Utility Theory Ontology

flow "UtilityTheoryOntology" v4.0

# UTILITY THEORY ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/risk-decision-mathematics/utility-theory-ontology.html"
  title: "Utility Theory Ontology"
  domain: "Probability and Statistics"
  category: "Risk & Decision Mathematics"
  entity: UTILITY_THEORY
  entity_type: RISK_DECISION_ONTOLOGY

DEFINE:
  UTILITY_THEORY :=
    FORMAL_PREFERENCE_FRAMEWORK(
      OUTCOMES,
      PREFERENCES,
      UTILITY_FUNCTIONS,
      UNCERTAINTY,
      EXPECTED_UTILITY,
      RISK_ATTITUDES,
      DECISION_RULES
    )

PURPOSE:
  REPRESENT:
    PREFERENCE_ORDERING
    OUTCOME_VALUATION
    CHOICE_UNDER_CERTAINTY
    CHOICE_UNDER_UNCERTAINTY
    RISK_ATTITUDE
    TRADEOFFS

FOUNDATION:
  OUTCOME_SPACE: X
  OUTCOMES: x,y,z IN X
  PREFERENCE_RELATION: x >=_pref y
  STRICT_PREFERENCE: x >_pref y
  INDIFFERENCE: x ~_pref y
  UTILITY_FUNCTION: U(X) -> REALS
  REPRESENTATION:
    x >=_pref y IFF U(x) >= U(y)

ENTITIES:
  UTILITY: NUMERICAL_PREFERENCE_REPRESENTATION
  PREFERENCE: ORDERING_BETWEEN_OUTCOMES
  OUTCOME: CONSEQUENCE_OR_RESULT
  UTILITY_FUNCTION: MAPPING_FROM_OUTCOMES_TO_UTILITY
  EXPECTED_UTILITY: PROBABILITY_WEIGHTED_UTILITY
  MARGINAL_UTILITY: CHANGE_IN_UTILITY_PER_UNIT_CHANGE
  RISK_AVERSION: PREFERENCE_AGAINST_MEAN_PRESERVING_RISK
  CERTAINTY_EQUIVALENT: CERTAIN_OUTCOME_EQUIVALENT_TO_LOTTERY
  RISK_PREMIUM: EXPECTED_VALUE_MINUS_CERTAINTY_EQUIVALENT
  LOTTERY: PROBABILITY_DISTRIBUTION_OVER_OUTCOMES
  DECISION_RULE: ACTION_SELECTION_CRITERION

RELATIONSHIPS:
  OUTCOME -> HAS -> UTILITY
  PREFERENCE -> REPRESENTED_BY -> UTILITY_FUNCTION
  LOTTERY -> HAS -> EXPECTED_UTILITY
  UTILITY_CURVATURE -> RELATES_TO -> RISK_ATTITUDE
  EXPECTED_UTILITY -> SUPPORTS -> DECISION_SELECTION
  CERTAINTY_EQUIVALENT -> CONNECTS_TO -> RISK_PREMIUM

ORDINAL_UTILITY:
  property:
    PRESERVES_PREFERENCE_ORDER
  transformation:
    V(x) := f(U(x))
  requirement:
    f_STRICTLY_INCREASING

CARDINAL_UTILITY:
  context:
    EXPECTED_UTILITY_THEORY
  admissible_transformation:
    V(x) := aU(x)+b
  requirement:
    a > 0

PREFERENCE_COMPLETENESS:
  condition:
    x >=_pref y OR y >=_pref x OR_BOTH

PREFERENCE_TRANSITIVITY:
  IF:
    x >=_pref y
    AND:
      y >=_pref z
  THEN:
    x >=_pref z

LOTTERY:
  L := {(x_1,p_1),...,(x_n,p_n)}
  requirements:
    p_i >= 0
    SUM_i p_i = 1

EXPECTED_UTILITY:
  EU(L) := SUM_i p_i U(x_i)
  continuous:
    EU := INTEGRAL U(x)f_X(x)dx
  decision_rule:
    SELECT_OPTION_WITH_MAXIMUM_EXPECTED_UTILITY

VON_NEUMANN_MORGENSTERN_AXIOMS:
  commonly_include:
    - COMPLETENESS
    - TRANSITIVITY
    - CONTINUITY
    - INDEPENDENCE

INDEPENDENCE_AXIOM:
  IF:
    L_1 >=_pref L_2
  THEN:
    alpha*L_1+(1-alpha)*L_3
      >=_pref
    alpha*L_2+(1-alpha)*L_3
  for:
    alpha IN (0,1)

