Asymmetric Decision Problems Research Articles

Decision analysis networks

This paper presents decision analysis networks (DANs) as a new type of probabilistic graphical model. Like influence diagrams (IDs), DANs are much more compact and easier to build than decision trees and can represent conditional independencies. In fact, for every ID there is an equivalent symmetric DAN, but DANs can also represent asymmetric problems involving partial orderings of the decisions (order asymmetry), restrictions between the values of the variables (domain asymmetry), and conditional observability (information asymmetry). Symmetric DANs can be evaluated with the same algorithms as IDs. Every asymmetric DAN can be evaluated by converting it into an equivalent decision tree or, much more efficiently, by decomposing it into a tree of symmetric DANs. Given that DANs can solve symmetric problems as easily and as efficiently as IDs, and are more appropriate for asymmetric problems—which include virtually all real-world problems—DANs might replace IDs as the standard type of probabilistic graphical model for decision support and decision analysis. We also argue that DANs compare favorably with other formalisms proposed for asymmetric decision problems. In practice, DANs can be built and evaluated with OpenMarkov, a Java open-source package for probabilistic graphical models.

A new method for tackling asymmetric decision problems

Chain Event Graphs are probabilistic graphical models designed especially for the analysis of discrete statistical problems which do not admit a natural product space structure. We show here how they can be used for decision analysis through designation of some nodes as decision nodes, and the addition of utilities. We provide a local propagation algorithm for finding an optimal decision strategy and maximising expected utility. We also compare CEGs with Influence diagrams, Valuation Networks, Sequential decision diagrams, Sequential influence diagrams and Decision circuits for the representation and analysis of asymmetric decision problems.

About the Authors

About the Authors

Formulating Asymmetric Decision Problems as Decision Circuits

Decision analysis problems have traditionally been solved using either decision trees or influence diagrams. Although decision trees are better at handling asymmetry, prevalent in many reliability and risk analysis problems, influence diagrams can solve larger real-world problems by exploiting conditional independence. Decision circuits are graphical representations that combine the computational benefits of both graphical models. They are syntactic representations, i.e., they depict the summation, multiplication, and maximization operations required to solve a decision analysis problem. Previous work on decision circuits has focused on compiling them automatically from influence diagrams and describing the ways in which they can be used for efficient solution and sensitivity analysis. In this paper, we show how a decision circuit can be formulated directly, with or without the preprocessing of numbers that are assessed from the decision maker. By constructing two decision circuits for a nuclear reactor example, one using probabilities in inferred form and the other using probabilities in assessed form, we show how decision circuits generalize decision trees. The notion of coalescence is also made more explicit because computations for decision analysis can be saved and then reused in several ways. Because of their generality, decision circuits provide the analyst with a great deal of flexibility in problem formulation.

Optimality proofs of quantum weight decision algorithms

Recently two quantum algorithms have been proposed for the symmetric and asymmetric weight decision problems. Although they show the quadratic speedup over the classical approaches, their optimality has not been proven formally. In this article, we prove their optimality in a formal fashion.

Sequential valuation networks for asymmetric decision problems

This paper deals with representation and solution of asymmetric decision problems. We describe a new representation called sequential valuation networks that is a hybrid of Covaliu and Oliver’s sequential decision diagrams and Shenoy’s valuation networks. The solution algorithm is based on the idea of decomposing a large asymmetric problem into smaller sub-problems and then using the fusion algorithm of valuation networks to solve the sub-problems. Sequential valuation networks inherit many of the strengths of sequential decision diagrams and valuation networks while overcoming many of their shortcomings. We illustrate our technique by representing and solving a modified version of Covaliu and Oliver’s [Manage. Sci. 41(12) (1995) 1860] Reactor problem in complete detail.

Sequential influence diagrams: A unified asymmetry framework

We describe a new graphical language for specifying asymmetric decision problems. The language is based on a filtered merge of several existing languages including sequential valuation networks, asymmetric influence diagrams, and unconstrained influence diagrams. Asymmetry is encoded using a structure resembling a clustered decision tree, whereas the representation of the uncertainty model is based on the (unconstrained) influence diagram framework. We illustrate the proposed language by modeling several highly asymmetric decision problems, and we describe an efficient solution procedure.

REPRESENTING AND SOLVING ASYMMETRIC DECISION PROBLEMS

This paper deals with the representation and solution of asymmetric Bayesian decision problems. We present a formal framework, termed asymmetric influence diagrams. The framework is based on the syntax and semantics of the traditional influence diagram, and allows an efficient representation of asymmetric decision problems. As opposed to existing frameworks, the asymmetric influence diagram primarily encodes asymmetry at the qualitative level and it can therefore be read directly from the model. We give an algorithm for solving asymmetric influence diagrams. The algorithm initially decomposes the asymmetric decision problem into a structure of symmetric subproblems organized as a tree. A solution to the decision problem can then be found by propagating from the leaves towards the root using existing evaluation methods to solve the subproblems.

Representing asymmetric decision problems using coarse valuations

A valuation-based system approach to knowledge representation has shown its advantages in improving computational efficiency and in allowing many decision models including belief networks. This study applies the Dempster–Shafer theory of belief functions and extends its framework to allow coarse valuations, which admit incomplete specification of probabilities and utilities and, therefore, are more flexible in representing asymmetric decision problems. It presents an algorithm for making inferences and decisions in systems using coarse valuations. It shows that a coarse valuation-based system provides a most natural and compact representation of decision problems.

Valuation network representation and solution of asymmetric decision problems

Abstract This paper deals with asymmetric decision problems. We describe a generalization of the valuation network representation and solution technique to enable efficient representation and solution of asymmetric decision problems. The generalization includes the concepts of indicator valuations and effective frames. We illustrate our technique by solving Howard's used car buyer's problem in complete detail. We highlight the contribution of this paper over the symmetric valuation network technique.

A Comparison of Graphical Techniques for Asymmetric Decision Problems

We compare four graphical techniques for representation and solution of asymmetric decision problems—decision trees, influence diagrams, valuation networks, and sequential decision diagrams. We solve a modified version of Covaliu and Oliver's Reactor problem using each of the four techniques. For each technique, we highlight the strengths, weaknesses, and some open issues that perhaps can be resolved with further research.

Structuring Conditional Relationships in Influence Diagrams

An influence diagram is a graphical representation of a decision problem that is at once a formal description of a decision problem that can be treated by computers and a representation that is easily understood by decision makers who may be unskilled in the art of complex probabilistic modeling. The power of an influence diagram, both as an analysis tool and a communication tool, lies in its ability to concisely summarize the structure of a decision problem. However, when confronted with highly asymmetric problems in which particular acts or events lead to very different possibilities, many analysts prefer decision trees to influence diagrams. In this paper, we extend the definition of an influence diagram by introducing a new representation for its conditional probability distributions. This extended influence diagram representation, combining elements of the decision tree and influence diagram representations, allows one to clearly and efficiently represent asymmetric decision problems and provides an attractive alternative to both the decision tree and conventional influence diagram representations.