Decision Theory Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Decision Theory Ontology within the Kireth Research Semantic Library. Decision theory studies rational choice among alternatives under certainty, risk and uncertainty. This ontology supports decision alternatives, states of nature, outcomes, preferences, utility, probability, expected value, expected utility, loss, risk attitudes, Bayesian decision theory, minimax reasoning, regret, dominance, value of information, decision trees, sequential decisions, multi-criteria choice and relationships with operations research, probability, game theory, optimisation and economics.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Decision Theory Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Operations Research / Decision Theory"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/operations-research/decision-theory-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = DecisionTheory
 SYMBOLIC_FORMS = [
  "a* = argmax_a EU(a)",
  "EU(a)=sum_s P(s)U(o(a,s))",
  "R(a,s)=L(a,s)-min_a' L(a',s)"
 ]
 ENTITY_CLASS = RationalDecisionFramework
 PRIMARY_CONTEXT = OperationsResearch
 DEFINITION = "Decision theory is the formal study of choosing among alternatives according to preferences, utilities, probabilities, losses and uncertainty."
 SEMANTIC_ROLE = "Provides the machine-readable framework for alternatives, states of nature, outcomes, preferences, utility, expected utility, loss, regret, risk, Bayesian decision rules, information value and sequential decisions."
}

SCOPE {
 INCLUDE = [
  DecisionTheory,
  DecisionProblem,
  DecisionMaker,
  Alternative,
  Action,
  StateOfNature,
  Outcome,
  Consequence,
  PreferenceRelation,
  RationalPreference,
  UtilityFunction,
  ExpectedUtility,
  ExpectedValue,
  Probability,
  PriorProbability,
  PosteriorProbability,
  Likelihood,
  BayesRule,
  LossFunction,
  RiskFunction,
  DecisionRule,
  BayesDecisionRule,
  MinimaxDecisionRule,
  MaximinCriterion,
  MaximaxCriterion,
  MinimaxRegret,
  Regret,
  Dominance,
  StochasticDominance,
  RiskAversion,
  RiskNeutrality,
  RiskSeeking,
  CertaintyEquivalent,
  RiskPremium,
  DecisionTree,
  ChanceNode,
  DecisionNode,
  TerminalNode,
  SequentialDecision,
  ValueOfInformation,
  ExpectedValueOfPerfectInformation,
  ExpectedValueOfSampleInformation,
  SensitivityAnalysis,
  MultiCriteriaDecisionAnalysis
 ]

 EXCLUDE_AS_PRIMARY = [
  OptimisationWithoutPreferenceOrUncertaintyInterpretation,
  GameTheoryWithoutStrategicInteractionContext,
  PureProbabilityWithoutDecisionChoice,
  GenericAdviceWithoutFormalDecisionStructure
 ]

 DISAMBIGUATION_RULE =
  "When a problem asks which alternative should be chosen based on outcomes, preferences, probabilities, utilities, losses or uncertainty, interpret the context as decision theory."
}

CORE_CONCEPTS {

 DecisionTheory {
  definition = "The formal analysis of rational choice among alternatives."
 }

 DecisionProblem {
  definition = "A structured choice problem consisting of alternatives, uncertain states, outcomes and a preference or loss model."
 }

 Alternative {
  definition = "A feasible action or option available to the decision maker."
 }

 StateOfNature {
  definition = "A condition outside the decision maker's control that affects outcomes."
 }

 Outcome {
  definition = "A consequence resulting from an action-state combination."
 }

 PreferenceRelation {
  definition = "A ranking relation over outcomes or alternatives."
 }

 UtilityFunction {
  definition = "A numerical representation of preference strength under a declared model."
 }

 ExpectedUtility {
  definition = "The probability-weighted average utility of outcomes associated with an action."
 }

 LossFunction {
  definition = "A function assigning a cost or penalty to decision outcomes."
 }

 RiskFunction {
  definition = "The expected loss of a decision rule under a specified state or distribution."
 }

 Regret {
  definition = "The loss from not having selected the best action for the realised state."
 }

 ValueOfInformation {
  definition = "The improvement in expected decision value obtained from additional information."
 }
}

