Sampling and Representation Complexity of Revenue Maximization

We consider (approximate) revenue maximization in auctions where the distribution on input valuations is given via "black box" access to samples from the distribution. We observe that the number of samples required -- the sample complexity -- is tightly related to the representation complexity of an approximately revenue-maximizing auction. Our main results are upper bounds and an exponential lower bound on these complexities

Lottery pricing equilibria

We extend the notion of Combinatorial Walrasian Equilibrium, as defined by Feldman et al. [2013], 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 [Dobzinski and Paes Leme 2014] 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-√5/2 W (≈ 0.3819-W). Next, given access to a demand oracle and an α-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 α 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. © Copyright 2016 ACM

On Revenue Monotonicity in Combinatorial Auctions

Along with substantial progress made recently in designing near-optimal mechanisms for multi-item auctions, interesting structural questions have also been raised and studied. In particular, is it true that the seller can always extract more revenue from a market where the buyers value the items higher than another market? In this paper we obtain such a revenue monotonicity result in a general setting. Precisely, consider the revenue-maximizing combinatorial auction for $m$ items and $n$ buyers in the Bayesian setting, specified by a valuation function $v$ and a set $F$ of $nm$ independent item-type distributions. Let $REV(v, F)$ denote the maximum revenue achievable under $F$ by any incentive compatible mechanism. Intuitively, one would expect that $REV(v, G)\geq REV(v, F)$ if distribution $G$ stochastically dominates $F$. Surprisingly, Hart and Reny (2012) showed that this is not always true even for the simple case when $v$ is additive. A natural question arises: Are these deviations contained within bounds? To what extent may the monotonicity intuition still be valid? We present an {approximate monotonicity} theorem for the class of fractionally subadditive (XOS) valuation functions $v$, showing that $REV(v, G)\geq c\,REV(v, F)$ if $G$ stochastically dominates $F$ under $v$ where $c>0$ is a universal constant. Previously, approximate monotonicity was known only for the case $n=1$: Babaioff et al. (2014) for the class of additive valuations, and Rubinstein and Weinberg (2015) for all subaddtive valuation functions.Comment: 10 page

Optimal Multi-Unit Mechanisms with Private Demands

In the multi-unit pricing problem, multiple units of a single item are for sale. A buyer's valuation for $n$ units of the item is $v \min \{ n, d\}$, where the per unit valuation $v$ and the capacity $d$ are private information of the buyer. We consider this problem in the Bayesian setting, where the pair $(v,d)$ is drawn jointly from a given probability distribution. In the \emph{unlimited supply} setting, the optimal (revenue maximizing) mechanism is a pricing problem, i.e., it is a menu of lotteries. In this paper we show that under a natural regularity condition on the probability distributions, which we call \emph{decreasing marginal revenue}, the optimal pricing is in fact \emph{deterministic}. It is a price curve, offering $i$ units of the item for a price of $p_i$, for every integer $i$. Further, we show that the revenue as a function of the prices $p_i$ is a \emph{concave} function, which implies that the optimum price curve can be found in polynomial time. This gives a rare example of a natural multi-parameter setting where we can show such a clean characterization of the optimal mechanism. We also give a more detailed characterization of the optimal prices for the case where there are only two possible demands

Mixture Selection, Mechanism Design, and Signaling

We pose and study a fundamental algorithmic problem which we term mixture selection, arising as a building block in a number of game-theoretic applications: Given a function $g$ from the $n$-dimensional hypercube to the bounded interval $[-1,1]$, and an $n \times m$ matrix $A$ with bounded entries, maximize $g(Ax)$ over $x$ in the $m$-dimensional simplex. This problem arises naturally when one seeks to design a lottery over items for sale in an auction, or craft the posterior beliefs for agents in a Bayesian game through the provision of information (a.k.a. signaling). We present an approximation algorithm for this problem when $g$ simultaneously satisfies two smoothness properties: Lipschitz continuity with respect to the $L^\infty$ norm, and noise stability. The latter notion, which we define and cater to our setting, controls the degree to which low-probability errors in the inputs of $g$ can impact its output. When $g$ is both $O(1)$-Lipschitz continuous and $O(1)$-stable, we obtain an (additive) PTAS for mixture selection. We also show that neither assumption suffices by itself for an additive PTAS, and both assumptions together do not suffice for an additive FPTAS. We apply our algorithm to different game-theoretic applications from mechanism design and optimal signaling. We make progress on a number of open problems suggested in prior work by easily reducing them to mixture selection: we resolve an important special case of the small-menu lottery design problem posed by Dughmi, Han, and Nisan; we resolve the problem of revenue-maximizing signaling in Bayesian second-price auctions posed by Emek et al. and Miltersen and Sheffet; we design a quasipolynomial-time approximation scheme for the optimal signaling problem in normal form games suggested by Dughmi; and we design an approximation algorithm for the optimal signaling problem in the voting model of Alonso and C\^{a}mara