Bounds on the value of information in uncertain decision problems II†

In most stochastic decision problems one has the opportunity to collect information that would partially or totally eliminate the inherent uncertainty. One wishes to compare the cost and value of such information in terms of the decision maker's preferences to determine an optimal information gathering plan. The calculation of the value of information generally involves oneor more stochastic recourse problems as well as one or more expected value distribution problems. The difficulty and costs of obtaining solutions to these problems has led to a focus on the development of upper and lower bounds on the various subproblems that yield bounds on the value of information. In this paper we discuss published and new bounds for static problems with linear and concave preference functions for partial and perfect information. We also provide numerical examples utilizing simple production and investment problems that illustrate the calculations involved in the computation of the various bounds and provide a setting for a comparison of the bounds that yields some tentative guidelines for their use. The bounds compared are the Jensen's Inequality bound,the Conditional Jensen's Inequality bound and the Generalized Jensen and Edmundson-Madansky bounds.     

An Extended Model for Establishing Dominance in Multiattribute Decisionmaking

This paper is concerned with the use of incomplete information about utilities and weights in multiattribute decisionmaking. Because of time pressure and/or lack of knowledge, a decision maker may only be able to provide incomplete information which might be expressed as a set of linear inequalities. If the decision maker's information on both weights and utilities is imprecisely identified, then the model for establishing pairwise dominance becomes a non-linear program. A method for obtaining non-dominated alternatives without solving the non-linear program is proposed using a simple weighted-additive function.     

Risk-averse decision making in overbooking problem

Traditional literature studying overbooking problems focuses on risk-neutral decision makers. In this paper, we propose a multi-period overbooking model incorporating risk-aversion and extend well-known structural results (the 3-region policy) under the risk-neutral case to the risk-averse one on the basis of an exponential utility function. We also show that the optimal policy for the risk-neutral decision maker can be obtained by letting the risk-aversion parameter approach to zero under the risk-averse case. Therefore, the extant results under the risk-neutral case can be interpreted as a special case of ours. We also investigate how the optimal policy changes with some cost parameters and the decision maker's degree of risk-aversion. Numerical results suggest that the optimal bounds in the 3-region policy may increase or decrease with the decision maker's degree of risk-aversion.     

Preference ranking of discrete multiattribute instances

This paper presents a procedure for preference ranking of discrete multiattribute instances based on the observations that (1) a difficult question could be answered in terms of answers to a sequence of easier ones, (2) in general, the decision maker's preferences can easily be expressed only relative to pairs of instances that differ over one attribute, and (3) the decision maker is able and willing to express his uncertainty regarding a quantity via a subjective probability distribution. The procedure is illustrated through a hypothetical situation.     

Improved Analytical Bounds for Gambler’s Ruin Probabilities

Given integer-valued wagers Feller (1968) has established upper and lower bounds on the probability of ruin, which often turn out to be very close to each other. However, the exact calculation of these bounds depends on the unique non-trivial positive root of the equation () = 1, where is the probability generating function for the wager. In the situation of incomplete information about the distribution of the wager, one is interested in bounds depending only on the first few moments of the wager. Ethier and Khoshnevisan (2002) derive bounds depending explicitly on the first four moments. However, these bounds do not make the best possible use of the available information. Based on the theory of s-convex extremal random variables among arithmetic and real random variables, a substantial improvement can be given. By fixed first four moments of the wager, the obtained new bounds are nearly perfect analytical approximations to the exact bounds of Feller.AMS 2000 Subject Classification: 60E15, 60G40, 91A60     

Interval estimations of global weights in AHP by upper approximation

In analytic hierarchy process (AHP) structured hierarchically as several criteria and alternatives, the priority of an alternative is obtained by using the pairwise comparisons based on a decision maker's intuition. Thus, the given comparisons are uncertain and inconsistent with each other. We use the interval approach for obtaining interval evaluations which are suitable for handling uncertain data. Since the given comparisons are ratio measures and too large intervals are not useful information, the intervals should be normalized and their redundancy should be reduced. We introduce interval probability which fills the role of interval normalization instead of crisp normalization in the estimations at each hierarchy. Then, as a final decision, the interval global weights reflecting the decision maker's uncertain judgements as their widths without redundancy are obtained.     