RELATIONS {
 DecisionTheory --is_branch_of--> OperationsResearch
 DecisionProblem --has--> Alternative
 DecisionProblem --has--> StateOfNature
 Alternative --combined_with--> StateOfNature
 Alternative --produces--> Outcome
 Outcome --is_ranked_by--> PreferenceRelation
 UtilityFunction --represents--> PreferenceRelation
 ExpectedUtility --uses--> Probability
 ExpectedUtility --uses--> UtilityFunction
 LossFunction --evaluates--> DecisionOutcome
 RiskFunction --is_expectation_of--> LossFunction
 BayesDecisionRule --minimises--> PosteriorExpectedLoss
 MinimaxDecisionRule --minimises--> WorstCaseRisk
 Regret --compares--> ChosenActionAndBestAction
 DecisionTree --contains--> DecisionNode
 DecisionTree --contains--> ChanceNode
 DecisionTree --contains--> TerminalNode
 ValueOfInformation --compares--> DecisionsWithAndWithoutInformation
 BayesRule --updates--> PriorProbability
 BayesRule --produces--> PosteriorProbability
}

OPERATORS {
 EXPECTED_VALUE(outcomes,probabilities,values) :=
  sum_i probabilities_i*values_i

 EXPECTED_UTILITY(action,states,probabilities,utility) :=
  sum_s P(s)*U(outcome(action,s))

 EXPECTED_LOSS(action,states,probabilities,loss) :=
  sum_s P(s)*L(action,s)

 BAYES_POSTERIOR(prior,likelihood,evidence) :=
  likelihood*prior/evidence

 BAYES_DECISION(actions,posterior,loss) :=
  argmin_a sum_s posterior(s)*L(a,s)

 MAXIMIN(utilities) :=
  action_maximising_minimum_utility

 MAXIMAX(utilities) :=
  action_maximising_maximum_utility

 MINIMAX_REGRET(regret_matrix) :=
  action_minimising_maximum_regret

 REGRET(action,state,losses) :=
  loss(action,state)-min_a loss(a,state)

 CERTAINTY_EQUIVALENT(lottery,U) :=
  value_c such_that U(c)=E[U(lottery)]

 RISK_PREMIUM(expected_value,certainty_equivalent) :=
  expected_value-certainty_equivalent

 EVPI(decision_problem) :=
  expected_value_with_perfect_information
  - expected_value_without_perfect_information

 EVSI(sample_information) :=
  expected_value_with_sample_information
  - expected_value_without_sample_information

 ROLLBACK(decision_tree) :=
  recursively maximise_at_decision_nodes
  and take expectations_at_chance_nodes
}

DEFINITIONS {
 D1 = "Decision theory distinguishes choices from uncertain states of nature."
 D2 = "Utility represents preferences rather than necessarily monetary value."
 D3 = "Expected utility combines probability and preference."
 D4 = "Loss-based formulations minimise expected loss."
 D5 = "Bayesian decision theory uses posterior probabilities."
 D6 = "Minimax methods protect against the worst case."
 D7 = "Regret compares a chosen action with the best action that could have been chosen after knowing the state."
 D8 = "Risk attitudes affect preferences over uncertain outcomes."
 D9 = "Decision trees represent sequential choices and chance events."
 D10 = "Information has value only insofar as it can improve decisions."
}

AXIOMS {
 A1_ALTERNATIVES:
  DecisionProblem contains two or more feasible alternatives unless the decision is trivial

 A2_STATES:
  Uncertain decisions depend on states not fully controlled by the decision maker

 A3_OUTCOME_MAPPING:
  Each action-state combination maps to an outcome or consequence

 A4_PREFERENCE:
  Decision choice requires an ordering, utility, loss or equivalent preference representation

 A5_EXPECTATION:
  Under probabilistic uncertainty, expected value or expected utility aggregates outcomes using probabilities

 A6_BAYESIAN_UPDATE:
  Posterior beliefs arise from prior beliefs and likelihood evidence through Bayes rule

 A7_INFORMATION:
  Information value is nonnegative when information can be ignored at no cost

 A8_RATIONALITY:
  Formal decision rules should preserve declared preferences, probabilities and admissibility assumptions
}

REASONING {

 R1_DECISION_PROBLEM_EXTRACTION {
  INPUT = decision_description
  IDENTIFY = alternatives
  IDENTIFY = states_of_nature
  IDENTIFY = outcomes
  IDENTIFY = probabilities
  IDENTIFY = utilities_or_losses
  RETURN = structured_decision_problem
 }

