Assaf Zeevi

Dynamic Pricing Without Knowing the Demand Function: Risk Bounds and Near-Optimal Algorithms

We consider a single-product revenue management problem where, given an initial inventory, the objective is to dynamically adjust prices over a finite sales horizon to maximize expected revenues. Realized demand is observed over time, but the underlying functional relationship between price and mean demand rate that governs these observations (otherwise known as the demand function or demand curve) is not known. We consider two instances of this problem: (i) a setting where the demand function is assumed to belong to a known parametric family with unknown parameter values; and (ii) a setting where the demand function is assumed to belong to a broad class of functions that need not admit any parametric representation. In each case we develop policies that learn the demand function “on the fly,” and optimize prices based on that. The performance of these algorithms is measured in terms of the regret: the revenue loss relative to the maximal revenues that can be extracted when the demand function is known prior to the start of the selling season. We derive lower bounds on the regret that hold for any admissible pricing policy, and then show that our proposed algorithms achieve a regret that is “close” to this lower bound. The magnitude of the regret can be interpreted as the economic value of prior knowledge on the demand function, manifested as the revenue loss due to model uncertainty.

Security of quantum key distribution with entangled photons against individual attacks

We investigate the security of quantum key distribution with entangled photons, focusing on the two-photon variation of the Bennett-Brassard 1984 (BB84) protocol proposed in 1992 by Bennett, Brasard, and Mermin (BBM92). We present a proof of security which applies to realistic sources, and to untrustable sources which can be placed outside the labs of the two receivers. The proof is restricted to individual eavesdropping attacks, and assumes that the detection apparatus is trustable. We find that the average collision probability for the BBM92 protocol is the same as that of the BB84 protocol with an ideal single-photon source. This indicates that there is no analog in BBM92 to photon splitting attacks, and that the source can be placed between the two receivers without changing the form of the collision probability. We then compare the communication rate of both protocols as a function of distance, and show that BBM92 has potential for much longer communication distances, up to 170 km, in the presence of realistic experimental imperfections. Finally, we propose a scheme based on entanglement swapping that can lead to even longer distance communication. The limiting factor in this scheme is the channel loss, which imposes very slow communication rates at longer distances.

A Method for Staffing Large Call Centers Based on Stochastic Fluid Models

We consider a call center model withm input flows andr pools of agents; them-vector ? of instantaneous arrival rates is allowed to be time dependent and to vary stochastically. Seeking to optimize the trade-off between personnel costs and abandonment penalties, we develop and illustrate a practical method for sizing ther agent pools. Using stochastic fluid models, this method reduces the staffing problem to a multidimensional newsvendor problem, which can be solved numerically by a combination of linear programming and Monte Carlo simulation. Numerical examples are presented, and in all cases the pool sizes derived by means of the proposed method are very close to optimal.

Design and Control of a Large Call Center: Asymptotic Analysis of an LP-Based Method

This paper analyzes a call center model with m customer classes and r agent pools. The model is one with doubly stochastic arrivals, which means that the m-vector of instantaneous arrival rates is allowed to vary both temporally and stochastically. Two levels of call center management are considered: staffing the r pools of agents, and dynamically routing calls to agents. The system managers objective is to minimize the sum of personnel costs and abandonment penalties. We consider a limiting parameter regime that is natural for call centers and relatively easy to analyze, but apparently novel in the literature of applied probability. For that parameter regime, we prove an asymptotic lower bound on expected total cost, which uses a strikingly simple distillation of the original system data. We then propose a method for staffing and routing based on linear programming (LP), and show that it achieves the asymptotic lower bound on expected total cost; in that sense the proposed method is asymptotically optimal.

Bayesian Dynamic Pricing Policies: Learning and Earning Under a Binary Prior Distribution

Motivated by applications in financial services, we consider a seller who offers prices sequentially to a stream of potential customers, observing either success or failure in each sales attempt. The parameters of the underlying demand model are initially unknown, so each price decision involves a trade-off between learning and earning. Attention is restricted to the simplest kind of model uncertainty, where one of two demand models is known to apply, and we focus initially on performance of the myopic Bayesian policy (MBP), variants of which are commonly used in practice. Because learning is passive under the MBP (that is, learning only takes place as a by-product of actions that have a different purpose), it can lead to incomplete learning and poor profit performance. However, under one additional assumption, a constrained variant of the myopic policy is shown to have the following strong theoretical virtue: the expected performance gap relative to a clairvoyant who knows the underlying demand model is bounded by a constant as the number of sales attempts becomes large. This paper was accepted by Gerard P. Cachon, stochastic models and simulation.

