Full-information Best-choice Problem Research Articles

Optimal rules for the sequential selection of uniform spacings

ABSTRACT The unit interval is broken at random into n spacings with the breaking points given by a random sample of size n−1 from the uniform distribution on . A decision-maker observes the lengths of these spacings sequentially and must decide either to select the present spacing or to reject it and continue to observe the next one. No recall of preceding observations is permitted. In this paper, we first find an optimal stopping rule to maximize the probability of selecting the largest spacing. Furthermore, we establish a connection to the classical secretary problem, from which we derive a lower bound for the maximum probability. Moreover, we conjecture that the limiting optimal probability is the same as in the full-information best-choice problem. Next, we find an optimal stopping rule to maximize the expected length of the selected spacing. It is shown that the maximum obtainable expected length lies between the expected value of the second-largest and the largest spacing.

Cost-Efficient Strategy in Clouds with Spot Price Uncertainty

We address a cloud spot bidding problem for user cost optimization. We propose stochastic optimization model to minimize the expected resource rental cost in the presence of spot price uncertainty. The model is based on the well-known full-information best-choice problem. Based on the model, we derive the strategy for cloud spots bidding. The strategy allows to minimize the expected cost for a spot instance in a specific period of time with quality of service guarantee. Our simulation analysis based on realistic settings clearly demonstrates the advantages of the proposed optimization solutions.

Iterated full information secretary problem

We consider a full information best-choice problem where an administrator who has only one on-line choice in m consecutive searches has to choose the best candidate in one of them.

The full-information best-choice problem with uniform or gamma horizons

A decision-maker has to choose one from among a Poisson stream of i.i.d. bids, with no recall. The stream stops at a random time with a uniform (in the first case) or Erlang (in the second case) distribution. We solve the problem explicitly for maximal expected gain for bids that may take on any finite number of values. A fast procedure to solve the problem for fixed horizon is presented as well.

Optimal Choice of the Best Available Applicant in Full-Information Models

The problem we consider here is a full-information best-choice problem in which n applicants appear sequentially, but each applicant refuses an offer independently of other applicants with known fixed probability 0≤q<1. The objective is to maximize the probability of choosing the best available applicant. Two models are distinguished according to when the availability can be ascertained; the availability is ascertained just after the arrival of the applicant (Model 1), whereas the availability can be ascertained only when an offer is made (Model 2). For Model 1, we can obtain the explicit expressions for the optimal stopping rule and the optimal probability for a given n. A remarkable feature of this model is that, asymptotically (i.e. n→∞), the optimal probability becomes insensitive to q and approaches 0.580 164. The planar Poisson process (PPP) model provides more insight into this phenomenon. For Model 2, the optimal stopping rule depends on the past history in a complicated way and seems to be intractable. We have not solved this model for a finite n but derive, via the PPP approach, a lower bound on the asymptotically optimal probability.

Optimal Choice of the Best Available Applicant in Full-Information Models

The problem we consider here is a full-information best-choice problem in which n applicants appear sequentially, but each applicant refuses an offer independently of other applicants with known fixed probability 0≤q<1. The objective is to maximize the probability of choosing the best available applicant. Two models are distinguished according to when the availability can be ascertained; the availability is ascertained just after the arrival of the applicant (Model 1), whereas the availability can be ascertained only when an offer is made (Model 2). For Model 1, we can obtain the explicit expressions for the optimal stopping rule and the optimal probability for a given n. A remarkable feature of this model is that, asymptotically (i.e. n→∞), the optimal probability becomes insensitive to q and approaches 0.580 164. The planar Poisson process (PPP) model provides more insight into this phenomenon. For Model 2, the optimal stopping rule depends on the past history in a complicated way and seems to be intractable. We have not solved this model for a finite n but derive, via the PPP approach, a lower bound on the asymptotically optimal probability.

