Expected Search Cost Research Articles

Modeling the search for the least costly opportunity

Abstract With the continuing growth in the number of opportunities available at virtual stores over the Internet there is also a growing demand for the services of computer programs capable of scanning a large number of stores in a very short time. We assume that the cost associated with each scan is linear in the number of stores scanned, and that the resulting list of price quotes is not always satisfactory to the customer, in which case an additional scan is performed, and so on. In such a reality the customer, wishing to minimize her expected cost, must specify the requested sample size and a rule (control limit) to stop the search. In the context of search theory, the above search model can be categorized as “fixed-sample-size, sequential, with infinite horizon”. According to this model the expected search cost is a function of two decision variables: the sample size and the control limit. We prove that for arbitrary sample size the expected search cost is either quasi-convex or strictly decreasing in the control limit, and that the optimal expected search cost is quasi-convex in the sample size. These properties allow an efficient calculation of the optimal policy. We also develop analytic formulas to calculate the cost’s variance, allowing customers to choose a slightly higher expected cost if there is a considerable decrease in the variance. Finally, we present detailed examples for price quotes that are distributed uniformly or exponentially.

Ranking hypotheses to minimize the search cost in probabilistic inference models

Suppose that we are given n mutually exclusive hypotheses, m mutually exclusive possible observations, the conditional probabilities for each of these observations under each hypothesis, and a method to probe each hypothesis whether it is the true one. We consider the problem of efficient searching for the true (target) hypothesis given a particular observation. Our objective is to minimize the expected search cost for a large number of instances, and for the worst-case distribution of targets. More precisely, we wish to rank the hypotheses so that probing them in the chosen order is optimal in this sense. Costs grow monotonic with the number of probes. While it is straightforward to formulate this problem as a linear program, we can solve it in polynomial time only after a certain reformulation: We introduce m n 2 the so-called rank variables and arrive at another linear program whose solution can be translated afterwards into an optimal mixed strategy of low description complexity: For each observation, at most n rankings, i.e., permutations of hypotheses, appear with positive probabilities. Dimensionality arguments yield further combinatorial bounds. Possible applications of the optimization goal are discussed.

Going Where the Ad Leads You

The search literature assumes that consumers know which firms sell products they are looking for, but are unaware of the particular variety and the prices at which each firm sells. In this paper, we consider the situation where consumers are uncertain whether a firm carries the product at all by proposing a model where in the first stage firms decide on whether or not to carry the product. Firms may advertise, informing consumers not only of the price they charge, but also of the basic fact that they sell the product. In this way, advertising lowers the expected search cost. We show that this role of advertising can lead to a situation where advertised prices are higher than non—advertised prices in equilibrium.

Going Where the Ad Leads You: On High Advertised Prices and Searching Where to Buy

An important role of informative advertising is to inform consumers of the simple fact that the shop that advertises sells a particular product. This information may help consumers to save on their search activities: instead of wandering around, a consumer can simply visit the shop that has advertised, knowing that there he can find the commodity he is looking for. The implications of this simple fact have not been studied before. Using game theoretic reasoning in a model that combines consumer search and firms' advertising we show that firms may find it optimal to advertise prices that are higher than nonadvertised prices. The important mechanism underlying this result is that advertising lowers the expected search cost for consumers. Through this analysis we provide a new insight into the role of informative advertising.

Efficient multicast search under delay and bandwidth constraints

The issue of a multicast search for a group of users is discussed in this study. Given the condition that the search is over only after all the users in the group are found, this problem is called the Conference Call Search (CCS) problem. The goal is to design efficient CCS strategies under delay and bandwidth constraints. While the problem of tracking a single user has been addressed by many studies, to the best of our knowledge, this study is one of the first attempts to reduce the search cost for multiple users. Moreover, as oppose to the single user tracking, for which one can always reduce the expected search delay by increasing the expected search cost, for a multicast search the dependency between the delay and the search cost is more complicated, as demonstrated in this study. We identify the key factors affecting the search efficiency, and the dependency between them and the search delay. Our analysis shows that under tight bandwidth constraints, the CCS problem is NP-hard. We therefore propose a search method that is not optimal, but has a low computational complexity. In addition, the proposed strategy yields a low search delay as well as a low search cost. The performance of the proposed search strategy is superior to the implementation of an optimal single user search on a group of users.

