We consider the problem of designing revenue maximizing online posted-price mechanisms when the seller has limited supply. A seller has k identical items for sale and is facing n potential buyers ("agents") that are arriving sequentially. Each agent is interested in buying one item. Each agent's value for an item is an independent sample from some fixed (but unknown) distribution with support [0, 1]. The seller offers a take-it-or-leave-it price to each arriving agent (possibly different for different agents), and aims to maximize his expected revenue.We focus on mechanisms that do not use any information about the distribution; such mechanisms are called detail-free (or prior-independent). They are desirable because knowing the distribution is unrealistic in many practical scenarios. We study how the revenue of such mechanisms compares to the revenue of the optimal offline mechanism that knows the distribution ("offline benchmark").We present a detail-free online posted-price mechanism whose revenue is at most O((k log n) 2/3 ) less than the offline benchmark, for every distribution that is regular. In fact, this guarantee holds without any assumptions if the benchmark is relaxed to fixed-price mechanisms. Further, we prove a matching lower bound. The performance guarantee for the same mechanism can be improved to O( √ k log n), with a distribution-dependent constant, if the ratio k n is sufficiently small. We show that, in the worst case over all demand distributions, this is essentially the best rate that can be obtained with a distribution-specific constant.On a technical level, we exploit the connection to multi-armed bandits (MAB). While dynamic pricing with unlimited supply can easily be seen as an MAB problem, the intuition behind MAB approaches breaks when applied to the setting with limited supply. Our high-level conceptual contribution is that even the limited supply setting can be fruitfully treated as a bandit problem.Consider a promoter that is interested in selling k tickets for a given concert. The seller is interested in maximizing her revenue from selling these tickets, and is offering the tickets on a website such as Ticketmaster. Potential buyers ("agents") arrive one after another, each with the goal of purchasing a ticket if the price is smaller than the agent's valuation. The seller expects n such agents to arrive. Whenever an agent arrives the seller presents to him a take-it-or-leave-it price, and the agent makes a purchasing decision according to that price. The seller can update the price taking into account the observed history and the number of remaining items and agents.We adopt a Bayesian view that the valuations of the buyers are IID samples from a fixed distribution, called demand distribution. A standard assumption in a Bayesian setting is that the demand distribution is known to the seller, who can design a specific mechanism tailored to this knowledge. (For example, the Myerson optimal auction for one item sets a reserve price that is a function of the distribution). However, i...

Persuasion, defined as the act of exploiting an informational advantage in order to effect the decisions of others, is ubiquitous. Indeed, persuasive communication has been estimated to account for almost a third of all economic activity in the US. This paper examines persuasion through a computational lens, focusing on what is perhaps the most basic and fundamental model in this space: the celebrated Bayesian persuasion model of Kamenica and Gentzkow [34]. Here there are two players, a sender and a receiver. The receiver must take one of a number of actions with a-priori unknown payoff, and the sender has access to additional information regarding the payoffs of the various actions for both players. The sender can commit to revealing a noisy signal regarding the realization of the payoffs of various actions, and would like to do so as to maximize her own payoff in expectation assuming that the receiver rationally acts to maximize his own payoff. When the payoffs of various actions follow a joint distribution (the common prior), the sender's problem is nontrivial, and its computational complexity depends on the representation of this prior.We examine the sender's optimization task in three of the most natural input models for this problem, and essentially pin down its computational complexity in each. When the payoff distributions of the different actions are i.i.d. and given explicitly, we exhibit a polynomial-time (exact) algorithmic solution, and a "simple" (1 − 1/e)-approximation algorithm. Our optimal scheme for the i.i.d. setting involves an analogy to auction theory, and makes use of Border's characterization of the space of reduced-forms for single-item auctions. When action payoffs are independent but non-identical with marginal distributions given explicitly, we show that it is #P-hard to compute the optimal expected sender utility. In doing so, we rule out a generalized Border's theorem, as defined by Gopalan et al [30], for this setting. Finally, we consider a general (possibly correlated) joint distribution of action payoffs presented by a black box sampling oracle, and exhibit a fully polynomial-time approximation scheme (FPTAS) with a bi-criteria guarantee. Our FPTAS is based on Monte-Carlo sampling, and its analysis relies on the principle of deferred decisions. Moreover, we show that this result is the best possible in the black-box model for information-theoretic reasons.A somewhat less artificial example of persuasion is in the context of providing financial advice. Here, the receiver is an investor, actions correspond to stocks, and the sender is a stockbroker or financial adviser with access to stock return projections which are a-priori unknown to the investor. When the adviser's commission or return is not aligned with the investor's returns, this is a nontrivial Bayesian persuasion problem. In fact, interesting examples exist when stock returns are independent from each other, or even i.i.d. Consider the following simple example which fits into the i.i.d. model considered in Section 3: the...

