Computing the value of information of quadratic decision problems and its non-negativity conditions

Abstract

There are many examples in the literature of non-cooperative games in which players prefer not to have additional information in order to improve their payoff. We present a general quadratic game in which, if one of the players improves his payoff upon obtaining more information, the other player’s payoff worsens in such a way that there is a net social loss due to having more information. How can we ensure this does not occur? The results of this paper are (1) the mathematical expression of the (social) value of information in a quadratic non-cooperative game, and (2) the conditions that ensure the social value of information is non-negative.     

Bounds for solutions of multipoint boundary value problems

Solutions of an nth order differential inequality with n boundary conditions are compared to solutions of an nth order disconjugate differential equation. It is shown the ratios of certain of these solutions are monotone.     

Designing information structures for quadratic decision problems

This paper investigates the problem of unconstrained optimization when there is only partial information on the random parameters in the objective function. The relation between the optimization performance and the available information is established. The best information structure design with fixed rank is described. The designing procedure is set up in such a way that successive information augmentation or deletion can be considered. The procedure can also be extended to multiperson decision problems.     

Estimates of the indeterminacy set for elliptic boundary value problems with uncertain data

Estimates are derived for the so-called indeterminacy set formed by solutions to an elliptic type boundary value problem with not fully determined coefficients. A two-sided estimate for the diameter of the indeterminacy set is obtained in the energy norm. It is shown that this estimate depends on parameters defining the variability range of the coefficients. The analysis is based on functional a posteriori estimates that provide guaranteed bounds of the difference between an approximate solution and any admissible function in the energy space. The estimates are obtained for the diffusion equation. However, the proposed tools can be used for other classes of partial differential equations if functional a posteriori estimates are established. Bibliography: 2 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 77–80.     

Bounds on stochastic convex allocation problems

We consider a stochastic convex program arising in a certain resource allocation problem. The uncertainty is in the demand for a resource which is to be allocated among several competing activities under convex inventory holding and shortage costs. The problem is cast as a two–period stochastic convex program and we derive tight upper and lower bounds to the problem using marginal distributions of the demands, which may be stochastically dependent. It turns out that these bounds are tighter than the usual bounds in the literature which are based on limited moment information of the underlying random variables. Numerical examples illustrate the bounds.     

Making the information systems outsourcing decision: A transaction cost approach to analyzing outsourcing decision problems

During the last several years outsourcing has emerged as a major issue in information systems management. As competitive forces impinge on business firms, senior managers are re-structuring their organizations with an eye on attaining or maintaining competitive advantage. Various strategies to IS outsourcing have emerged, for example, some outsourcers contract with a sole vendor while others contract with several. To date no studies have been done to determine which strategies are appropriate under what conditions. And while some firms have achieved varying degrees of success with any of these strategies, many have encountered significant difficulties. How then are managers to choose from a set of options that which is most appropriate for their firm? Outsourcing problems are complex and entail considerable implications for the strategy of the firm. A wrong decision can result in loss of core competencies and capabilities, and exposure to unexpected risks. Although many articles have appeared on outsourcing, few have extended the discussion beyond simple cost–benefit analysis. In this paper we discuss a transaction cost theory approach to the analysis of outsourcing decision making. Our approach provides managers with a strategy and techniques for analyzing some of the more subtle issues they may face when dealing with complex outsourcing decisions problems.     

Set choice problems with incomplete information about the preferences of the decision maker

We consider the problem of selecting a predetermined number of objects from a given finite set. It is assumed that the preferences of the decision maker on this set are only partially known. Our solution approach is based on the notions of optimal and non-dominated subsets. The properties of such subsets and the objects they contain are investigated. The implementation of the developed approach is discussed and illustrated by various examples.     

A value of information in FLP problems via sensitivity analysis

A Fuzzy Linear Programming (FLP) Problem is formulated and the value of information is discussed. If the fuzziness of coefficients is decreased by using information about the coefficients, we can expect to obtain a more satisfactory solution than without information. With this view, the prior value of information is discussed. Using the sensitivity analysis technique, a method of allocating the given investigation cost to each fuzzy coefficient is proposed.     

The value of information to decision makers: an experimental approach using card-based decision gaming

This paper describes a novel experimental method for determining the value of different types of information to military decision makers. The experimental method used a simple scenario and a set of serials constructed from cards, each presenting a single piece of information and presented sequentially. Each of a number of pairs of players were taken through the scenario and asked to judge when they would make each of a pair of escalating responses to the situation. The data proved well suited to analysis using a probit model and is consistent with the hypothesis of a Bayesian decision mechanism with normally distributed ‘action points’. The methodology allowed the determination of weights for each of a number of different classes of information, together with estimates of the human and situational elements of variation, including estimates of the ‘prior belief’ of the different pairs of players.     