Emerging behavior as binary search trees are symmetrically updated

When repeated updates are made to a binary search tree, the expected search cost tends to improve, as observed by Knott. For the case in which the updates use an asymmetric deletion algorithm, the Knott effect is swamped by the behavior discovered by Eppinger. The Knott effect applies also to updates using symmetric deletion algorithms, and it remains unexplained, along with several other trends in the tree distribution. It is believed that updates using symmetric deletion do not cause search cost to deteriorate, but the evidence is all experimental. The contribution of this paper is to model separately several different trends which may contribute to or detract from the Knott effect, including a previously unreported centripetal tendency.

On Binary Searching with Nonuniform Costs

Let us consider an ordered vector A[1:n]. If the cost of testing each position is similar, then the standard binary search is the best strategy to search the vector. This is true in both the average and worst case. However, if the costs are nonuniform, then the best strategy is not necessarily the standard binary search. The best algorithm to construct a strategy that minimizes the expected search cost runs in O(n3 ) time and requires O(n2 ) space. The same complexities hold for the best algorithm to construct a strategy that minimizes the worst case search cost. Here, we show how to efficiently construct search strategies that are at most at a constant factor from the optimal one. These constructions take linear time and use only linear space. For both the problem of minimizing the expected search cost, under uniform access probabilities, and the problem of minimizing the worst case search cost, we present algorithms that require O(n) space and give a $(2+\epsilon+o(1))$-approximated solution in O(n) time for any fixed value of $\epsilon > 0$.

A search game when a search path is given

In this paper, we investigate a search game in discrete time and space. A searcher is given a search path in advance and his look on the path is determined by a randomized look strategy. A target selects a path from some options. The searcher gains a value on the detection of the target but expends search cost by the look. A pay-off function of the game for the searcher is the expected reward which is defined as the expected value minus the expected search cost. First, we show a recursive relation for the conditional optimal look strategy of the searcher given a target path. We prove its NP-completeness, though it looks simple, and clarify some characteristics of the solution. Then our original continuous game is converted to a matrix game. From these facts, we consider a relationship between the game and the one-sided optimizing problem and examine some examples.

A SEARCH GAME WITH REWARD CRITERION

This paper investigates a search game of a searcher and a target. At the beginning of the search, the target selects his path from some options and the searcher determines the distribution of his available search resources into a search space which consists of discrete cells and discrete time points. The searcher gains a value on detection of the target while he expends the search cost depending on the allocation of the search resource. The payoff of the search is the expected reward which is defined as the expected value minus the expected search cost. The searcher wants to maximize the expected reward and the target wants to minimize it. We formulate the problem as a two-person zero-sum game and reduce it to a concave maximization problem. We propose a computational method to obtain an optimal solution of the game. Our method proceeds in such a way that one-sided problems generated from the original game are repeatedly solved and their solutions converge asymptotically to an optimal solution of the game. By some examples, we examine the effect of parameters included in the problem upon an optimal solution to elucidate some characteristics of the solution and the computational time of the proposed method.

On the distribution of search cost for the move-to-front rule

A file of records, each with an associated request probability, is dynamically maintained as a serial list. Successive requests are mutually independent. The list is reordered according to the move-to-front (MTF) rule: The requested record is moved to the front of the list. We derive the stationary distribution of search cost (=depth of requested item) by embedding in Poisson processes and derive certain finite-time stochastic ordering results for the MTF chain so embedded. A connection with cache fault probabilities is discussed. We also establish a Schur-concavity result for stationary expected search cost. © 1996 John Wiley & Sons, Inc.

An exact formula for the move-to-front rule for self-organizing lists

A collection of items (e.g., books), each with an associated weight (or popularity), is arranged in a row. At each unit of time an item is removed with probability proportional to its weight and replaced at the left end of the row. Thismove-to-front rule gives a Markov chain on permutations often known as theTsetlin library. We derive an exact and tractable formula for the probability of any permutation after any number of moves. From the formula we read off previously studied quantities of interest associated with the chain, such as the stationary distribution and eigenvalues. Measuring discrepancy from stationarity by separation, we use the formula to find the initial arrangement giving the slowest convergence to stationarity. The time to stationarity in this case is a convolution of geometric random variables which we analyze for three natural choices of weights. We also assess the time required for an important functional, namely, expected search cost, to approach its stationary value.

