Lottery (decision theory)¶
A discrete probability distribution over mutually exclusive outcomes or states, treated as an object of preference and choice under risk and often evaluated by probability-weighted utility in expected-utility theory.
Core Idea¶
A lottery in decision theory is a discrete probability distribution over possible outcomes. The lottery is the risky option as a whole, not the random mechanism, ticket, or outcome that eventually occurs.
Expected-utility theory evaluates a lottery by assigning a utility to each outcome and taking the probability-weighted sum. It then models choice through preferences over lotteries, often with completeness and transitivity plus stronger axioms. Behavioral results can reject that model of ranking without changing what the lottery object is.
Structural Signature¶
Sig role-phrases:
- outcome set. Lists mutually exclusive possible consequences. Constitutive support. If altered: A vague uncertain prospect with no outcomes specified is not a formal lottery.
- probability assignment. Gives each outcome a nonnegative weight summing to one. Identity-bearing distribution. If altered: Utility weights are not probabilities.
- risky option. Treats the distribution itself as one available choice. Constitutive decision object. If altered: One realized outcome is not the lottery that preceded it.
- preference relation. Ranks lotteries for a decision maker. Diagnostic theory layer. If altered: A lottery exists before any rationality assumptions about ranking.
- utility evaluation. Maps outcomes to utilities and, under expected utility, averages them by probability. Conditional analytic rule. If altered: Not every decision theory evaluates by expected utility.
What It Is Not¶
- Gambling lottery. Is the term colloquial or a formal distribution?
- Realized outcome. Is the distribution being confused with its result?
- Utility function. Are values being confused with probabilities?
- Mixed strategy. Is randomness over actions rather than consequences the object?
Scope of Application¶
Use lottery for formal risky options whose outcome support and probability weights are explicit.
- Expected utility. Computes probability-weighted utility.
- Choice under risk. Ranks alternative distributions.
- Game theory. Represents random outcomes and mixed consequences.
- Behavioral economics. Tests departures from expected-utility predictions.
- Policy analysis. Compares probabilistic consequences.
Clarity¶
Calling any uncertainty a lottery loses the formal object. Probabilities must be attached to mutually exclusive outcomes, and the whole distribution must be one option.
Manages Complexity¶
The representation separates beliefs about chance from values assigned to consequences. This allows two people to agree on probabilities yet rank the same lottery differently because their utilities differ.
Abstract Reasoning¶
- List the mutually exclusive outcomes.
- Assign probabilities and verify they sum to one.
- Treat the distribution as one option among alternatives.
- If using expected utility, map outcomes to utilities separately.
- Compare rankings without assuming observed behavior satisfies every rationality axiom.
Knowledge Transfer¶
Probability-weighted consequence representation transfers to medicine and engineering, but choice among risky options distinguishes a decision lottery from generic uncertainty. The nearest stopping boundary is explicit: A probability distribution is closest: it becomes a decision lottery when its outcomes constitute one option whose preference can be compared with other options. The inclusion test remains: An object is a decision-theoretic lottery when it specifies a discrete probability distribution over outcomes and is treated as a risky option available for comparison. The structure no longer applies when the case exits when probabilities do not sum to one, outcomes are not the option's alternatives, or only the realized result remains.
Examples¶
Canonical¶
Drug A is represented by {cured .90, dead .10}; this full distribution is one lottery to compare with Drug B's different outcome probabilities.
Mapped back: outcome set → cured and dead; probability assignment → .90 and .10; risky option → Drug A; preference relation → compared with Drug B; utility evaluation → weighted outcome utilities.
Applied / In Practice¶
After Drug A is taken and the patient recovers, the observed recovery is one realized outcome, not the original lottery of possible consequences.
Mapped back: outcome set → collapsed after realization; probability assignment → no longer the prospective option; risky option → past choice; preference relation → not an outcome ranking; utility evaluation → realized utility only.