MARGINAL_UTILITY:
  definition:
    MU(x) := dU(x)/dx

DIMINISHING_MARGINAL_UTILITY:
  condition:
    U_double_prime(x) < 0
  connection:
    CONCAVE_UTILITY

RISK_NEUTRALITY:
  utility:
    LINEAR

RISK_AVERSION:
  utility:
    CONCAVE
  condition:
    U_double_prime(w) < 0
  Jensen_relation:
    U(E[W]) >= E[U(W)]

RISK_SEEKING:
  utility:
    CONVEX
  condition:
    U_double_prime(w) > 0

CERTAINTY_EQUIVALENT:
  CE:
    U(CE) = E[U(W)]

RISK_PREMIUM:
  RP := E[W]-CE

ARROW_PRATT_ABSOLUTE_RISK_AVERSION:
  A(w) := -U_double_prime(w)/U_prime(w)

ARROW_PRATT_RELATIVE_RISK_AVERSION:
  R(w) := -w*U_double_prime(w)/U_prime(w)

CARA:
  utility_example:
    U(w) := -exp(-a*w)
  parameter:
    a > 0

CRRA:
  utility:
    IF gamma != 1:
      U(w) := w^(1-gamma)/(1-gamma)
    IF gamma = 1:
      U(w) := log(w)
  domain:
    w > 0

LOG_UTILITY:
  U(w) := log(w)
  domain:
    w > 0
  risk_attitude:
    RISK_AVERSE

EXPONENTIAL_UTILITY:
  U(w) := -exp(-a*w)
  property:
    CONSTANT_ABSOLUTE_RISK_AVERSION

UTILITY_OF_MONEY:
  caution:
    MONEY != UTILITY
  implication:
    E[U(MONEY)] != U(E[MONEY])
    IN_GENERAL

EXPECTED_VALUE_CONNECTION:
  expected_value:
    E[X]
  expected_utility:
    E[U(X)]
  risk_neutral_case:
    LINEAR_U:
      EXPECTED_UTILITY_ORDERING_EQUIVALENT_TO_EXPECTED_VALUE_ORDERING
  rule:
    EXPECTED_VALUE != EXPECTED_UTILITY_BY_DEFAULT

DECISION_THEORY_CONNECTION:
  action:
    a
  uncertain_outcome:
    X_a
  expected_utility:
    EU(a) := E[U(X_a)]
  optimal_action:
    a_star := argmax_a EU(a)

RISK_ANALYSIS_CONNECTION:
  role:
    MAP_RISK_OUTCOMES_TO_DECISION_PREFERENCES
  caution:
    SAME_RISK_DISTRIBUTION_CAN_PRODUCE_DIFFERENT_DECISIONS_FOR_DIFFERENT_UTILITY_FUNCTIONS

LOSS_FUNCTION_CONNECTION:
  common_relation:
    U(theta,a) := -L(theta,a)
  rule:
    LOSS_MINIMISATION_CAN_BE_EXPRESSED_AS_UTILITY_MAXIMISATION

MULTI_ATTRIBUTE_UTILITY:
  attributes:
    x_1,...,x_m
  utility:
    U(x_1,...,x_m)
  purpose:
    REPRESENT_PREFERENCES_OVER_MULTIPLE_OUTCOME_DIMENSIONS

ADDITIVE_MULTI_ATTRIBUTE_UTILITY:
  form:
    U(x) := SUM_j w_j u_j(x_j)
  requirements:
    DEPEND_ON_PREFERENCE_INDEPENDENCE_ASSUMPTIONS
  caution:
    ADDITIVE_FORM_IS_NOT_UNIVERSALLY_VALID

MULTI_CRITERIA_CONNECTION:
  criteria:
    c_1,...,c_m
  utilities:
    u_1,...,u_m
  aggregate:
    U(a)
  caution:
    WEIGHTS_AND_SINGLE_ATTRIBUTE_UTILITIES_ENCODE_PREFERENCES

PARETO_CONNECTION:
  dominated_option:
    WORSE_OR_EQUAL_ON_ALL_RELEVANT_ATTRIBUTES
    AND:
      STRICTLY_WORSE_ON_AT_LEAST_ONE

UTILITY_ELICITATION:
  methods:
    - CERTAINTY_EQUIVALENT
    - STANDARD_GAMBLE
    - PAIRWISE_CHOICE
    - TRADEOFF_METHOD
    - DIRECT_RATING
  purpose:
    ESTIMATE_PREFERENCE_FUNCTION

STANDARD_GAMBLE:
  choice:
    CERTAIN_OUTCOME
    VS:
      LOTTERY_BETWEEN_BEST_AND_WORST_OUTCOMES
  indifference_probability:
    USED_TO_INFER_UTILITY_OF_CERTAIN_OUTCOME

