Approximate Value Functions

Exact methods assume you can list every state and store a value for each. For any real project, that assumption dies immediately.

Consider a modest project: a dozen work packages, each in one of several statuses, a budget position, an approvals queue, a handful of live risks. The number of distinct states — every combination at once — runs into the billions. This is the curse of dimensionality: state counts multiply as factors combine, and no table could ever hold them. Continuous quantities such as cost variance make it formally infinite. Approximate value functions are how sequential decision-making survives contact with reality.

1Value as a function, not a lookup

The idea is to stop treating value as a giant lookup table and start treating it as a function of a few meaningful features. Rather than storing a number for every conceivable project state, you describe each state by a compact set of indicators — schedule variance, cost variance, open high-severity risks, approvals outstanding — and fit a function that maps those features to an estimated value. Compute the value at a manageable set of sampled states, then let the function generalise to the vast majority you never explicitly visited.

Figure 1. Value at a few sampled states is computed directly; a fitted function fills in everywhere else. The estimate is no longer exact, but it is available for states you could never afford to enumerate.

2A spectrum of approximators

The techniques range from simple to sophisticated, trading transparency against flexibility:

Nearest neighbour and kernel smoothing estimate a new state's value from the nearby sampled states — intuitive and local.
Linear and simplex interpolation blend values across a grid of reference points.
Linear regression on features fits value as a weighted sum of your chosen indicators — compact, fast, and interpretable (you can read off which features drive value).
Neural network regression learns rich, non-linear value surfaces when the relationships are complex and you have enough data to fit them.

Key idea

Approximation trades exactness for reach. You give up a perfect value for every state in exchange for a usable estimate at any state — the only way sequential decisions become tractable on a project of real size.

3The feature choice is the model

The quality of an approximate value function lives or dies by its features. Choose indicators that genuinely capture what makes a project state better or worse, and a simple linear fit will serve you well. Choose poorly, and no amount of modelling sophistication rescues it. For risk practitioners this is oddly familiar territory: deciding which few metrics summarise project health is exactly the judgement behind any good dashboard.

What it means in practice

You will never have a value for every project state, and you don't need one. Identify the handful of features that actually characterise how healthy a project is, and estimate value as a function of those. Start simple and interpretable — a weighted combination of familiar variance and exposure metrics — before reaching for anything heavier. Your effort belongs in choosing the right features, not in chasing a more elaborate approximator on top of the wrong ones.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Approximate Value Functions. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to convert the subject into a governed decision, owned work, usable evidence and a reviewable outcome. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Use Approximate Value Functions as a decision instrument rather than an administrative form. The subject terms—value, function, feature, approximate, functions—need an explicit connection to the project objective, business value and stakeholder commitments. Before completing the artefact, write one sentence stating who will use it, what decision it supports and when that decision is required.

Apply a disciplined information model. Separate facts supported by evidence, forecasts derived from a method, assumptions awaiting validation, constraints that limit choice, risks that may occur, issues that already exist and actions assigned to people. Each material entry should have an owner, date, status and next review point. Where probability or impact scores are used, define the scale so different reviewers interpret it consistently.

A baseline is useful only when changes are visible. Give the artefact an identifier, version, approval state and effective date. Define which changes require reapproval, how superseded versions are retained and where supporting evidence is stored. During reviews, focus on exceptions, decisions and trends rather than reading every field aloud. Record the decision and rationale, not merely that a meeting occurred.

Close the loop beyond delivery. Confirm acceptance criteria, unresolved items, transferred responsibilities and operational ownership. Where benefits are expected, identify the outcome measure, baseline, target, observation period and owner who remains accountable after the project team disbands. Lessons should describe the condition, consequence and reusable action; a generic statement such as “communicate better” cannot improve the next project.

Step-by-step operating method

Clarify the decision. Name the outcome, sponsor, affected stakeholders and decision that this work must enable.
Set boundaries. Record scope, assumptions, constraints, dependencies, tolerances and escalation conditions.
Plan the evidence. Define deliverables, measures, owners, due dates and acceptance criteria before execution.
Control delivery. Compare actual performance with the baseline, assess changes and manage risks and issues explicitly.
Close the loop. Confirm acceptance, transfer ownership, capture lessons and track benefits beyond handover.

Completion and governance protocol

Start with a short drafting workshop involving the accountable owner and the people who hold the evidence. Complete high-consequence fields first: objective, scope, owner, baseline, acceptance, dependencies and escalation. Mark unknowns as assumptions or actions rather than hiding them behind vague prose. Circulate a review draft, resolve conflicting interpretations, baseline the approved version and place the next review date in an owned schedule.

Information type	Minimum useful content	Review test
Outcome	Observable change and intended recipient	Not merely a deliverable or activity
Measure	Definition, baseline, target, frequency and source	Two reviewers would calculate it the same way
Ownership	One accountable role plus contributors and approver	Authority matches responsibility
Uncertainty	Assumption, risk or issue with response and trigger	Status reflects current reality
Control	Version, approval, review date and change rule	Current baseline is identifiable

Common failure modes and recovery actions

1. Watch for

Producing a document with no named decision or accountable owner.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Mixing risks, current issues, assumptions and actions in one unstructured list.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Measuring activity or output while leaving the intended outcome undefined.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Accepting changes without evaluating effects on value, scope, schedule, cost and risk.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Closing the project at delivery even though benefit ownership has not transferred.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Which decision or commitment does this artefact support?
Who owns each action, risk, acceptance and post-project benefit?
What is the baseline and what variance triggers escalation?
Where is the evidence that the result was accepted and transferred?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Approximate Value Functions?