Efficiency Guarantees in Auctions with Budgets

In settings where players have a limited access to liquidity, represented in the form of budget constraints, efficiency maximization has proven to be a challenging goal. In particular, the social welfare cannot be approximated by a better factor then the number of players. Therefore, the literature has mainly resorted to Pareto-efficiency as a way to achieve efficiency in such settings. While successful in some important scenarios, in many settings it is known that either exactly one incentive-compatible auction that always outputs a Pareto-efficient solution, or that no truthful mechanism can always guarantee a Pareto-efficient outcome. Traditionally, impossibility results can be avoided by considering approximations. However, Pareto-efficiency is a binary property (is either satisfied or not), which does not allow for approximations. In this paper we propose a new notion of efficiency, called \emph{liquid welfare}. This is the maximum amount of revenue an omniscient seller would be able to extract from a certain instance. We explain the intuition behind this objective function and show that it can be 2-approximated by two different auctions. Moreover, we show that no truthful algorithm can guarantee an approximation factor better than 4/3 with respect to the liquid welfare, and provide a truthful auction that attains this bound in a special case. Importantly, the liquid welfare benchmark also overcomes impossibilities for some settings. While it is impossible to design Pareto-efficient auctions for multi-unit auctions where players have decreasing marginal values, we give a deterministic $O(\log n)$-approximation for the liquid welfare in this setting

Single-Sample Prophet Inequalities via Greedy-Ordered Selection

We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.</p

Optimal auctions through deep learning

Designing an incentive compatible auction that maximizes expected revenue is an intricate task. The single-item case was resolved in a seminal piece of work by Myerson in 1981. Even after 30-40 years of intense research the problem remains unsolved for seemingly simple multibidder, multi-item settings. In this work, we initiate the exploration of the use of tools from deep learning for the automated design of optimal auctions. We model an auction as a multi-layer neural network, frame optimal auction design as a constrained learning problem, and show how it can be solved using standard pipelines. We prove generalization bounds and present extensive experiments, recovering essentially all known analytical solutions for multi-item settings, and obtaining novel mechanisms for settings in which the optimal mechanism is unknown

Prophet Inequalities for IID Random Variables from an Unknown Distribution

A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: given a sequence of random variables X1, . . . , Xn drawn independently from a distribution F , the goal is to choose a stopping time τ so as to maximize α such that for all distributions F we have E[Xτ ] ≥ α · E[maxt Xt ]. What makes this problem challenging is that the decision whether τ = t may only depend on the values of the random variables X1, . . . , Xt and on the distribution F . For a long time the best known bound for the problem had been α ≥ 1 − 1/e ≈ 0.632, but quite recently a tight bound of α ≈ 0.745 was obtained. The case where F is unknown, such that the decision whether τ = t may depend only on the values of the random variables X1, . . . , Xt , is equally well motivated but has received much less attention. A straightforward guarantee for this case of α ≥ 1/e ≈ 0.368 can be derived from the solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F , and show that even with o(n) samples α ≤ 1/e. On the other hand, n samples allow for a significant improvement, while O(n2) samples are equivalent to knowledge of the distribution: specifically, with n samples α ≥ 1 − 1/e ≈ 0.632 and α ≤ ln(2) ≈ 0.693, and with O(n2) samples α ≥ 0.745 − ε for any ε > 0

Differential Expression of miRNAs in Response to Topping in Flue-Cured Tobacco (Nicotiana tabacum) Roots

Topping is an important cultivating measure for flue-cured tobacco, and many genes had been found to be differentially expressed in response to topping. But it is still unclear how these genes are regulated. MiRNAs play a critical role in post-transcriptional gene regulation, so we sequenced two sRNA libraries from tobacco roots before and after topping, with a view to exploring transcriptional differences in miRNAs.Two sRNA libraries were generated from tobacco roots before and after topping. Solexa high-throughput sequencing of tobacco small RNAs revealed a total of 12,104,207 and 11,292,018 reads representing 3,633,398 and 3,084,102 distinct sequences before and after topping. The expressions of 136 conserved miRNAs (belonging to 32 families) and 126 new miRNAs (belonging to 77 families) were determined. There were three major conserved miRNAs families (nta-miR156, nta-miR172 and nta-miR171) and two major new miRNAs families (nta-miRn2 and nta-miRn26). All of these identified miRNAs can be folded into characteristic miRNA stem-loop secondary hairpin structures, and qRT-PCR was adopted to validate and measure the expression of miRNAs. Putative targets were identified for 133 out of 136 conserved miRNAs and 126 new miRNAs. Of these miRNAs whose targets had been identified, the miRNAs which change markedly (>2 folds) belong to 53 families and their targets have different biological functions including development, response to stress, response to hormone, N metabolism, C metabolism, signal transduction, nucleic acid metabolism and other metabolism. Some interesting targets for miRNAs had been determined.The differential expression profiles of miRNAs were shown in flue-cured tobacco roots before and after topping, which can be expected to regulate transcripts distinctly involved in response to topping. Further identification of these differentially expressed miRNAs and their targets would allow better understanding of the regulatory mechanisms for flue-cured tobacco response to topping

