Optimal Competitive Auctions

We study the design of truthful auctions for selling identical items in unlimited supply (e.g., digital goods) to n unit demand buyers. This classic problem stands out from profit-maximizing auction design literature as it requires no probabilistic assumptions on buyers' valuations and employs the framework of competitive analysis. Our objective is to optimize the worst-case performance of an auction, measured by the ratio between a given benchmark and revenue generated by the auction. We establish a sufficient and necessary condition that characterizes competitive ratios for all monotone benchmarks. The characterization identifies the worst-case distribution of instances and reveals intrinsic relations between competitive ratios and benchmarks in the competitive analysis. With the characterization at hand, we show optimal competitive auctions for two natural benchmarks. The most well-studied benchmark $\mathcal{F}^{(2)}(\cdot)$ measures the envy-free optimal revenue where at least two buyers win. Goldberg et al. [13] showed a sequence of lower bounds on the competitive ratio for each number of buyers n. They conjectured that all these bounds are tight. We show that optimal competitive auctions match these bounds. Thus, we confirm the conjecture and settle a central open problem in the design of digital goods auctions. As one more application we examine another economically meaningful benchmark, which measures the optimal revenue across all limited-supply Vickrey auctions. We identify the optimal competitive ratios to be $(\frac{n}{n-1})^{n-1}-1$ for each number of buyers n, that is $e-1$ as $n$ approaches infinity

Correlations for subsets of particles in symmetric states: what photons are doing within a beam of light when the rest are ignored

Given a state of light, how do its properties change when only some of the constituent photons are observed and the rest are neglected (traced out)? By developing formulae for mode-agnostic removal of photons from a beam, we show how the expectation value of any operator changes when only $q$ photons are inspected from a beam, ignoring the rest. We use this to reexpress expectation values of operators in terms of the state obtained by randomly selecting $q$ photons. Remarkably, this only equals the true expectation value for a unique value of $q$: expressing the operator as a monomial in normally ordered form, $q$ must be equal to the number of photons annihilated by the operator. A useful corollary is that the coefficients of any $q$-photon state chosen at random from an arbitrary state are exactly the $q$th order correlations of the original state; one can inspect the intensity moments to learn what any random photon will be doing and, conversely, one need only look at the $n$-photon subspace to discern what all of the $n$th order correlation functions are. The astute reader will be pleased to find no surprises here, only mathematical justification for intuition. Our results hold for any completely symmetric state of any type of particle with any combination of numbers of particles and can be used wherever bosonic correlations are found.Comment: 11+3 pages, 1 figure, comments always welcom

Teleamplification on the Borealis boson-sampling device

A recent theoretical proposal for teleamplification requires preparation of Fock states, programmable interferometers, and photon-number resolving detectors to herald the teleamplification of an input state. These enable teleportation and heralded noiseless linear amplification of a photonic state up to an arbitrarily large energy cutoff. We report on adapting this proposal for Borealis and demonstrating teleamplification of squeezed-vacuum states with variable amplification factors. The results match the theoretical predictions and exhibit features of amplification in the teleported mode, with fidelities from 50 to 93%. This demonstration motivates the continued development of photonic quantum computing hardware for noiseless linear amplification's applications across quantum communication, sensing, and error correction.Comment: 9+5 pages, 6+7 figures; close to published versio

Properties of Nucleon Resonances by means of a Genetic Algorithm

We present an optimization scheme that employs a Genetic Algorithm (GA) to determine the properties of low-lying nucleon excitations within a realistic photo-pion production model based upon an effective Lagrangian. We show that with this modern optimization technique it is possible to reliably assess the parameters of the resonances and the associated error bars as well as to identify weaknesses in the models. To illustrate the problems the optimization process may encounter, we provide results obtained for the nucleon resonances $\Delta$(1230) and $\Delta$(1700). The former can be easily isolated and thus has been studied in depth, while the latter is not as well known experimentally.Comment: 12 pages, 10 figures, 3 tables. Minor correction