Optimal Shopping When the Sales Are on—a Markovian Full-Information Best-Choice Problem

We study a full-information best-choice problem viewed in a shopping context. A certain commodity can be found at certain random times with stochastically fluctuating prices. While the prices may have a tendency to decrease, the instants at which items are offered become less frequent and it is possible that the item currently found will be the last one. The prospective customer's objective is to buy at the right time so as to minimize the expected price of the acquired item. We propose a two-dimensional Markov chain model with a rather general continuous-time point process structure and dependence of the random prices on the availability times of the items. The value function v of the associated optimal stopping problem is characterized as the smallest solution of a two-dimensional integral equation; this allows us to find the optimal policy under certain conditions. In particular, we consider a nonhomogeneous Poisson model for which more specific results can be obtained. We derive a differential equation of which v is the uniformly smallest nonnegative solution. This way v is determined up to a boundary condition at infinity. We provide criteria for identifying a solution as the value function and also for the natural stopping rule to be optimal. Several examples are given.

Winning Rate in the Full-Information Best-Choice Problem

Following a long-standing suggestion by Gilbert and Mosteller, we derive an explicit formula for the asymptotic winning rate in the full-information best-choice problem.

Winning Rate in the Full-Information Best-Choice Problem

Following a long-standing suggestion by Gilbert and Mosteller, we derive an explicit formula for the asymptotic winning rate in the full-information best-choice problem.

Winning Rate in the Full-Information Best-Choice Problem

Following a long-standing suggestion by Gilbert and Mosteller, we derive an explicit formula for the asymptotic winning rate in the full-information best-choice problem.

Why do these quite different best-choice problems have the same solutions?

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

Why do these quite different best-choice problems have the same solutions?

The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.

On the full information best-choice problem

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

On the full information best-choice problem

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

Optimal Stopping with Random Horizon with Application to the Full-Information Best-Choice Problem with Random Freeze

Abstract A general theorem is given showing that an optimal stopping problem with independent random horizon is essentially equivalent to a “discounted” fixed horizon optimal stopping problem derived from it. The result is used to treat the “full information” secretary problem, where a known number of applicants for a job are potentially available. Applicants are interviewed sequentially with no recall, and their X i -values are observed, where the X i are iid random variables from a known continuous distribution. The goal is to pick the applicant with the highest X i value. The “payoff” for this applicant is 1, and for any other applicant is zero. A random “freeze-time” variable M, with known distribution independent of the X i 's makes it impossible to pick an applicant after time M. The optimal rule is described and necessary and sufficient conditions for it to have a “monotone structure” are given. Uniform and geometric freeze variables are discussed, and some asymptotic results are given.

A Full-Information Best-Choice Problem with Allowance

This paper considers a variation of the classical full-information best-choice problem. The problem allows success to be obtained even when the best item is not selected, provided the item that is selected is within the allowance of the best item. Under certain regularity conditions on the allowance function, the general nature of the optimal strategy is given as well as an algorithm to determine it exactly. It is also examined how the success probability depends on the allowance function and the underlying distribution of the observed values of the items.

A full-information best-choice problem with finite memory

n i.i.d. random variables with known continuous distribution are observed sequentially with the objective of selecting the largest. This paper considers the finite-memory case which, at each stage, allows a solicitation of anyone of the last m observations as well as of the present one. If the (k – t)th observation with value x is solicited at the k th stage, the probability of successful solicitation is p1(x) or p2(x) according to whether t = 0 or 1 ≦ t ≦ m. The optimal procedure is shown to be characterized by the double sequences of decision numbers. A simple algorithm for calculating the decision numbers and the probability of selecting the largest is obtained in a special case.

A full-information best-choice problem with finite memory

n i.i.d. random variables with known continuous distribution are observed sequentially with the objective of selecting the largest. This paper considers the finite-memory case which, at each stage, allows a solicitation of anyone of the last m observations as well as of the present one. If the (k – t)th observation with value x is solicited at the k th stage, the probability of successful solicitation is p 1(x) or p 2(x) according to whether t = 0 or 1 ≦ t ≦ m. The optimal procedure is shown to be characterized by the double sequences of decision numbers. A simple algorithm for calculating the decision numbers and the probability of selecting the largest is obtained in a special case.