Structural Tensions¶
T1: probability vs. utility. Chance and value combine in expected utility but represent different judgments. Diagnostic: Is disagreement about likelihood or consequence value?
T2: normative axioms vs. descriptive behavior. Coherent preference models simplify analysis while people violate them systematically. Diagnostic: Is the model prescribing or predicting choice?
Structural–Framed Character¶
Description turns on outcome set, probability assignment, risky option, preference relation, utility evaluation. Skeletal core. One option is a normalized distribution over mutually exclusive consequences. Domain-bound accent. Risk, outcomes, probabilities, preferences, utility, and choice axioms define the decision object. Transfer remains bounded because Why not prime. Weighted alternatives are portable; this is a formal object in decision theory. The negative boundary is concrete: Any gambling ticket, random event, uncertain belief, probability estimate, realized outcome, mixed strategy, utility function, or stochastic process is not automatically a lottery in this sense. A decision-theory lottery is structural-formal as a probability distribution, while utilities and preference adequacy are agent-relative and empirical. Its character: a risky option represented by weighted alternative outcomes.
Structural Core vs. Domain Accent¶
Skeletal core. One option is a normalized distribution over mutually exclusive consequences.
Domain-bound accent. Risk, outcomes, probabilities, preferences, utility, and choice axioms define the decision object.
Why not prime. Weighted alternatives are portable; this is a formal object in decision theory.
Instantiates / Related Primes¶
This entry is a kind of Probability Distribution.
- Distribution. Probabilities allocate unit mass over outcomes.
- Preference. Decision makers rank whole lotteries.
- No strict parent is asserted.
Relationships to Other Abstractions¶
Current abstraction Lottery (decision theory) Domain-specific
Parents (1) — more general patterns this builds on
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Lottery (decision theory) is a kind of Probability Distribution Domain-specific
Lottery (decision theory) is a strict kind of Probability Distribution: its frozen identity entails the parent's defining structure while adding domain-specific restrictions.Every reviewed Lottery (decision theory) instance satisfies Probability Distribution because the child identity—A discrete probability distribution over mutually exclusive outcomes or states, treated as an object of preference and choice under risk and often evaluated by probability-weighted utility in expected-utility theory—entails the parent identity—The complete specification of how probability mass or density is spread over a random variable's possible values — a measure that, once compressed to a named parametric family, encodes shape, moments, tails, and a generative claim about the process producing the data. Probability Distribution can occur without the domain, mechanism, population, or boundary conditions that distinguish Lottery (decision theory).
Hierarchy paths (5) — routes to 3 parentless roots
- Lottery (decision theory) → Probability Distribution → Random Variable → Function (Mapping)
- Lottery (decision theory) → Probability Distribution → Probability → Measure → Set and Membership
- Lottery (decision theory) → Probability Distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Lottery (decision theory) → Probability Distribution → Random Variable → Probability → Measure → Set and Membership
- Lottery (decision theory) → Probability Distribution → Random Variable → Probability → Measure → Aggregation → Micro Macro Linkage
Neighborhood in Abstraction Space¶
Lottery (decision theory) sits in a moderately populated region (47th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Strategic Decision Biases & Mechanisms (29 abstractions)
Nearest neighbors
- Participation constraint (mechanism design) — 0.87
- Correlated equilibrium — 0.87
- Evaluation function — 0.86
- Evidential Decision Theory — 0.86
- Certainty Effect — 0.86
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Gambling lottery. Tell: Is the term colloquial or a formal distribution?
- Realized outcome. Tell: Is the distribution being confused with its result?
- Utility function. Tell: Are values being confused with probabilities?
- Mixed strategy. Tell: Is randomness over actions rather than consequences the object?
References¶
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Lottery_(decision_theory) (revision 1301118179).
- Preserved source candidate: https://link.springer.com/book/10.1007/978-94-017-5040-0
- Preserved source candidate: https://www.stanford.edu/~jdlevin/Econ%20202/Uncertainty.pdf
The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.