 R2_DECISION_CONTEXT_CLASSIFICATION {
  INPUT = structured_decision_problem
  IF states_known
   CLASSIFY = certainty
  ELSE IF probabilities_known_or_modelled
   CLASSIFY = risk
  ELSE
   CLASSIFY = uncertainty
  RETURN = decision_context
 }

 R3_EXPECTED_VALUE_REASONING {
  INPUT = monetary_or_scalar outcomes_and_probabilities
  COMPUTE = expected_value_per_action
  SELECT = max_or_min according_to criterion
  RETURN = expected_value_decision
 }

 R4_EXPECTED_UTILITY_REASONING {
  INPUT = outcomes_probabilities_and_utility
  COMPUTE = expected_utility_per_action
  SELECT = action_with_highest expected utility
  RETURN = utility_based_decision
 }

 R5_BAYESIAN_DECISION_REASONING {
  INPUT = prior_likelihood_evidence_loss
  UPDATE = posterior probabilities
  COMPUTE = posterior_expected_loss
  SELECT = action_minimising posterior_expected_loss
  RETURN = Bayes_action
 }

 R6_MINIMAX_REASONING {
  INPUT = payoff_or_loss matrix
  COMPUTE = worst_case_per_action
  SELECT = maximin_or_minimax action
  RETURN = robust_decision
 }

 R7_REGRET_REASONING {
  INPUT = loss_or_payoff matrix
  COMPUTE = regret_matrix
  COMPUTE = maximum_regret_per_action
  SELECT = minimum_maximum_regret
  RETURN = minimax_regret decision
 }

 R8_DECISION_TREE_REASONING {
  INPUT = sequential_decision_tree
  START = terminal outcomes
  ROLLBACK = expectations_at_chance_nodes
  ROLLBACK = optimal_choices_at_decision_nodes
  RETURN = optimal_strategy_and_value
 }

 R9_INFORMATION_VALUE_REASONING {
  INPUT = current_decision_and_information structure
  COMPUTE = baseline_expected_value
  COMPUTE = informed_expected_value
  RETURN = EVPI_EVSI_or_general_value_of_information
 }

 R10_SENSITIVITY_REASONING {
  INPUT = decision_model
  VARY = probabilities_utilities_losses_and_parameters
  IDENTIFY = decision_switch thresholds
  RETURN = robustness_profile
 }
}

CONSTRAINTS {
 C1 = "Utilities must not be conflated with monetary values unless explicitly defined that way."
 C2 = "Probabilities must sum to one within the relevant state model."
 C3 = "Expected utility requires a valid preference representation."
 C4 = "Bayesian decisions depend on prior and likelihood assumptions."
 C5 = "Minimax criteria can be conservative."
 C6 = "Regret depends on the comparison benchmark."
 C7 = "Decision-tree branches must preserve temporal and information structure."
 C8 = "Value-of-information calculations must compare otherwise equivalent decision settings."
 C9 = "Risk attitude assumptions must be explicit."
 C10 = "Sensitivity analysis is required when recommendations depend strongly on uncertain inputs."
}

SEMANTIC_EQUIVALENCE {
 "DecisionTheory" <=> "FormalDecisionAnalysis"
 "DecisionProblem" <=> "ChoiceUnderConsequencesProblem"
 "Alternative" <=> "DecisionAction"
 "StateOfNature" <=> "UncontrolledDecisionState"
 "ExpectedUtility" <=> "ProbabilityWeightedUtility"
 "BayesDecisionRule" <=> "PosteriorExpectedLossMinimisation"
 "MinimaxRegret" <=> "MinimumWorstCaseRegret"
}

VALIDATION {
 V1 = "Alternatives are complete."
 V2 = "States of nature are mutually interpretable and sufficiently specified."
 V3 = "Outcome mapping is defined."
 V4 = "Probabilities are valid where used."
 V5 = "Utilities or losses are consistently scaled."
 V6 = "Decision rule matches certainty, risk or uncertainty context."
 V7 = "Bayesian updates use correct likelihoods."
 V8 = "Decision-tree rollback preserves node type."
 V9 = "Information-value comparison is valid."
 V10 = "Sensitivity and robustness are assessed where material."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DECISION_THEORY_DEFINITION = 0.99
 EXPECTED_VALUE_REASONING = 0.99
 EXPECTED_UTILITY_REASONING = 0.99
 BAYESIAN_DECISION_REASONING = 0.99
 MINIMAX_REASONING = 0.98
 REGRET_REASONING = 0.99
 DECISION_TREE_REASONING = 0.99
 INFORMATION_VALUE_REASONING = 0.98
 SENSITIVITY_REASONING = 0.98