Dynamic Routing and Admission Control in High-Volume Service Systems: Asymptotic Analysis via Multi-Scale Fluid Limits

Motivated by applications in telephone call centers, we consider a service system model with m customer classes and r server pools. The model is one with doubly stochastic arrivals, which means that the m-vector λ of instantaneous arrival rates is allowed to vary both temporally and stochastically. Two levels of dynamic control are considered: customers may be either blocked or accepted at the time of their arrival, and then accepted customers of each class must be routed, either immediately upon acceptance or after some period of waiting, to a server pool that is qualified to handle that class. Customers who are made to wait before commencement of their service are liable to defect. The objective is to minimize the expected sum of blocking costs, waiting costs and defection costs over a fixed and finite planning horizon. We consider an asymptotic parameter regime in which (i) the arrival rates, service rates and defection rates are uniformly accelerated by a large factor κ, then (ii) arrival rates are increased by an additional factor g(κ), and the number of servers in each pool is increased by g(κ) as well. This produces a separation of time scales, justifying a pointwise stationary stochastic fluid approximation for our original system model. In the stochastic fluid approximation, optimal admission control and routing decisions are determined by a simple linear program that uses the current arrival rate vector λ as data. We explain how to implement the fluid models optimal control policy in our original service system context, and prove that the proposed implementation is asymptotically optimal in the first-order sense.

The Hough Transform Estimator

This paper pursues a statistical study of the Hough transform; the celebrated computer vision algorithm used to detect the presence of lines in a noisy image. We first study asymptotic properties of the Hough transform estimator, whose objective is to find the line that “best” fits a set of planar points. In particular, we establish strong consistency, rates of convergence and characterize the limiting distribution of the Hough transform estimator. While the convergence rates are seen to be slower than those found in some standard regression methods, the Hough transform estimator is shown to be more robust as measured by its breakdown point. We next study the Hough transform in the context of the problem of detecting multiple lines. This is addressed via the framework of excess mass functionals and modality testing. Throughout, several numerical examples help illustrate various properties of the estimator. Relations between the Hough transform and more mainstream statistical paradigms and methods are discussed as well. Short Title: The Hough transform estimator

Portfolio Credit Risk with Extremal Dependence: Asymptotic Analysis and Efficient Simulation

We consider the risk of a portfolio comprising loans, bonds, and financial instruments that are subject to possible default. In particular, we are interested in performance measures such as the probability that the portfolio incurs large losses over a fixed time horizon, and the expected excess loss given that large losses are incurred during this horizon. Contrary to the normal copula that is commonly used in practice (e.g., in the CreditMetrics system), we assume a portfolio dependence structure that is semiparametric, does not hinge solely on correlation, and supports extremal dependence among obligors. A particular instance within the proposed class of models is the so-called t-copula model that is derived from the multivariate Student t distribution and hence generalizes the normal copula model. The size of the portfolio, the heterogeneous mix of obligors, and the fact that default events are rare and mutually dependent make it quite complicated to calculate portfolio credit risk either by means of exact analysis or naive Monte Carlo simulation. The main contributions of this paper are twofold. We first derive sharp asymptotics for portfolio credit risk that illustrate the implications of extremal dependence among obligors. Using this as a stepping stone, we develop importance-sampling algorithms that are shown to be asymptotically optimal and can be used to efficiently compute portfolio credit risk via Monte Carlo simulation.

Blind Network Revenue Management

We consider a general class of network revenue management problems, where mean demand at each point in time is determined by a vector of prices, and the objective is to dynamically adjust these prices so as to maximize expected revenues over a finite sales horizon. A salient feature of our problem is that the decision maker can only observe realized demand over time but does not know the underlying demand function that maps prices into instantaneous demand rate. We introduce a family of “blind” pricing policies that are designed to balance trade-offs between exploration demand learning and exploitation pricing to optimize revenues. We derive bounds on the revenue loss incurred by said policies in comparison to the optimal dynamic pricing policy that knows the demand function a priori, and we prove that asymptotically, as the volume of sales increases, this gap shrinks to zero.

Capacity Sizing Under Parameter Uncertainty: Safety Staffing Principles Revisited

We study a capacity sizing problem in a service system that is modeled as a single-class queue with multiple servers and where customers may renege while waiting for service. A salient feature of the model is that the mean arrival rate of work is random (in practice this is a typical consequence of forecasting errors). The paper elucidates the impact of uncertainty on the nature of capacity prescriptions, and relates these to well established rules-of-thumb such as the square-root safety staffing principle. We establish a simple and intuitive relationship between the incoming load (measured in Erlangs) and the extent of uncertainty in arrival rates (measured via the coefficient of variation) that characterizes the extent to which uncertainty dominates stochastic variability or vice versa. In the former case it is shown that traditional square-root safety staffing logic is no longer valid, yet simple capacity prescriptions derived via a suitable newsvendor problem are surprisingly accurate.