Influence diagram

Influence diagram is a graphical model for representing and analyzing decision problems under uncertainty, using nodes and directed arcs to show relationships between decisions, chance events, and objectives (Howard R.A., Matheson J.E. 1984)^[1]. Three shapes tell the whole story: rectangles for decisions you control, ovals for uncertainties you face, and diamonds for values you care about. Arrows connect them, showing what affects what. A complex business decision with dozens of variables—market conditions, competitor moves, technology risks, customer preferences—becomes a single-page picture you can actually reason about.

Decision trees do similar work but grow exponentially with each new variable. Add five binary uncertainties and you have 32 branches. Add five more and you have 1,024. Influence diagrams avoid this explosion by representing structure rather than enumerating scenarios. They've become standard tools in decision analysis, risk assessment, and strategic planning.

Origins

Ronald Howard and James Matheson introduced influence diagrams in 1984, working at Stanford and the Strategic Decisions Group. Their goal: provide a higher-level representation than decision trees while maintaining the same analytical rigor.

The timing mattered. Computing power was expanding rapidly. Decision analysts wanted tools that could scale to real business complexity without drowning in combinatorial explosion. Influence diagrams offered a way to specify problems compactly while letting algorithms handle the computational expansion^[2].

The approach drew on probability theory, decision theory, and graph theory. It formalized what good analysts already did intuitively—focus on structure and dependencies before diving into numbers.

Node types

Influence diagrams use four distinct node types:

Decision nodes (rectangles)

These represent choices the decision maker controls. "Launch the product now or delay?" "Invest $10 million or $20 million?" "Hire internally or externally?"

Each decision node has a set of alternatives—the options available at that decision point. The diagram doesn't specify which alternative to choose; that emerges from analysis.

Multiple decision nodes can appear in sequence, representing multi-stage decisions. The ordering matters—some decisions happen before others, with information arriving between them^[3].

Chance nodes (ovals)

These represent uncertainties—things the decision maker doesn't control and doesn't know with certainty. "Will the economy grow or shrink?" "Will the competitor respond aggressively?" "Will the technology work?"

Each chance node has possible outcomes with associated probabilities. The probabilities may be unconditional (base rates) or conditional on other nodes (dependencies).

Value nodes (diamonds)

Also called utility nodes. These represent the objective—what the decision maker is trying to achieve. Typically, profit, net present value, or some utility function.

A standard influence diagram has one value node, appearing at the end. Complex models sometimes use intermediate value nodes that combine into a final objective.

Deterministic nodes (double ovals)

These represent quantities that are completely determined by their inputs—no uncertainty involved. If you know the inputs, you know the output exactly.

Example: Total cost equals fixed cost plus variable cost times quantity. Given the inputs, total cost follows deterministically.

Arc types

Arrows between nodes convey different meanings depending on what they connect:

Informational arcs (pointing into decision nodes) indicate what information is available when the decision is made. An arrow from a chance node to a decision node means the outcome of that uncertainty will be known before choosing.

Conditional arcs (pointing into chance nodes) indicate probabilistic dependence. An arrow from Node A to chance node B means that B's probability distribution depends on A's outcome^[4].

Functional arcs (pointing into value or deterministic nodes) indicate that the target node's value is calculated from the source nodes.

The absence of arrows is equally informative—it means no direct relationship. Influence diagrams make independence assumptions explicit.

Reading an influence diagram

Consider a simple pharmaceutical R&D decision:

• Decision: Proceed with clinical trials (yes/no)?
• Uncertainties: Will trials succeed? Will FDA approve? Will competitors launch first?
• Value: Expected net present value of the project

The diagram shows:

• An arrow from "trials decision" to "success" (the decision affects outcomes)
• An arrow from "success" to "FDA approval" (success influences approval chances)
• An arrow from "competitor launch" to "value" (competitive timing affects value)
• No arrow from "trials decision" to "competitor launch" (your decision doesn't influence competitor timing)

The structure reveals assumptions. If stakeholders disagree about whether your decision affects competitors, the disagreement becomes visible and debatable^[5].

Building an influence diagram

The construction process typically follows these steps:

1. Identify decisions. What choices does the decision maker face? List all relevant decision points.

2. Identify uncertainties. What don't you know that matters? What could go differently than expected?

3. Identify objectives. What are you trying to achieve? How will you measure success?

4. Draw nodes. Place decisions, uncertainties, and objectives on the diagram.

5. Add arcs. For each node, ask: "What directly influences this?" Draw arrows from influences.

6. Check structure. The diagram must be acyclic (no circular dependencies). Decisions must be ordered consistently with information flow.

7. Quantify. Add probabilities for chance nodes, payoffs for value nodes. This step can be deferred—the structure stands alone as a conceptual model^[6].

Solving influence diagrams

Once quantified, influence diagrams can be "solved" to find optimal strategies and expected values.

