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Standard lottery
Definition
A standard lottery is a reference gamble used in expected-utility assessment, typically offering a defined best outcome with probability p and a defined worst outcome with probability one minus p. By finding the probability at which a decision-maker is indifferent between that lottery and a certain outcome, the analyst can place the certain outcome on a utility scale. The term concerns preference elicitation under uncertainty, not a government lottery or a statistical sampling method.
What it means
Utility numbers represent preferences in a decision model. Fixing the best reference outcome at utility one and the worst at zero provides a convenient scale. If a person is indifferent between an outcome and a standard lottery with best-outcome probability p, expected-utility theory assigns that outcome utility p on the chosen scale. This is a statement about preferences under the model's assumptions. It is not the probability that the certain outcome will happen or an objective measurement of its monetary worth.
How it works
Specify reference outcomes that bound the relevant consequences and make their meaning clear. Present a certain outcome and a lottery between those references, then adjust p to identify an indifference point through a suitable elicitation process. Repeat for other outcomes and check the consistency and sensitivity of the responses. The interpretation relies on expected-utility assumptions, including appropriate consistency of preferences over lotteries. Framing effects, misunderstanding probabilities and changing preferences can affect the assessment, so the resulting numbers should not be treated as exact physical measurements.
Three practical examples
Example 1: A planning exercise compares project outcomes using clearly defined best and worst operational consequences. Example 2: An AI decision-support workshop elicits how a stakeholder trades a guaranteed moderate outcome against uncertain extremes. Example 3: A research demonstration explores why two people can assign different utilities to the same consequence. These examples concern preference modeling. They do not require participants to take the gamble, and the chosen references should be meaningful and appropriate to the decision context.
Think of it this way
Think of using two fixed marks on a ruler to create a scale, except that the positions are determined by choices under uncertainty rather than physical distance. A standard lottery provides the comparison that places an intermediate outcome on that scale. Changing the reference outcomes changes the ruler, so utility values should not be compared casually across unrelated elicitation exercises.
Worked example
Suppose the best reference outcome has utility one and the worst zero. A decision-maker is indifferent between certain outcome A and a lottery that gives the best outcome with probability 0.7, so U(A) = 0.7 on this scale. They are indifferent between certain outcome B and a corresponding lottery with probability 0.4, so U(B) = 0.4. Under expected utility and the stated consistency assumptions, a fifty-fifty lottery between A and B has utility 0.5 times 0.7 plus 0.5 times 0.4 = 0.55. It is therefore equivalent in this model to a standard lottery with best-outcome probability 0.55. If every utility is transformed by U new = 2 + 3U, the mixed lottery's utility becomes 3.65 and preference ordering is preserved. The numbers are stipulated preferences, not measured business returns.
Illustrative case study
In this fictional case study, Birch Planning compares service-expansion outcomes. Facilitators define the reference consequences carefully and ask stakeholders to consider several standard lotteries. Responses vary when descriptions are changed, so the team revisits wording and reports a range of plausible utilities rather than one supposedly exact value. The analysis distinguishes differences in preferences from disagreements about outcome probabilities. Managers use the exercise to understand trade-offs and test decision sensitivity, while recognizing that the model is an aid to deliberation.
Common misconceptions and mistakes
Common mistakes include interpreting utility 0.7 as a 70 percent chance that the certain outcome occurs, equating utility with money, and assuming everyone shares the same utility scale. Another error is averaging utilities across people without an explicit basis for interpersonal comparison. Expected-utility representation is preserved by positive affine transformations, not by every increasing transformation when lotteries are involved. Elicited preferences can also depend on the chosen reference outcomes, context and how clearly the uncertain alternatives are understood.
Business use
Businesses may use standard lotteries to make preference assumptions visible in decision analysis, especially when outcomes combine several consequences. The process requires care with framing, reference selection and stakeholder interpretation. Keep preference elicitation separate from estimating probabilities of real outcomes: one concerns what is valued, the other what may happen. Sensitivity analysis can show whether a recommendation changes across plausible utility assessments. A result that depends on a narrowly estimated indifference point should be presented with that dependence clearly explained.
People also ask
Does a utility of 0.4 mean a forty-percent likelihood of receiving that outcome?
No; it is an elicited preference value on the reference scale.
Must utility equal financial value?
No.
Can different people give different indifference probabilities?
Yes.
Does the method prove that people always obey expected-utility axioms?
No.
Can the same preferences be represented on a scale other than zero to one?
Yes, through an appropriate positive affine transformation.
Check your understanding
What utility is assigned to A in the example?
Answer: 0.7.
What is the expected utility of the fifty-fifty A/B lottery?
Answer: 0.55.
What does that value represent?
Answer: A preference comparison under the specified expected-utility model.
Why must the best and worst reference outcomes be stated?
Answer: They define the scale and the alternatives used in the indifference assessment.
Related terms
- Certainty equivalent
- Related terms include expected utility
- utility elicitation
- indifference
- risk attitude
- preference axioms and lottery
Learn next
Learn next how expected-utility axioms justify probability-weighted utility calculations and why monetary expectation differs from expected utility. Examine sensitivity to reference outcomes and framing. Then distinguish individual preference models from collective decision rules, where combining different stakeholders' values requires additional assumptions and explicit choices.
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Sources
Sources: Russell and Norvig, Artificial Intelligence: A Modern Approach, Chapter 16 author slides on utilities, https://aima.cs.berkeley.edu/slides-pdf/chapter16.pdf. This primary teaching source explains standard-lottery assessment using best and worst reference outcomes. The utility arithmetic and affine transformation are original and independently checked; the planning case is fictional.
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