Complements between goods -where one good takes on added value in the presence of another -have been a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this intractability has stymied theoretical progress in the area. For example, there are no known positive results for combinatorial auctions in which bidder valuations are multi-parameter and non-complement-free, other than the relatively weak results known for general valuations.To make inroads on the problem of combinatorial auction design in the presence of complements, we propose a model for valuations with complements that is parameterized by the "size" of the complements. The model permits a succinct representation, a variety of computationally efficient queries, and non-trivial welfare-maximization algorithms and mechanisms. Specifically, a hypergraph-r valuation v for a good set M is represented by a hypergraph H = (M, E), where every (hyper-)edge e ∈ E contains at most r vertices and has a nonnegative weight w e . Each good j ∈ M also has a nonnegative weight w j . The value v(S) for a subset S ⊆ M of goods is defined as the sum of the weights of the goods and edges entirely contained in S.We design the following polynomial-time approximation algorithms and truthful mechanisms for welfare maximization with bidders with hypergraph valuations.2. We give a polynomial-time, r-approximation algorithm for welfare maximization with hypergraph-r valuations. Our algorithm randomly rounds a compact linear programming relaxation of the problem.3. We design a different approximation algorithm and use it to give a polynomial-time, truthful-in-expectation mechanism that has an approximation factor of O(log r m).

We design an expected polynomial time, truthful in expectation, (1 − 1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions and matroid weighted-rank functions. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P = NP . Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP -hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations.Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm, rather than over the input to the rounding algorithm. This * A full version of this paper is available at

We study the algorithmics of information structure design -a.k.a. persuasion or signaling -in a fundamental special case introduced by Arieli and Babichenko: multiple agents, binary actions, and no inter-agent externalities. Unlike prior work on this model, we allow many states of nature. We assume that the principal's objective is a monotone set function, and study the problem both in the public signal and private signal models, drawing a sharp contrast between the two in terms of both efficacy and computational complexity.When private signals are allowed, our results are largely positive and quite general. First, we use linear programming duality and the equivalence of separation and optimization to show polynomial-time equivalence between (exactly) optimal signaling and the problem of maximizing the objective function plus an additive function. This yields an efficient implementation of the optimal scheme when the objective is supermodular or anonymous. Second, we exhibit a (1 − 1/e)-approximation of the optimal private signaling scheme, modulo an additive loss of ǫ, when the objective function is submodular. These two results simplify, unify, and generalize results of [3,4], extending them from a binary state of nature to many states (modulo the additive loss in the latter result). Third, we consider the binary-state case with a submodular objective, and simplify and slightly strengthen the result of [4] to obtain a (1 − 1/e)-approximation via a scheme which (i) signals independently to each receiver and (ii) is "oblivious" in that it does not depend on the objective function so long as it is monotone submodular.When only a public signal is allowed, our results are negative. First, we show that it is NP-hard to approximate the optimal public scheme, within any constant factor, even when the objective is additive. Second, we show that the optimal private scheme can outperform the optimal public scheme, in terms of maximizing the sender's objective, by a polynomial factor.