BEHAVIOURAL_UTILITY_CONNECTION:
  observed_behaviour:
    MAY_VIOLATE_EXPECTED_UTILITY_AXIOMS
  examples:
    - ALLAIS_PARADOX
    - ELLSBERG_PARADOX
    - FRAMING_EFFECT
    - LOSS_AVERSION

PROSPECT_THEORY_CONNECTION:
  reference_point:
    r
  value_function:
    v(x-r)
  features:
    - REFERENCE_DEPENDENCE
    - LOSS_AVERSION
    - DIMINISHING_SENSITIVITY
    - PROBABILITY_WEIGHTING
  distinction:
    PROSPECT_THEORY != EXPECTED_UTILITY_THEORY

PROBABILITY_WEIGHTING:
  objective_probability:
    p
  decision_weight:
    w(p)
  distinction:
    DECISION_WEIGHT != PROBABILITY

STATE_DEPENDENT_UTILITY:
  utility:
    U(x,s)
  implication:
    VALUE_OF_OUTCOME_MAY_DEPEND_ON_STATE

TIME_PREFERENCE_CONNECTION:
  discounted_utility:
    SUM_t beta^t u(c_t)
  parameter:
    beta IN (0,1]

GAME_THEORY_CONNECTION:
  player:
    i
  utility:
    U_i(a_1,...,a_n)
  best_response:
    MAXIMISE_PLAYER_i_UTILITY_GIVEN_OTHER_ACTIONS

REINFORCEMENT_LEARNING_CONNECTION:
  reward:
    R_t
  return:
    G_t := SUM_k gamma^k R_(t+k)
  objective:
    MAXIMISE_EXPECTED_RETURN
  caution:
    REWARD_FUNCTION_NEED_NOT_EQUAL_HUMAN_UTILITY

VALUE_FUNCTION_CONNECTION:
  state_value:
    V_pi(s) := E_pi[G_t|S_t=s]
  action_value:
    Q_pi(s,a) := E_pi[G_t|S_t=s,A_t=a]

INFORMATION_VALUE_CONNECTION:
  value:
    EXPECTED_MAXIMUM_UTILITY_WITH_ADDITIONAL_INFORMATION
      -
    EXPECTED_MAXIMUM_UTILITY_WITHOUT_ADDITIONAL_INFORMATION

BELIEF_PREFERENCE_SEPARATION:
  beliefs:
    REPRESENTED_BY_PROBABILITIES
  preferences:
    REPRESENTED_BY_UTILITIES
  decision:
    COMBINES_BELIEFS_AND_PREFERENCES
  rule:
    PROBABILITY != UTILITY

APPLICATIONS:
  UTILITY_THEORY -> SUPPORTS -> DECISION_THEORY
  UTILITY_THEORY -> SUPPORTS -> RISK_ANALYSIS
  UTILITY_THEORY -> SUPPORTS -> ECONOMICS
  UTILITY_THEORY -> SUPPORTS -> FINANCE
  UTILITY_THEORY -> SUPPORTS -> GAME_THEORY
  UTILITY_THEORY -> SUPPORTS -> MULTI_CRITERIA_DECISION_ANALYSIS
  UTILITY_THEORY -> SUPPORTS -> REINFORCEMENT_LEARNING

MODEL_CONNECTIONS:
  UTILITY_THEORY -> CONNECTS_TO -> EXPECTED_VALUE
  UTILITY_THEORY -> CONNECTS_TO -> DECISION_THEORY
  UTILITY_THEORY -> CONNECTS_TO -> RISK_ANALYSIS
  UTILITY_THEORY -> CONNECTS_TO -> BAYESIAN_DECISION_THEORY
  UTILITY_THEORY -> CONNECTS_TO -> GAME_THEORY
  UTILITY_THEORY -> CONNECTS_TO -> OPTIMISATION
  UTILITY_THEORY -> CONNECTS_TO -> BEHAVIOURAL_DECISION_THEORY

SEMANTIC_EQUIVALENCE:
  "utility"
    ~= "numerical representation of preference"
    UNDER:
      UTILITY_REPRESENTATION_CONTEXT

  "expected utility"
    ~= "probability-weighted utility"

  "certainty equivalent"
    ~= "certain outcome indifferent to specified risky prospect"

  "risk premium"
    ~= "expected wealth minus certainty equivalent"
    UNDER:
      STANDARD_MONETARY_GAIN_CONVENTION

