Multi-dimensional Virtual Values and Second-degree Price Discrimination

We consider a multi-dimensional screening problem of selling a product with multiple quality levels and design virtual value functions to derive conditions that imply optimality of only selling highest quality. A challenge of designing virtual values for multi-dimensional agents is that a mechanism that pointwise optimizes virtual values resulting from a general application of integration by parts is not incentive compatible, and no general methodology is known for selecting the right paths for integration by parts. We resolve this issue by first uniquely solving for paths that satisfy certain necessary conditions that the pointwise optimality of the mechanism imposes on virtual values, and then identifying distributions that ensure the resulting virtual surplus is indeed pointwise optimized by the mechanism. Our method of solving for virtual values is general, and as a second application we use it to derive conditions of optimality for selling only the grand bundle of items to an agent with additive preferences

Optimal Auctions for Correlated Buyers with Sampling

Cr\'emer and McLean [1985] showed that, when buyers' valuations are drawn from a correlated distribution, an auction with full knowledge on the distribution can extract the full social surplus. We study whether this phenomenon persists when the auctioneer has only incomplete knowledge of the distribution, represented by a finite family of candidate distributions, and has sample access to the real distribution. We show that the naive approach which uses samples to distinguish candidate distributions may fail, whereas an extended version of the Cr\'emer-McLean auction simultaneously extracts full social surplus under each candidate distribution. With an algebraic argument, we give a tight bound on the number of samples needed by this auction, which is the difference between the number of candidate distributions and the dimension of the linear space they span

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

Buying from a Group

A buyer procures a good owned by a group of sellers whose heterogeneous cost of trade is private information. The buyer must either buy the whole good or nothing, and sellers share the transfer in proportion to their share of the good. We characterize the optimal mechanism: trade occurs if and only if the buyer’s benefit of trade exceeds a weighted average of sellers’ virtual costs. These weights are endogenous, with sellers who are ex ante less inclined to trade receiving higher weight. This mechanism always outperforms posted-price mechanisms. An extension characterizes the entire Pareto frontier

Optimal Multi-parameter Auction Design

This thesis studies the design of Bayesian revenue-optimal auctions for a class of problems in which buyers have general (non-linear and multi-parameter) preferences. This class includes the classical linear single-parameter problem considered by Myerson (1981), for which he provided a simple characterization of revenue proving optimality of a mechanism, leading to numerous applications in theory and practice. However, for fully general preferences no generic and practical solution is known (various negative computational or structural results exist for special cases), even for the problem of designing a mechanism for a single agent. This thesis sets to identifies key conditions implying that the optimal mechanism is practical. Our main results are different in that they identify different conditions implying different notions of practicality, but are all similar in adopting a modular view to the problem that separates the task of designing a solution for the single-agent problem as the main module, from the task of combining these modules to form an optimal multi-agent mechanism. For multi-parameter linear settings, we specify a large class of distributions over values that implies that the optimal single-agent mechanism is posted pricing, and the optimal multi-agent mechanism maximizes \emph{virtual values} for players' favorite items (e.g., when agents are identical, second price auction with reserve for favorite items). More generally, we specify a condition called revenue-linearity (defined beyond multi-parameter linear settings) that implies that optimizing agents' marginal revenue maximizes revenue. Finally, adopting efficient computability as the notion of practicality, we show that for any setting in which single-agent solutions are efficiently computable, multi-agent solutions are also computable

Sequential Mechanisms with Ex-post Participation Guarantees

How should one sell an item to a buyer whose value for the item will only be realized next week? E.g. consider selling a flight to some executive who may or may not have a meeting with a client next week. Suppose that both the seller and the buyer only know a distribution, F, from which the buyer's value, v, for the item will be drawn. One way the seller could go about this sale is to make a take-it-or-leave-it offer today. The offer reads "pay the expected value today to get the item next week". A risk-neutral buyer would find this offer attractive, hence the seller would extract the full surplus. The unsettling feature of the afore-described mechanism is that, for some realizations of v, the bidder ends up with negative utility. In particular, while our mechanism is interim Individually Rational (IR), it is not ex-post IR. How could we fix this? One way is to wait until next week when the value is realized and make a take-it-or-leave-it offer of the item at an optimal monopoly price. The new mechanism is clearly ex-post IR, but its revenue could be much smaller than that of the previous one. Still, this trivial mechanism extracts the best possible revenue among all ex-post IR mechanisms, as a simple argument can establish. However, this optimality argument fails when several items are to be sold over consecutive periods. In this paper, we provide a characterization of the revenue-optimal, ex-post IR, dynamic mechanism selling k items over k periods to a bidder whose values are independent. In particular, we optimize the seller's revenue subject to the following strong individual rationality condition: at each period, the stage utility of the agent, defined to be the surplus from that period's allocation minus the agent's payment, must be non-negative. In particular, the non-negativity of the stage utilities implies that, at the end of each period, the agent's realized utility from participating in the mechanism so far is non-negative. We provide extensions to multiple bidders and an infinite horizon with discount factors.United States. Office of Naval Research (grant N0 0014-12-1-0999)National Science Foundation (U.S.) (Award CCF-0953960 (CAREER))National Science Foundation (U.S.). Division of Computing and Communication Foundations (CCF-1551875)National Science Foundation (U.S.). Division of Computing and Communication Foundations (SES-1254768

Optimal Auctions with Positive Network Externalities

We consider the problem of designing auctions in social networks for goods that exhibit single-parameter submodular network externalities in which a bidder’s value for an outcome is a fixed private type times a known submodular function of the allocation of his friends. Externalities pose many issues that are hard to address with traditional techniques; our work shows how to resolve these issues in a specific setting of particular interest. We operate in a Bayesian environment and so assume private values are drawn according to known distributions. We prove that the optimal auction is APX-hard. Thus we instead design auctions whose revenue approximates that of the optimal auction. Our main result considers step-function externalities in which a bidder’s value for an outcome is either zero, or equal to his private type if at least one friend has the good. For these e e+1 settings, we provide a-approximation. We also give a 0.25-approximation auction for general single-parameter submodular network externalities, and discuss optimizing over a class of simple pricing strategies