least log(m). We extend our results to bidders with additive valuations subject to downward-closed constraints, showing that these significantly more general valuations increase the competition complexity by at most an additive m-1 factor. We further improve this bound for the special case of matroid constraints, and provide additional extensions as well.

Lottery Pricing Equilibria

We extend the notion of Combinatorial Walrasian Equilibrium, as defined by \citet{FGL13}, to settings with budgets. When agents have budgets, the maximum social welfare as traditionally defined is not a suitable benchmark since it is overly optimistic. This motivated the liquid welfare of \cite{DP14} as an alternative. Observing that no combinatorial Walrasian equilibrium guarantees a non-zero fraction of the maximum liquid welfare in the absence of randomization, we instead work with randomized allocations and extend the notions of liquid welfare and Combinatorial Walrasian Equilibrium accordingly. Our generalization of the Combinatorial Walrasian Equilibrium prices lotteries over bundles of items rather than bundles, and we term it a lottery pricing equilibrium. Our results are two-fold. First, we exhibit an efficient algorithm which turns a randomized allocation with liquid expected welfare W into a lottery pricing equilibrium with liquid expected welfare 3-\sqrt{5}{2}\cdot W approx 0.3819\cdot W. Next, given access to a demand oracle and an alpha-approximate oblivious rounding algorithm for the configuration linear program for the welfare maximization problem, we show how to efficiently compute a randomized allocation which is (a) supported on polynomially-many deterministic allocations and (b) obtains [nearly] an alpha fraction of the optimal liquid expected welfare. In the case of subadditive valuations, combining both results yields an efficient algorithm which computes a lottery pricing equilibrium obtaining a constant fraction of the optimal liquid expected welfare.

Online Random Sampling for Budgeted Settings

We study online multi-unit auctions in which each agent’s private type consists of the agent’s arrival and departure times, valuation function and budget. Similarly to secretary settings, the different attributes of the agents’ types are determined by an adversary, but the arrival process is random. We establish a general framework for devising truthful random sampling mechanisms for online multi-unit settings with budgeted agents. We demonstrate the applicability of our framework by applying it to different objective functions (revenue and liquid welfare), and a range of assumptions about the agents’ valuations (additive or general) and the items’ nature (divisible or indivisible). Our main result is the design of mechanisms for additive bidders with budget constraints that extract a constant fraction of the optimal revenue, for divisible and indivisible items (under a standard large market assumption). We also show a mechanism that extracts a constant fraction of the optimal liquid welfare for general valuations over divisible items.

Interdependent Values without Single-Crossing

We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation of each agent is a known function of all n private signals. This captures settings such as valuations for oil drilling rights, broadcast rights, pieces of art, and many more. In the interdependent value setting, all previous work has assumed a so-called single-crossing condition. Single-crossing means that the impact of a private signal, si , on the valuation of agent i , is greater than the impact of si on the valuation of any other agent. It is known that without the single-crossing condition, an efficient outcome cannot be obtained. We ask what approximation to the optimal social welfare can be obtained when valuations do not exhibit single-crossing. We show that, in general, without the single-crossing condition, one cannot hope to approximate the optimal social welfare any better than assigning the item to a random bidder. Consequently, we consider a relaxed version of single-crossing, c-single-crossing, with some parameter c≥1 , which means that the impact of si on the valuation of agent i is at least 1/ c times the impact of si on the valuation of any other agent ( c =1 is single-crossing). Using this relaxed notion, we obtain a host of positive results. These include a prior-free universally truthful 2√ nc3/2 -approximation to welfare, and a prior-free deterministic ( n -1) c -approximation to welfare. Under appropriate concavity conditions, we improve this to a prior-free universally truthful 2 c -approximation to welfare as well as a universally truthful O(c2)-approximation to the optimal revenue.