Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Sample Complexity of Up-to-$varepsilon$ Multi-Dimensional Revenue Maximization
61
0
0.0
(
0
)
Added by
Yannai A. Gonczarowski
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of $n$ additive bidders whose values for $m$ heterogeneous items are drawn independently. For any such instance and any $varepsilon>0$, we show that it is possible to learn an $varepsilon$-Bayesian Incentive Compatible auction whose expected revenue is within $varepsilon$ of the optimal $varepsilon$-BIC auction from only polynomially many samples. Our fully nonparametric approach is based on ideas that hold quite generally, and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive, and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well-understood, our corollary for this case extends slightly the state-of-the-art.
rate research
Read More
Consider a monopolist selling $n$ items to an additive buyer whose item values are drawn from independent distributions $F_1,F_2,ldots,F_n$ possibly having unbounded support. Unlike in the single-item case, it is well known that the revenue-optimal selling mechanism (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. Also known is that simple mechanisms with a bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, whether an arbitrarily high fraction of the optimal revenue can be extracted via a bounded menu size remained open. We give an affirmative answer: for every $n$ and $varepsilon>0$, there exists $C=C(n,varepsilon)$ s.t. mechanisms of menu size at most $C$ suffice for obtaining $(1-varepsilon)$ of the optimal revenue from any $F_1,ldots,F_n$. We prove upper and lower bounds on the revenue-approximation complexity $C(n,varepsilon)$ and on the deterministic communication complexity required to run a mechanism achieving such an approximation.
Most work in mechanism design assumes that buyers are risk neutral; some considers risk aversion arising due to a non-linear utility for money. Yet behavioral studies have established that real agents exhibit risk attitudes which cannot be captured by any expected utility model. We initiate the study of revenue-optimal mechanisms under buyer behavioral models beyond expected utility theory. We adopt a model from prospect theory which arose to explain these discrepancies and incorporates agents under-weighting uncertain outcomes. In our model, an event occurring with probability $x < 1$ is worth strictly less to the agent than $x$ times the value of the event when it occurs with certainty. In contrast to the risk-neutral setting, the optimal mechanism may be randomized and appears challenging to find, even for a single buyer and a single item for sale. Nevertheless, we give a characterization of the optimal mechanism which enables positive approximation results. In particular, we show that under a reasonable bounded-risk-aversion assumption, posted pricing obtains a constant approximation. Notably, this result is risk-robust in that it does not depend on the details of the buyers risk attitude. Finally, we examine a dynamic setting in which the buyer is uncertain about his future value. In contrast to positive results for a risk-neutral buyer, we show that the buyers risk aversion may prevent the seller from approximating the optimal revenue in a risk-robust manner.
We study mechanisms for selling a single item when buyers have private values for their outside options, which they forego by participating in the mechanism. This substantially changes the revenue maximization problem. For example, the seller can strictly benefit from selling lotteries already in the single-buyer setting. We bound the menu size and the sample complexity for the optimal single-buyer mechanism. We then show that posting a single price is in fact optimal under the assumption that either (1) the outside option value is independent of the item value, and the item value distribution has decreasing marginal revenue or monotone hazard rate; or (2) the outside option value is a concave function of the item value. Moreover, when there are multiple buyers, we show that sequential posted pricing guarantees a large fraction of the optimal revenue under similar conditions.
We present a polynomial-time algorithm that, given samples from the unknown valuation distribution of each bidder, learns an auction that approximately maximizes the auctioneers revenue in a variety of single-parameter auction environments including matroid environments, position environments, and the public project environment. The valuation distributions may be arbitrary bounded distributions (in particular, they may be irregular, and may differ for the various bidders), thus resolving a problem left open by previous papers. The analysis uses basic tools, is performed in its entirety in value-space, and simplifies the analysis of previously known results for special cases. Furthermore, the analysis extends to certain single-parameter auction environments where precise revenue maximization is known to be intractable, such as knapsack environments.
The literature on mechanism design from samples, which has flourished in recent years at the interface of economics and computer science, offers a bridge between the classic computer-science approach of worst-case analysis (corresponding to no samples) and the classic economic approach of average-case analysis for a given Bayesian prior (conceptually corresponding to the number of samples tending to infinity). Nonetheless, the two directions studied so far are two extreme and almost diametrically opposed directions: that of asymptotic results where the number of samples grows large, and that where only a single sample is available. In this paper, we take a first step toward understanding the middle ground that bridges these two approaches: that of a fixed number of samples greater than one. In a variety of contexts, we ask what is possibly the most fundamental question in this direction: are two samples really better than one sample?. We present a few surprising negative results, and complement them with our main result: showing that the worst-case, over all regular distributions, expected-revenue guarantee of the Empirical Revenue Maximization algorithm given two samples is greater than that of this algorithm given one sample. The proof is technically challenging, and provides the first result that shows that some deterministic mechanism constructed using two samples can guarantee more than one half of the optimal revenue.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