Regulation of early steps of GPVI signal transduction by phosphatases: a systems biology approach

We present a data-driven mathematical model of a key initiating step in platelet activation, a central process in the prevention of bleeding following Injury. In vascular disease, this process is activated inappropriately and causes thrombosis, heart attacks and stroke. The collagen receptor GPVI is the primary trigger for platelet activation at sites of injury. Understanding the complex molecular mechanisms initiated by this receptor is important for development of more effective antithrombotic medicines. In this work we developed a series of nonlinear ordinary differential equation models that are direct representations of biological hypotheses surrounding the initial steps in GPVI-stimulated signal transduction. At each stage model simulations were compared to our own quantitative, high-temporal experimental data that guides further experimental design, data collection and model refinement. Much is known about the linear forward reactions within platelet signalling pathways but knowledge of the roles of putative reverse reactions are poorly understood. An initial model, that includes a simple constitutively active phosphatase, was unable to explain experimental data. Model revisions, incorporating a complex pathway of interactions (and specifically the phosphatase TULA-2), provided a good description of the experimental data both based on observations of phosphorylation in samples from one donor and in those of a wider population. Our model was used to investigate the levels of proteins involved in regulating the pathway and the effect of low GPVI levels that have been associated with disease. Results indicate a clear separation in healthy and GPVI deficient states in respect of the signalling cascade dynamics associated with Syk tyrosine phosphorylation and activation. Our approach reveals the central importance of this negative feedback pathway that results in the temporal regulation of a specific class of protein tyrosine phosphatases in controlling the rate, and therefore extent, of GPVI-stimulated platelet activation

Target dependence of chick retinal ganglion cells during embryogenesis: cell survival and dendritic development

The survival of retinal ganglion cells and the dendritic development were investigated a) in normal chick embryos, b) in embryos whose primordial optic lobes and adjacent areas were removed (target reduced embryos), and c) in embryos whose optic nerves were transected (target deprived embryos) in order to study the influences of central targets on developing ganglion cells. The ganglion cells were stained postmortem with the carbocyanine dye DiI. Cell body and dendritic field diameters were measured in whole-mounted retinae before and after the period of cell death at embryonic day 10 (E10) and E16. The cell densities within the ganglion cell layer were counted in cresyl violet/thionine stained retinae. The central retinal projection in target reduced embryos was studied with the anterogradely transported fluorescent marker rhodamine-B-isothiocyanate (RITC). In normal embryos, the earliest dendritic processes were observed at E6 in the central retina, whereas at E10 elaborate dendritic branching was found across the retina. Different morphological types of ganglion cells could be identified at E16. In both, target reduced embryos and target deprived embryos, the initial dendritic growth and pattern of ramification were indistinguishable from those of normal embryos up to E10. Cell body diameters, dendritic tree diameters, and cell densities were not significantly different. At the end of the naturally occurring cell death period (E16), the ganglion cell density was strongly reduced in both experimental groups compared to controls. In particular, when the optic nerve was transected, it resulted in the almost complete degeneration of ganglion cells. In target reduced embryos, a small population (about 5% of the normal number) of ganglion cells survived. The proportion of large cells was increased within the total population compared to normal retinae. Displaced ganglion cells were not affected by partial target removal but strongly affected by transection of the optic nerve. Anterograde labelling from the retina revealed that in target reduced embryos the remaining ganglion cells innervated non-tectal primary visual nuclei. The present results suggest the following: a) Before the onset of the cell death period, the growth and ramification of ganglion cell dendrites occur independently of central visual targets. b) In target reduced embryos, a small population of ganglion cells survives, namely, those cells that project to remaining central areas. Complete disconnection from central targets by transecting the optic nerve leads to the degeneration of almost all ganglion cells. c) The surviving ganglion cell population consists mainly of large ganglion cells