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

Pricing Ad Slots with Consecutive Multi-unit Demand

We consider the optimal pricing problem for a model of the rich media advertisement market, as well as other related applications. In this market, there are multiple buyers (advertisers), and items (slots) that are arranged in a line such as a banner on a website. Each buyer desires a particular number of {\em consecutive} slots and has a per-unit-quality value $v_i$ (dependent on the ad only) while each slot $j$ has a quality $q_j$ (dependent on the position only such as click-through rate in position auctions). Hence, the valuation of the buyer $i$ for item $j$ is $v_iq_j$. We want to decide the allocations and the prices in order to maximize the total revenue of the market maker. A key difference from the traditional position auction is the advertiser's requirement of a fixed number of consecutive slots. Consecutive slots may be needed for a large size rich media ad. We study three major pricing mechanisms, the Bayesian pricing model, the maximum revenue market equilibrium model and an envy-free solution model. Under the Bayesian model, we design a polynomial time computable truthful mechanism which is optimum in revenue. For the market equilibrium paradigm, we find a polynomial time algorithm to obtain the maximum revenue market equilibrium solution. In envy-free settings, an optimal solution is presented when the buyers have the same demand for the number of consecutive slots. We conduct a simulation that compares the revenues from the above schemes and gives convincing results.Comment: 27page

Budget Constrained Auctions with Heterogeneous Items

In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders can have arbitrary demand and budget constraints. Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with discrete correlated type space for each bidder. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles. We believe this approach will be useful in other Bayesian mechanism design contexts.Comment: Final version accepted to STOC '10. Incorporates significant reviewer comment

A dominant strategy, double clock auction with estimation-based tatonnement

The price mechanism is fundamental to economics but difficult to reconcile with incentive compatibility and individual rationality. We introduce a double clock auction for a homogeneous good market with multidimensional private information and multiunit traders that is deficit‐free, ex post individually rational, constrained efficient, and makes sincere bidding a dominant strategy equilibrium. Under a weak dependence and an identifiability condition, our double clock auction is also asymptotically efficient. Asymptotic efficiency is achieved by estimating demand and supply using information from the bids of traders that have dropped out and following a tâtonnement process that adjusts the clock prices based on the estimates

Multi-Unit Open Ascending Price Efficient Auction

This paper presents an open ascending price mechanism that allocates efficiently M units of the same good among N bidders with interdependent values The mechanism consists of a number of sequential English auctions with reentry and has the following attributes. In each of the individual auctions all the bidders compete simultaneously in the open ascending price format. The most distinctive feature of the mechanism is that winners are determined first, and then additional auxillary auctions are conducted to determine prices. The total number of auctions depends only on the number of goods to be allocated and not on the number of bidders.Multiple units, Interdependent values, Sequential auctions, Ascending price auction

On the Concentration of Allocations and Comparisons of Auctions in Large Economies

We analyze competitive pressures in a sequence of auctions with a growing number of bidders, in a model that includes private and common valuations as special cases. We show that the key determinant of bidders' surplus (and implicitly auction revenue) is how the goods are distributed. In any setting and sequence of auctions where the allocation of good(s) is concentrated among a shrinking proportion of the population, the winning bidders enjoy no surplus in the limit. If instead the good(s) are allocated in a dispersed manner so that a non- vanishing proportion of the bidders obtain objects, then in any of a wide class of auctions bidders enjoy a surplus that is bounded away from zero. Moreover, under dispersed allocations, the format of the auction matters. If bidders have constant marginal utilities for objects up to some limit, then uniform price auctions lead to higher revenue than discriminatory auctions. If agents have decreasing marginal utilities for objects, then uniform price auctions are asymptotically efficient, while discriminatory auctions are asymptotically {\sl in}efficient. Finally, we show that in some cases where dispersed allocations are efficient, revenue may increase by bundling goods at the expense of efficiency.Auction, Competition, Mechanism, Asymptotic Efficiency, Revenue Equivalence