Rates of Convergence for the Move-to-Root Markov Chain for Binary Search Trees

The move-to-root heuristic is a self-organizing rule that attempts to keep a binary search tree in near-optimal form. It is a tree analogue of the move-to-front scheme (also known as the weighted random-to-top card shuffle or Tsetlin library) for self-organizing lists. We study convergence of the move-to-root Markov chain to its stationary distribution and show that move-to-root converges two to four times faster than move-to-front for many examples. We also discuss asymptotics for expected search cost. For equal weights, $\operatorname{cn}/\ln n$ steps are necessary and sufficient to drive the maximum relative error to 0.

An Analysis of Spiral Hashing

Spiral hashing is an interesting extendible hashing method. Only one approximate analysis of its performance was given, which was based on a heuristic approach and did not specify the error. In this paper, we present a discrete analysis of the expected search cost and the expected amortized rehashing cost in spiral hashing for which a big-O term can be given to describe the error

Average search and update costs in skip lists

Skip lists, introduced by Pugh, provide an alternative to search trees, although a precise analysis of their behaviour had been elusive. The exact value of the expected cost for the search of themth element in a skip list ofn elements is derived first in terms of previously studied functions, and secondly as an asymptotic expression. The latter suggests that Pugh's upper bound of the expected search cost is fairly tight for the interesting cases. Assuming a uniform query distribution, the exact and an asymptotic value of the average (over allm) expected search cost in a skip list ofn elements is also derived. Finally, all insert and delete costs are obtained.

Optimal search plan minimizing the expected risk of the search for a target with conditionally deterministic motion

This article deals with a search problem for a moving target with a rather simple type of motion called factorable conditionally deterministic. A search plan is characterized by (ϕ, T), the elements of which specify how to search and when to stop the search, respectively. The problem is to find the optimal search plan which minimizes the expected risk (the expected search cost minus the expected reward). We obtain conditions for the optimal search plan, and applying the theorems, we derive the optimal search plan in a closed form for the case in which the target moves straight from a fixed point selecting his course and speed randomly.

連続的探索における最適探索停止規則

This paper investigates an optimal search policy with stopping for a stationary target being in one of n boxes. It is assumed that the search is conducted continuously with a total search cost C per unit time and the search in box i costs ci per unit search effort. The conditional probability of detecting the target with unit search effort is Oli and a reward Ri is given to the searcher when he successfully detects the target in box i. We derive conditions for the optimal search and stop policy which minimizes the expected risk of the search (the expected search cost minus the expected reward). The physical meaning of the conditions and several properties of the optimal policy are elucidated. The optimal policy for two-box case is examined in detail, and necessary and suffi­ cient conditions for the optimal policy and the closed form risk function are obtained. Lp.=1. ~ The search in box i costs C i ( > 0) per unit search effort and the conditional probability of detecting the target with unit search effort 1S assumed to be a i (> 0). irrespective of the history of the past search. This assumption implies that the random search is conducted in each box and if a search effort ~(i.t) is applied te box i until t. the conditional de­ tection probability when the target exists really in the box is given by 1 - exp(-a.~(i.t)).

Two Spectra of Self-Organizing Sequential Search Algorithms

Two sets of algorithms for searching and dynamic reorganization of linear lists are presented. Each set forms a spectrum, with the well-known move-to-front and transposition algorithms at the extrema. A linear ordering on the stationary expected search cost of the algorithms in each of the two spectra is established over a restricted set of probability distributions.

Sequential Search with Random Overlook Probabilities

Suppose an object is hidden in one of $N$ boxes. The initial prior probability that it is hidden in box $i$ is known to the searcher, who once a day must choose a box to be searched. The probability that the $j$th search of box $i$ will be unsuccessful, given that the object is in box $i$, is described by a random variable $\alpha_{ij}$. Assuming that each search of box $i$ costs $c_i > 0$, sufficient conditions on the joint distribution of the $\{\alpha_{ij}\}^\infty_{j=1}$ are found in order to guarantee that a particular search rule (analogous to earlier search rules found independently by R. Bellman, D. Blackwell, W. Black and J. Kadane) is optimal with respect to minimizing the total expected search cost of finding the object. An extension of a search problem studied by C. Sweat is also treated.