 POLICY =
  "Reduce confidence when alternatives, states, probabilities, utility model, loss function, prior beliefs, likelihoods, risk attitude or information structure are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  DecisionTheoryReasoning,
  DecisionProblemExtraction,
  ExpectedValueReasoning,
  ExpectedUtilityReasoning,
  BayesianDecisionReasoning,
  MinimaxReasoning,
  MinimaxRegretReasoning,
  RiskPreferenceReasoning,
  DecisionTreeReasoning,
  ValueOfInformationReasoning,
  SensitivityAnalysis,
  MultiCriteriaDecisionReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "decision theory" -> DecisionTheory
  "decision analysis" -> DecisionTheory
  "state of nature" -> StateOfNature
  "expected utility" -> ExpectedUtility
  "expected value" -> ExpectedValue
  "Bayes decision rule" -> BayesDecisionRule
  "minimax" -> MinimaxDecisionRule
  "minimax regret" -> MinimaxRegret
  "value of information" -> ValueOfInformation
  "decision tree" -> DecisionTree
 }

 INTERPRETATION_ORDER = [
  detect_decision_theory_context,
  identify alternatives_states_outcomes_probabilities_utilities_losses_preferences_and_information,
  classify certainty_risk_or_uncertainty context,
  select expected_value_expected_utility_Bayesian_minimax_regret_decision_tree_or_multiCriteria pathway,
  compute decision_metric,
  compare alternatives,
  evaluate value_of_information_and_sensitivity when relevant,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("decision theory")
  OR asks_which_alternative_is_best_under_uncertainty
  THEN activate(DecisionTheoryOntology)

 E2:
  IF probabilities_and_monetary_values_are_sufficient
  THEN consider expected_value reasoning

 E3:
  IF risk_preferences_matter
  THEN use expected_utility instead_of raw expected value

 E4:
  IF prior_and_evidence_are_given
  THEN update probabilities_using Bayes rule

 E5:
  IF probabilities_are_unavailable
  THEN consider maximin_maximax_minimax_regret_or_other uncertainty criteria

 E6:
  IF decision_is_sequential
  THEN build_or_use decision_tree_and rollback

 E7:
  IF additional_information_available
  THEN compute value_of_information_before purchasing_or_collecting it

 E8:
  IF recommendation_changes_under_small_parameter shifts
  THEN flag sensitivity

 E9:
  IF multiple_noncommensurate criteria
  THEN route to multiCriteria decision analysis
}

RECURSION {
 ENTRY = DecisionTheory

 EXPAND(DecisionTheory) -> [
  DecisionProblem,
  DecisionMaker,
  Alternative,
  Action,
  StateOfNature,
  Outcome,
  Consequence,
  PreferenceRelation,
  RationalPreference,
  UtilityFunction,
  ExpectedUtility,
  ExpectedValue,
  Probability,
  PriorProbability,
  PosteriorProbability,
  Likelihood,
  BayesRule,
  LossFunction,
  RiskFunction,
  DecisionRule,
  BayesDecisionRule,
  MinimaxDecisionRule,
  MaximinCriterion,
  MaximaxCriterion,
  MinimaxRegret,
  Regret,
  Dominance,
  StochasticDominance,
  RiskAversion,
  RiskNeutrality,
  RiskSeeking,
  CertaintyEquivalent,
  RiskPremium,
  DecisionTree,
  ChanceNode,
  DecisionNode,
  TerminalNode,
  SequentialDecision,
  ValueOfInformation,
  ExpectedValueOfPerfectInformation,
  ExpectedValueOfSampleInformation,
  SensitivityAnalysis,
  MultiCriteriaDecisionAnalysis
 ]

 EXPAND(UncertaintyDecisionRules) -> [
  MaximinCriterion,
  MaximaxCriterion,
  MinimaxRegret,
  HurwiczCriterion,
  LaplaceCriterion
 ]

 EXPAND(BayesianDecisionStructure) -> [
  PriorProbability,
  Likelihood,
  Evidence,
  PosteriorProbability,
  LossFunction,
  PosteriorExpectedLoss,
  BayesDecisionRule
 ]