  EXCLUDE_EQUIVALENCE:
    UTILITY != MONEY
    UTILITY != PROBABILITY
    EXPECTED_UTILITY != EXPECTED_VALUE_BY_DEFAULT
    RISK_AVERSION != LOSS_AVERSION
    ORDINAL_UTILITY != CARDINAL_UTILITY
    DECISION_WEIGHT != OBJECTIVE_PROBABILITY
    REWARD != HUMAN_UTILITY_BY_DEFAULT
    PROSPECT_THEORY != EXPECTED_UTILITY_THEORY

VALIDATION_RULES:
  RULE_1:
    OUTCOME_SPACE_AND_PREFERENCE_RELATION_MUST_BE_DEFINED
  RULE_2:
    UTILITY_FUNCTION_MUST_PRESERVE_SPECIFIED_PREFERENCE_ORDER
  RULE_3:
    LOTTERY_PROBABILITIES_MUST_BE_VALID_AND_NORMALISED
  RULE_4:
    ORDINAL_AND_CARDINAL_UTILITY_INTERPRETATIONS_MUST_NOT_BE_CONFLATED
  RULE_5:
    EXPECTED_UTILITY_REPRESENTATION_REQUIRES_APPROPRIATE_AXIOMS_AND_ASSUMPTIONS
  RULE_6:
    RISK_ATTITUDE_INTERPRETATION_MUST_USE_APPROPRIATE_UTILITY_DOMAIN_AND_CURVATURE
  RULE_7:
    MULTI_ATTRIBUTE_AGGREGATION_REQUIRES_EXPLICIT_PREFERENCE_ASSUMPTIONS
  RULE_8:
    PROBABILITY_BELIEFS_AND_UTILITY_PREFERENCES_MUST_REMAIN_SEMANTICALLY_DISTINCT

CONFIDENCE:
  ontology_identity: 0.99
  preference_representation: 0.99
  expected_utility_structure: 0.99
  risk_attitude_structure: 0.99
  utility_elicitation_structure: 0.98
  decision_theory_relationship: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF:
    PREFERENCES_CHANGE
  THEN:
    UPDATE:
      UTILITY_FUNCTION
      RISK_ATTITUDE
      CERTAINTY_EQUIVALENTS
      RISK_PREMIA
    RECOMPUTE:
      EXPECTED_UTILITIES
      OPTIMAL_ACTIONS

  IF:
    PROBABILITY_BELIEFS_CHANGE
  THEN:
    UPDATE:
      EXPECTED_UTILITIES
      VALUE_OF_INFORMATION
    RECOMPUTE:
      DECISION_RULE

  IF:
    OUTCOME_SPACE_OR_ATTRIBUTES_CHANGE
  THEN:
    UPDATE:
      UTILITY_DOMAIN
      MULTI_ATTRIBUTE_STRUCTURE
    REASSESS:
      PREFERENCE_REPRESENTATION

  PRESERVE:
    OUTCOME_DEFINITIONS
    PREFERENCE_RELATION
    UTILITY_SCALE_TYPE
    UTILITY_FUNCTION_VERSION
    PROBABILITY_MODEL
    RISK_ATTITUDE_ASSUMPTIONS
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_OUTCOME_SPACE
    - IDENTIFY_PREFERENCE_RELATION
    - DETERMINE_ORDINAL_OR_EXPECTED_UTILITY_CONTEXT
    - VERIFY_UTILITY_REPRESENTATION
    - IDENTIFY_PROBABILITY_DISTRIBUTION_FOR_LOTTERIES
    - COMPUTE_EXPECTED_UTILITY
    - ASSESS_UTILITY_CURVATURE_AND_RISK_ATTITUDE
    - COMPUTE_CERTAINTY_EQUIVALENT_OR_RISK_PREMIUM_IF_REQUIRED
    - EVALUATE_MULTI_ATTRIBUTE_TRADEOFFS_IF_PRESENT
    - CONNECT_TO_DECISION_RULE
    - RETURN_UTILITY_THEORETIC_INTERPRETATION

  constraints:
    - PRESERVE_PREFERENCE_ORDER
    - PRESERVE_UTILITY_SCALE_INTERPRETATION
    - PRESERVE_PROBABILITY_NORMALISATION
    - DISTINGUISH_UTILITY_FROM_MONEY
    - DISTINGUISH_UTILITY_FROM_PROBABILITY
    - DISTINGUISH_EXPECTED_UTILITY_FROM_EXPECTED_VALUE
    - DISTINGUISH_RISK_AVERSION_FROM_LOSS_AVERSION
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "utility-theory"
  namespace: "probability-statistics/risk-decision-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - PREFERENCE_RELATION
    - UTILITY_FUNCTION
    - EXPECTED_UTILITY
    - RISK_ATTITUDE

END_FLOW