The Sample Complexity of Auctions with Side Information

Traditionally, the Bayesian optimal auction design problem has been considered either when the bidder values are i.i.d, or when each bidder is individually identifiable via her value distribution. The latter is a reasonable approach when the bidders can be classified into a few categories, but there are many instances where the classification of bidders is a continuum. For example, the classification of the bidders may be based on their annual income, their propensity to buy an item based on past behavior, or in the case of ad auctions, the click through rate of their ads. We introduce an alternate model that captures this aspect, where bidders are a priori identical, but can be distinguished based (only) on some side information the auctioneer obtains at the time of the auction. We extend the sample complexity approach of Dhangwatnotai et al. and Cole and Roughgarden to this model and obtain almost matching upper and lower bounds. As an aside, we obtain a revenue monotonicity lemma which may be of independent interest. We also show how to use Empirical Risk Minimization techniques to improve the sample complexity bound of Cole and Roughgarden for the non-identical but independent value distribution case.Comment: A version of this paper appeared in STOC 201

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

Fairly Allocating Goods in Parallel

We initiate the study of parallel algorithms for fairly allocating indivisible goods among agents with additive preferences. We give fast parallel algorithms for various fundamental problems, such as finding a Pareto Optimal and EF1 allocation under restricted additive valuations, finding an EF1 allocation for up to three agents, and finding an envy-free allocation with subsidies. On the flip side, we show that fast parallel algorithms are unlikely to exist (formally, $CC$-hard) for the problem of computing Round-Robin EF1 allocations

On the Complexity of Dynamic Mechanism Design

We introduce a dynamic mechanism design problem in which the designer wants to offer for sale an item to an agent, and another item to the same agent at some point in the future. The agent's joint distribution of valuations for the two items is known, and the agent knows the valuation for the current item (but not for the one in the future). The designer seeks to maximize expected revenue, and the auction must be deterministic, truthful, and ex post individually rational. The optimum mechanism involves a protocol whereby the seller elicits the buyer's current valuation, and based on the bid makes two take-it-or-leave-it offers, one for now and one for the future. We show that finding the optimum deterministic mechanism in this situation - arguably the simplest meaningful dynamic mechanism design problem imaginable - is NP-hard. We also prove several positive results, among them a polynomial linear programming-based algorithm for the optimum randomized auction (even for many bidders and periods), and we show strong separations in revenue between non-adaptive, adaptive, and randomized auctions, even when the valuations in the two periods are uncorrelated. Finally, for the same problem in an environment in which contracts cannot be enforced, and thus perfection of equilibrium is necessary, we show that the optimum randomized mechanism requires multiple rounds of cheap talk-like interactions

Algorithmic Persuasion with Evidence

We consider a game of persuasion with evidence between a sender and a receiver. The sender has private information. By presenting evidence on the information, the sender wishes to persuade the receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender's utility depends solely on whether or not the receiver takes the action. The receiver's utility depends on both the action as well as the sender's private information. We study three natural variations. First, we consider sequential equilibria of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and then the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and then the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, as well as polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm for a natural problem in algorithmic economics.Comment: 31 page