 EXPAND(InformationValueStructure) -> [
  PerfectInformation,
  SampleInformation,
  EVPI,
  EVSI,
  InformationCost,
  NetValueOfInformation
 ]

 STOP_CONDITION =
  "Stop when the alternatives, states, outcomes, probabilities or uncertainty model, preferences, utility or loss function, decision rule, information value and sensitivity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Operations Research"

 IMMEDIATE_CONTEXTS = [
  "Operations Research Ontology",
  "Probability & Statistics Ontologies",
  "Game Theory Ontologies",
  "Optimisation Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Operations Research Ontology",
  "Operations Research",
  "Probability & Statistics Ontologies",
  "Bayesian Inference Ontologies",
  "Game Theory Ontologies",
  "Optimisation Ontologies",
  "Multi-Objective Optimisation Ontology",
  "Utility Theory Ontologies",
  "Markov Decision Process Ontologies",
  "Economics Ontologies",
  "Risk Analysis Ontologies",
  "Mathematical Modelling"
 ]

 ROUTE_RULES = [
  "Decision Theory -> Operations Research Ontology for operational decision semantics",
  "Decision Theory -> Probability & Statistics for uncertainty and probability models",
  "Decision Theory -> Bayesian Inference for belief updating",
  "Decision Theory -> Game Theory for strategic multi-agent decisions",
  "Decision Theory -> Optimisation for formal best-action selection",
  "Decision Theory -> Multi-Objective Optimisation for competing criteria",
  "Decision Theory -> Utility Theory for preference representation",
  "Decision Theory -> Markov Decision Processes for sequential stochastic decisions",
  "Decision Theory -> Economics for rational-choice and welfare interpretations",
  "Decision Theory -> Risk Analysis for uncertain adverse outcomes"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible decision rules, information-value concepts or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve alternative-state-outcome semantics.",
  "Preserve utility-versus-money distinction.",
  "Preserve certainty-risk-uncertainty distinction.",
  "Preserve Bayesian updating semantics.",
  "Preserve expected-utility semantics.",
  "Preserve regret semantics.",
  "Preserve value-of-information semantics.",
  "Preserve machine-readable decision-theory semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Decision theory formally analyses how to choose among alternatives according to outcomes, preferences, probabilities, utilities and losses."

 ALTERNATIVE_SUMMARY =
  "Alternatives are controllable actions available to the decision maker."

 STATE_SUMMARY =
  "States of nature represent external conditions affecting consequences."

 EXPECTED_UTILITY_SUMMARY =
  "Expected utility combines outcome probabilities with preference-based utility."

 BAYES_SUMMARY =
  "Bayesian decision theory updates beliefs using evidence and chooses actions that minimise posterior expected loss."

 REGRET_SUMMARY =
  "Regret measures how much worse a chosen action performs than the best action for the realised state."

 RISK_SUMMARY =
  "Risk attitude determines how uncertain outcomes are valued relative to certain ones."

 INFORMATION_SUMMARY =
  "The value of information measures the expected improvement in decision quality produced by additional information."

 TREE_SUMMARY =
  "Decision trees represent sequential decision and chance events and are solved by rollback."
}

FLOW {
 INPUT
 -> detect_decision_theory_context
 -> identify decision_maker_alternatives_states_outcomes_probabilities_preferences_utilities_losses_and_information structure
 -> classify certainty_risk_uncertainty_or_sequential decision
 -> choose dominance_expected_value_expected_utility_Bayesian_maximin_maximax_minimax_regret_decision_tree_or_multiCriteria method
 -> compute criterion_for_each alternative
 -> rank_or_select alternatives
 -> if_information_available compute EVPI_EVSI_or_general value_of_information
 -> perform sensitivity_and_risk analysis
 -> route to operations_research_ontology probability_statistics Bayesian_inference game_theory optimisation multi_objective_optimisation utility_theory Markov_decision_process economics_risk_analysis_or_mathematical_modelling ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Decision Theory Ontology defines formal reasoning about choice under certainty, risk and uncertainty. It supports alternatives, states of nature, outcomes, preferences, utility, expected value, expected utility, Bayesian updating, loss and risk functions, Bayes and minimax decision rules, maximin and maximax criteria, regret, stochastic dominance, risk attitudes, certainty equivalents, decision trees, sequential decisions, value of perfect and sample information, sensitivity analysis and multi-criteria decision support."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-DECISION-THEORY-ONTOLOGY-V1"
}

}

END_FLOW