The value of two-person zero-sum repeated games with incomplete information and uncertain duration

Fix a zero-sum repeated game Γ with incomplete information on both sides. It is known that the value of the infinitely repeated game Γ_∞ need not exist (Aumann and Maschler 1995). It is proved that any number between the minmax and the maxmin of Γ_∞ is the value of a long finitely repeated game Γ _n where players’ information about the uncertain number of repetitions n is asymmetric.     

Interactive decision making for uncertain minimum spanning tree problems with total importance based on a risk-management approach

This paper deals with a minimum spanning tree problem where each edge cost includes uncertainty and importance measure. In risk management to avoid adverse impacts derived from uncertainty, a d-confidence interval for the total cost derived from robustness is introduced. Then, by maximizing the considerable region as well as minimizing the cost-importance ratio, a biobjective minimum spanning tree problem is proposed. Furthermore, in order to satisfy the objects of the decision maker and to solve the proposed model in mathematical programming, fuzzy goals for the objects are introduced as satisfaction functions, and an exact solution algorithm is developed using interactive decision making and deterministic equivalent transformations. Numerical examples are provided to compare our proposed model with some previous models.     

Models for planning capacity expansion of convenience stores under uncertain demand and the value of information

Several stochastic optimization models for planning capacity expansion for convenience store chains (or other similar businesses) are developed that incorporate uncertainty in future demand. All of these models generate schedules for capacity expansion, specifying the size, location, and timing of these expansions in order to maximize the expected profit to the company and to remain within a budget constraint on available resources. The models differ in how uncertainty is incorporated, specifically they differ in the point in the decision-making process that the uncertainty in the demand is resolved. Several measures of the value of information are defined by comparing the results from the different models. Two sample problems are given and their solutions for the various approaches compared.     

投资决策中信息服务的研究

在一定假设条件下,建立了投资者对信息服务的价值分析和最佳投资的理论模型,并对文[1]中的结论进行修正,得到了与实际更加符合的结论.     

Bounds for aggregating nodes in network problems

It is often necessary or desirable in practice to replace a large, detailed optimization model with a smaller, approximate model. It would be useful to have bounds on the error resulting from this process of aggregation. This paper develops such bounds for a class of linear minimum-cost network-flow problems, where groups of nodes in a large problem are replaced by aggregate nodes. Two types of bounds are derived: A priori bounds are available before solving the aggregated problem; a posteriori bounds, which are generally tighter, may be computed afterwards.     

Robust optimization analysis for multiple attribute decision making problems with imprecise information

This paper considers ranking decision alternatives under multiple attributes with imprecise information on both attribute weights and alternative ratings. It is demonstrated that regret results from the decision maker??s inadequate knowledge about the true scenario to occur. Potential optimality analysis is a traditional method to evaluate alternatives with imprecise information. The essence of this approach is to identify any alternative that outperforms the others in its best-case scenario. Our analysis shows that potential optimality analysis is optimistic in nature and may lead to a significant loss if an unfavorable scenario occurs. We suggest a robust optimization analysis approach that ranks alternatives in terms of their worst-case absolute or relative regret. A robust optimal alternative performs reasonably well in all scenarios and is shown to be desirable for a risk-concerned decision maker. Linear programming models are developed to check robust optimality.     

Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients

This paper is concerned with obtaining upper and lower bounds for the eigenvalues of Sturm-Liouville problems with discontinuous coefficients. Such problems occur naturally in many areas of composite material mechanics.The problem is first transformed by using an analog of the classical Liouville transformation. Upper bounds are obtained by application of a Rayleigh-Ritz technique to the transformed problem. Explicit lower bounds in terms of the coefficients are established. Numerical examples illustrate the accuracy of the results.

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Résumé Dans cet article les bornes supérieures et inférieures sont détermineés pour les valeurs caractéristiques des problèmes de Sturm-Liouville avec des coefficients discontinus. De tels problèmes se trouvent naturellement dans la mécanique des materiaux composites.Après avoir transformé ce problème en utilisant un analogue de la transformation classique de Liouville, les bornes supérieures sont obtenues par l'application d'une technique de Rayleigh-Ritz au problème transformé. Les bornes inférieurs sont determinées en fonction des coefficients sous une forme explicite. Quelques exemples numériques montrent l'exactitude des résultats.

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This work was supported by the U.S. Army Research Office under Grants DAH C04-75-G-0059, DAAG 29-76-G-0063 and DAAG 29-77-G-0034.     

Bounds for the solution set of linear complementarity problems

We give here bounds for the feasible domain and the solution norm of Linear Complementarity Problems (LCP). These bounds are motivated by formulating the LCP as a global quadratic optimization problem and are characterized by the eigenstructure of the corresponding matrix. We prove boundedness of the feasible domain when the quadratic problem is concave, and give easily computable bounds for the solution norm for the convex case. We also obtain lower and upper bounds for the solution norm of the general nonconvex problem.     

Bounds for blow-up time in nonlinear parabolic problems

A first order differential inequality technique is used on suitably defined auxiliary functions to determine lower bounds for blow-up time in initial-boundary value problems for parabolic equations of the form

ut=div(ρ(u)gradu)+f(u)