The standard algorithm works backward from the value node:

• For each chance node with no successors except value: compute expected value, replace with equivalent certain value
• For each decision node with no successors except value: find the alternative maximizing expected value
• Repeat until all nodes are processed

This procedure, called "arc reversal" or "node elimination," is equivalent to folding back a decision tree but operates directly on the compact diagram representation.

Software tools—TreeAge, Analytica, GeNIe, Netica—implement these algorithms, allowing analysts to focus on modeling rather than computation.

Advantages over decision trees

Compactness. A problem with 10 binary variables might require 1,024 tree branches but only 10 nodes in an influence diagram.

Structural clarity. Independence assumptions appear explicitly. Dependencies are visible rather than hidden in branch structure.

Flexibility. Changing a probability affects one node. Adding a new variable adds one node and a few arcs. In a tree, such changes might require restructuring the entire tree^[7].

Communication. Executives understand node diagrams more easily than sprawling trees. The high-level picture aids discussion.

Modularity. Submodels can be developed separately and connected. Large models decompose into manageable pieces.

Limitations

Acyclicity requirement. The diagram must be a directed acyclic graph. Some problems involve genuine feedback loops that don't fit this structure.

Single decision maker. Standard influence diagrams assume one coherent decision maker. Game-theoretic situations with multiple players require extensions.

Computational limits. Though more compact than trees, very large influence diagrams still challenge computation. Approximation methods become necessary.

Modeling skill. Building good diagrams requires training. Untrained users may miss important dependencies or include irrelevant ones.

Static snapshots. Basic influence diagrams represent a single point in time. Dynamic problems evolving over time require more sophisticated variants^[8].

Applications

Medical decision making. Should a patient undergo surgery? The diagram captures diagnostic test results, procedure risks, disease progression uncertainties, and quality-of-life outcomes.

Oil and gas exploration. Should the company drill? Geological uncertainty, commodity price risk, regulatory uncertainty, and development costs all enter the analysis.

Product launch timing. Launch now or wait? Market readiness, competitive moves, technology completion, and revenue projections interact.

Investment decisions. Build the factory? Economic conditions, demand forecasts, cost estimates, and strategic options combine.

Environmental policy. Implement regulation? Emission impacts, economic costs, health benefits, and political feasibility factor in^[9].

Extensions

Researchers have developed numerous extensions to basic influence diagrams:

Multi-objective influence diagrams handle multiple value nodes representing different objectives (profit, safety, environmental impact).

Dynamic influence diagrams represent problems evolving over multiple time periods.

Limited memory influence diagrams relax the assumption that decision makers remember all past observations.

Multi-agent influence diagrams handle situations with multiple decision makers having potentially conflicting objectives.

Continuous influence diagrams allow continuous variables rather than discrete alternatives.

Relationship to other methods

Influence diagrams connect to several related approaches:

Bayesian networks use the same graphical structure for chance nodes but exclude decision and value nodes—pure probabilistic reasoning without decision optimization.

Decision trees represent the same problems but enumerate scenarios explicitly. Trees and diagrams are mathematically equivalent; the difference is representational.

System dynamics models feedback loops and continuous time—different assumptions and applications but overlapping domains.

Monte Carlo simulation can implement influence diagram structures for numerical solution, especially when closed-form analysis is intractable^[10].

Influence diagram — recommended articles

Decision making — Risk management — Strategic planning — Project management

References

• Clemen R.T., Reilly T. (2013), Making Hard Decisions with DecisionTools, 3rd Edition, Cengage Learning.
• Howard R.A., Matheson J.E. (1984), Influence Diagrams, in Readings on the Principles and Applications of Decision Analysis, Strategic Decisions Group.
• Shachter R.D. (1986), Evaluating Influence Diagrams, Operations Research, Vol. 34, No. 6, pp. 871-882.
• Smith J.E., von Winterfeldt D. (2004), Decision Analysis in Management Science, Management Science, Vol. 50, No. 5.

Footnotes

1. ↑ Howard R.A., Matheson J.E. (1984), Influence Diagrams
2. ↑ Shachter R.D. (1986), Evaluating Influence Diagrams, pp.871-882
3. ↑ Clemen R.T., Reilly T. (2013), Making Hard Decisions, pp.234-256
4. ↑ Howard R.A., Matheson J.E. (1984), Influence Diagrams
5. ↑ Smith J.E., von Winterfeldt D. (2004), Decision Analysis in Management Science
6. ↑ Clemen R.T., Reilly T. (2013), Making Hard Decisions, pp.278-295
7. ↑ Shachter R.D. (1986), Evaluating Influence Diagrams, pp.875-878
8. ↑ Smith J.E., von Winterfeldt D. (2004), Decision Analysis in Management Science
9. ↑ Clemen R.T., Reilly T. (2013), Making Hard Decisions, pp.312-334
10. ↑ Howard R.A., Matheson J.E. (1984), Influence Diagrams

Author: Sławomir Wawak