Alessandro Arlotto

Optimal Hiring and Retention Policies for Heterogeneous Workers Who Learn

We study the hiring and retention of heterogeneous workers who learn over time. We show that the problem can be analyzed as an infinite-armed bandit with switching costs, and we apply results from Bergemann and Valimaki [Bergemann D, Valimaki J 2001 Stationary multi-choice bandit problems. J. Econom. Dynam. Control 2510:1585--1594] to characterize the optimal hiring and retention policy. For problems with Gaussian data, we develop approximations that allow the efficient implementation of the optimal policy and the evaluation of its performance. Our numerical examples demonstrate that the value of active monitoring and screening of employees can be substantial. This paper was accepted by Yossi Aviv, operations management.

Optimal sequential selection of a unimodal subsequence of a random sequence

We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does so within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences.

Hessian orders and multinormal distributions

Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex cone H of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are equal and the difference of their covariance matrices belongs to the dual of H. Then we show that the same conditions are also sufficient for multinormal random vectors. We study several particular cases of this general result.

OPTIMAL ONLINE SELECTION OF A MONOTONE SUBSEQUENCE: A CENTRAL LIMIT THEOREM

Consider a sequence of n independent random variables with a common continuous distribution F, and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy πn∗ that is optimal in the sense that it maximizes the expected value of Ln(πn∗), the number of selected observations. We investigate the distribution of Ln(πn∗); in particular, we obtain a central limit theorem for Ln(πn∗) and a detailed understanding of its mean and variance for large n. Our results and methods are complementary to the work of Bruss and Delbaen (2004) where an analogous central limit theorem is found for monotone increasing selections from a finite sequence with cardinality N where N is a Poisson random variable that is independent of the sequence.

OPTIMAL ONLINE SELECTION OF AN ALTERNATING SUBSEQUENCE: A CENTRAL LIMIT THEOREM

We analyze the optimal policy for the sequential selection of an alternating subsequence from a sequence of n independent observations from a continuous distribution F, and we prove a central limit theorem for the number of selections made by that policy. The proof exploits the backward recursion of dynamic programming and assembles a detailed understanding of the associated value functions and selection rules.

Quickest Online Selection of an Increasing Subsequence of Specified Size

Given a sequence of independent random variables with a common continuous distribution, we consider the online decision problem where one seeks to minimize the expected value of the time that is needed to complete the selection of a monotone increasing subsequence of a prespecified length

Beardwood–Halton–Hammersley theorem for stationary ergodic sequences: A counterexample

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A Central Limit Theorem for Temporally Nonhomogenous Markov Chains with Applications to Dynamic Programming

. This problem is dual to some online decision problems that have been considered earlier, and this dual problem has some notable advantages. In particular, the recursions and equations of optimality lead with relative ease to asymptotic formulas for mean and variance of the minimal selection time.

An adaptive O(log n)-optimal policy for the online selection of a monotone subsequence from a random sample

We construct a stationary ergodic process X1;X2;::: such that each Xt has the uniform distribution on the unit square and the length Ln of the shortest path through the points X1;X2;:::;Xn is not asymptotic to a constant times the square root of n. In other words, we show that the Beardwood, Halton and Hammersley theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions.

A Central Limit Theorem for Costs in Bulinskaya’s Inventory Management Problem When Deliveries Face Delays

We prove a central limit theorem for a class of additive processes that arise naturally in the theory of finite horizon Markov decision problems. The main theorem generalizes a classic result of Dobrushin for temporally nonhomogeneous Markov chains, and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain. We show through several examples that this added flexibility gives one a direct path to asymptotic normality of the optimal total reward of finite horizon Markov decision problems. The same examples also explain why such results are not easily obtained by alternative Markovian techniques such as enlargement of the state space.