16.1 Combining Beliefs and Desires under Uncertainty
A rational choice maximizes expected utility: average the utility of each consequence weighted by its probability.
AI · DECISION THEORY
16 · Making Simple Decisions
Choose actions by combining beliefs about uncertain outcomes with preferences over those outcomes. Expected utility gives a consistent rule for comparing choices.
16.2 The Basis of Utility Theory
Preferences over lotteries can be represented by a utility scale when they satisfy consistency principles such as transitivity and independence.
16.3 Utility Functions
A utility function assigns a value to outcomes; only relative values matter, and positive affine transformations preserve the same choices.
16.4 Multiattribute Utility Functions
When outcomes have several dimensions, such as health and cost, a combined utility expresses the trade-offs among attributes.
16.5 Decision Networks
Decision networks extend Bayesian networks with decision and utility nodes, making dependencies among observations, actions, and outcomes explicit.
16.6 The Value of Information
Information is valuable when it can change an action. Its expected value is the improvement in utility after observing it, before paying for the observation.
16.7 Unknown Preferences
Preferences may be uncertain or incomplete. A decision maker can elicit trade-offs, compare robust choices, or defer a choice until more is known.
Key terms and worked example
- Expected utility
- For action a, EU(a) = ΣoP(o|a)U(o): multiply each consequence's utility by its probability, then add.
- Value of information
- Expected utility when choosing after an observation, minus the best expected utility before it. This is gross value, before information cost.
- Utility scale
- These illustrative utilities are negative costs. Adding a constant or multiplying all utilities by a positive constant preserves the preference ordering.
- Decision network
- A compact dependency model linking uncertain states, available actions, and utility consequences; the slider changes the prior belief.
At the default prior p = 0.30: waiting costs 100p = 30 units (EU = −30); treating costs 12 + 100(0.1p) = 15 units (EU = −15), so treat. With perfect diagnosis, act only in the illness case: expected utility = −22p = −6.6; gross information value = −6.6 − (−15) = 8.4 utils. The model assumes illness loss 100, fixed treatment cost 12, treatment leaves 10% of the prior risk, and diagnosis reveals illness status perfectly. These are teaching assumptions, not clinical advice.
Interactive decision network
Treat now or wait?
Illustrative utilities: untreated illness costs 100 units; treatment costs 12 and leaves 10% of the chosen prior illness risk.
Treatment leaves 10% of prior risk; utilities use negative costs.
Demo performance
The slider parameter p is evaluated against two actions and a fixed number of outcomes. One update computes both expected utilities, the best action, and a closed-form perfect-information value: O(1) time and O(1) auxiliary space. The bars and labels also have constant size. For an explicit decision table with A actions and O outcomes, exhaustive expected-utility comparison takes O(AO) time; this two-action toy model does not enumerate a larger table.