Simple influence diagram



An influence diagram is a simple visual representation of a decision problem. Influence diagrams offer an intuitive way to identify and display the essential elements, including decisions, uncertainties, and objectives, and how they influence each other.

This simple influence diagram depicts a variable describing the situation, a decision "What do we do?", a chance variable "What's the outcome?", and our final valuation "How do we like it?". These four node types are the building blocks of decision problems.  The influence diagram gives a high-level conceptual view on which the analyst may build a detailed quantitative model.

 What do the nodes mean?

A decision is a variable that you (or your organization) as the decision maker have the power to modify directly. It could be whether to invest in a new project, how much to invest, how much to bid, where to locate a new site, or, in this example, what budget to allocate for marketing.

A chance variable is uncertain, whose value you don't (yet) know, because you don't have complete information -- maybe it's in the future -- and which (unlike a decision) you cannot control directly.

An objective is a measure of your satisfaction with possible outcomes. It might be net present value, lives saved, or EBITDA, or more generally, "utility". Usually, the decision maker is trying to find decisions to maximize (or minimize) the objective.  Often an objective combines multiple subobjectives or attributes, which may be in conflict -- such as energy costs, benefits, and environmental and health risks. Usually, the objective is uncertain, decision analysts suggest maximizing the expected value, or more generally expected utility, based on risk preference.

A general variable is a deterministic function of the quantities it depends on.

What do the arrows mean?

An arrow denotes an influence. A influences B means that knowing A would directly affect our belief or expectation about the value of B. An influence expresses knowledge about relevance. It does not necessarily imply a causal relation, or a flow of material, data, or money.

How do influence diagrams compare to decision trees?

Decision trees are another common representation for decision problems. Decision trees display the set of alternative values for each decision and chance variable as branches coming out of each node.

Figure 1.
Decision tree for R&D and commercialization of a new product.

Figure 2.
Corresponding influence diagram.


The influence diagram and decision tree show different kinds of information. The influence diagram shows the dependencies among the variables more clearly than the decision tree. The decision tree shows more details of possible paths or scenarios as sequences of branches from left to right. But, this detail comes at a steep price: First, you must treat all variables as discrete (a small number of alternatives) even if they are actually continuous. Second, the number of nodes in a decision tree increases exponentially with the number of decision and chance variables. We would need 121 nodes to display the decision tree corresponding to the simple market-analysis influence diagram at the top of this page, even if we assume only three branches for each of the two decisions and two chance variables. The tree would be too complicated to display on this Web page. The influence diagram is a much more compact representation.

How do you create influence diagrams in Analytica®?

With Lumina's Analytica™ software, you can draw an influence diagram simply by selecting new nodes, placing them, and drawing arrows among them. When you specify a formula for a variable to define its quantitative definition, you can select directly from its inputs - the variables from which there are incoming arrows. Or, if you use other variables in the definition, the arrows will automatically redraw to keep the diagram consistent with the underlying relationships.

How does Analytica extend influence diagrams?

Analytica extends the standard influence diagram notation with additional types of node, to provide the power and flexibility to handle real-world problems of greater complexity than can be handled with conventional tools.

1. Hierarchy of modules
With module nodes, you can organize a complex model as a hierarchy of modules. Double-click on a module node, such as Costs, to display its details as another diagram:

Using Analytica's modules, you can organize a model containing hundreds, or even thousands, of variables into a hierarchy of diagrams, each of which is small enough to be easily comprehensible and manageable.

2. Variables as multidimensional arrays
Standard influence diagrams assume that variables are scalar quantities. In Analytica, a variable may be a vector, or a multidimensional array - for example, with the market size and sales for each region, each product, and each time period. Analytica employs index variables to identify the dimensions.

3. User-defined functions
You can use existing libraries of functions to extend the richness of Analytica's modeling language for particular problem domains. Or you can create your own functions.

4. Diagrams with feedback loops
Traditional influence diagrams don't permit feedback loops - like, where Marketing budget -> Market share -> Revenues -> Marketing budget. Analytica, however, does let you create loops like this in dynamic model, provided there is a time lag, denoted as a dashed arrow, somewhere in the loop.

Who invented influence diagrams?

Several people from the decision-analysis community were involved in the creation of influence diagrams. Professor Ronald Howard from Stanford University and his colleague, Dr. James Matheson, refined and popularized influence diagrams as a convenient notation for communicating about decision problems, that is complementary to decision trees.

Where can I learn more?

  • Uncertainty: A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis, M. Granger Morgan & Max Henrion, Cambridge University Press, reprinted 1998. (Relevant excerpt from Chapter 10 available as download.)
  • Analytica User Guide, especially chapter 5: Building Effective Models and Chapter 6: Creating Lucid Influence Diagrams, from Lumina Decisions Systems, 1998. (PDF file available for free download)
  • Making Hard Decisions: An Introduction to Decision Analysis, Second Edition, Duxbury Press, Belmont, CA, 1996.