Distributionally robust mixed integer linear programs: Persistency models with applications

In this paper, we review recent advances in the distributional analysis of mixed integer linear programs with random objective coefficients. Suppose that the probability distribution of the objective coefficients is incompletely specified and characterized through partial moment information. Conic programming methods have been recently used to find distributionally robust bounds for the expected optimal value of mixed integer linear programs over the set of all distributions with the given moment information. These methods also provide additional information on the probability that a binary variable attains a value of 1 in the optimal solution for 0–1 integer linear programs. This probability is defined as the persistency of a binary variable. In this paper, we provide an overview of the complexity results for these models, conic programming formulations that are readily implementable with standard solvers and important applications of persistency models. The main message that we hope to convey through this review is that tools of conic programming provide important insights in the probabilistic analysis of discrete optimization problems. These tools lead to distributionally robust bounds with applications in activity networks, vertex packing, discrete choice models, random walks and sequencing problems, and newsvendor problems.     

A generalized decomposable multiattribute utility function

In this paper a generalized decomposable multiattribute utility function (MAUF) is developed. It is demonstrated that this new MAUF structure is more general than other well-known MAUF structures, such as additive, multiplicative, and multilinear. Therefore, it is more flexible and does not require that the decision maker be consistent with restrictive assumptions such as preferential independence conditions about his/her preferences. We demonstrate that this structure does not require any underlying assumption and hence solves the interdependence among attributes. Hence there is no need for verification of its structure. Several useful extensions and properties for this generalized decomposable MAUF are developed which simplify its structure or assessment. The concept of utility efficiency is developed to identify efficient alternatives when there exists partial information on the scaling constants of an assumed MAUF. It is assumed that the structure (decomposition) of the MAUF is known and the partial information about the scaling constants of the decision maker is in the form of bounds or constraints. For the generalized decomposable structure, linear programming is sufficient to solve all ensuing problems. Some examples are provided.     

On the convergence of reference point methods in multiobjective programming

The purpose of this paper is to present some results about the convergence of interactive reference point methods in multiobjective programming. In particular, we describe how dual information may guide the decision maker in his choice of the successive reference points.In the literature different convergence models have been proposed. The analyst may induce convergence by selecting appropriate rules of the communication. Or he may rely on the learning process of the decision maker to induce some kind of ‘psychological’ convergence. In neither case are the activities of the decision maker precisely described. Consequently, the quality of the final decision cannot be established, and the question of convergence remains an unsolved issue.We describe different ways in which the decision maker may select his successive reference points, and we discuss the convergence of the resulting reference point procedures. Also, we comment on the relevance of these different assumptions about the decision maker's behavior. The procedures are illustrated by a small numerical example.     

Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples. Received: October 1998 / Accepted: June 2000?Published online October 18, 2000     

不完全信息下多目标决策的一种新方法

基于部分偏好信息（目标权重）,本提出了多目标决策的一种线性规划算法,该法避免了获取偏好信息的困难,在较少信息下,为决策提供更普遍,易操作且有效的方案排序结果。最后进行了算例分析。     

The value of the stochastic solution in multistage problems

We generalize the definition of the bounds for the optimal value of the objective function for various deterministic equivalent models in multistage stochastic programs. The parameters EVPI and VSS were introduced for two-stage models. The parameter EVPI, the expected value of perfect information, measures how much it is reasonable to pay to obtain perfect information about the future. The parameter VSS, the value of the stochastic solution, allows us to obtain the goodness of the expected solution value when the expected values are replaced by the random values for the input variables. We extend the definition of these parameters to the multistage stochastic model and prove a similar chain of inequalities with the lower and upper bounds depending substantially on the structure of the problem. This research has been partially supported by the grants, 1/BBVA 00038.16421/2004 from Fundación BBVA, SEJ2005-05549/ECON from Ministerio de Educación y Ciencia and the grant GRUPOS79/04 from the Generalitat Valenciana, Spain.     

Some Characteristic Properties of the Fisher Information Matrix via Cacoullos-Type Inequalities

Some lower bounds for the variance of a function g of a random vector X are extended to a wider class of distributions. Using these bounds, some useful inequalities for the Fisher information are obtained for convolutions and linear combinations of random variables. Finally, using these inequalities, simple proofs are given of classical characterizations of the normal distribution, under certain restrictions, including the matrix analogue of the Darmois-Skitovich result.     

Selecting the best choice in the full information group interview problem

We consider the problem of selecting the single best choice when several groups of choices are presented sequentially for evaluation. In the so-called group interview problem, we assume that the values of choices are random observations from a known distribution function and derive the optimal search strategy that maximizes the probability of selecting the best among all choices. Under the optimal search strategy derived by means of a dynamic programming technique, a decision maker simply selects the best choice in the group under consideration if its value is higher than the pre-specified decision value for that group. We also consider the optimal ordering strategy for the case where the decision maker is permitted to rearrange the sequence of groups for evaluation. We show that the optimal search and ordering strategies can be applied to many sequential decision problems such as the store location problem.     

A new generic model for applying MAUT

A new model for preferences is introduced that weakens the notion of preferential independence. With this model the decision maker's preferences over n real-valued attributes can be elicited with n−1 sets of indifference curves. A simple way of modelling probability distributions with the help of minimally informative joint distributions is introduced in the area of decision making. The aim of these methods was to provide a straightforward method for performing a decision analysis that can be easily explained to engineers and the management. These methods are illustrated in a case-study.     

Rank-based selection strategies for the random walk process

In many decision situations such as hiring a secretary, selling an asset, or seeking a job, the value of each offer, applicant, or choice is assumed to be an independent, identically distributed random variable. In this paper, we consider a special case where the observations are auto-correlated as in the random walk model for stock prices. For a given random walk process of n observations, we explicitly compute the probability that the j-th observation in the sequence is the maximum or minimum among all n observations. Based on the probability distribution of the rank, we derive several distribution-free selection strategies under which the decision maker's expected utility of selecting the best choice is maximized. We show that, unlike in the classical secretary problem, evaluating more choices in the random walk process does not increase the likelihood of successfully selecting the best.     

Pointwise Efficiency in Multiobjective Stochastic Linear Programming

In most of the approaches to the Multiobjective Stochastic Linear Programming problem that have been proposed in the literature, the notion of quality of a solution is not adequately defined. We reconsider this problem from a decision point of view, in contexts where either the decision maker's preference structure cannot be described by a utility function, or where this structure is expressed by an unknown non-decreasing utility function and the probability distribution of the random parameters is unknown. We define a fundamental set of ‘pointwise admissible’ solutions, as well as several subsets of particular interest. We discuss the relevance of these various pointwise efficient sets, their interrelations, and their practical identification.     

Expected utility for nonstochastic risk

Stochastic random phenomena considered in von Neumann–Morgenstern utility theory constitute only a part of all possible random phenomena (Kolmogorov, 1986). We show that any sequence of observed consequences generates a corresponding sequence of frequency distributions, which in general does not have a single limit point but a non-empty closed limit set in the space of finitely additive probabilities. This approach to randomness allows to generalize the expected utility theory in order to cover decision problems under nonstochastic random events. We derive the maxmin expected utility representation for preferences over closed sets of probability measures. The derivation is based on the axiom of preference for stochastic risk, i.e. the decision maker wishes to reduce a set of probability distributions to a single one. This complements Gilboa and Schmeidler’s (1989) consideration of the maxmin expected utility rule with objective treatment of multiple priors.