Luce (1959), Chapter 3: Applications to Utility Theory [Luc59] #
Choices among gambles: aρb is "outcome a if chance event ρ occurs, else
b" (p. 78), and S(A,E) = (A × E × A) ∪ A collects gambles and pure
alternatives. A decomposable preference structure ⟨A, E, P, Q⟩ (Definition 5)
couples a choice function P over S(A,E) with an event choice function Q
via Axiom 2: P(aρb, aσb) = P(a,b)·Q(ρ,σ) + P(b,a)·Q(σ,ρ).
Main results #
DecomposablePreference: Definition 5 (p. 78), Axiom 2 in full mixture form;gam_of_alt_eq_oneis the reduced formP(aρb, aσb) = Q(ρ,σ)under perfect discriminationP(a,b) = 1, as used inside the proofs of Theorems 13–14.Complementation: Axiom 3 (p. 83),P(aρb, x) = P(bρ̄a, x), withEa Boolean algebra as Luce requires.NontrivialPreference: Axiom 4 (p. 83).q_compl_compl: Lemma 9 (p. 84),Q(ρ, σ) = Q(σ̄, ρ̄);v_mul_v_compl_constderives from it the constancy ofv(ρ)·v(ρ̄), the source of theφ(ρ)φ(ρ̄) = constantclause of §3.D.3 (p. 89).Neutral/Favorable/Unfavorable: comparison of an event to its own complement — Luce's classesC(½),C(1),C(0)(Lemma 11, p. 85).neutral_indifferentis the first clause of Lemma 10 (p. 84);favorable_gt_unfavorableis the between-class ordering.- The §3.B.2 chain:
lemma5(the cubic constraint, p. 80),eventPref_trans(Lemma 6),lemma7(p. 81), andatMostThreeClasses— Lemma 8/Theorem 10 as a four-event pigeonhole.theorem11(p. 82):Qis constant across∼-classes; with Lemma 9 it yields both clauses of Lemma 10 (neutral_indifferent,neutral_of_indiff_neutral, p. 84). theorem12(p. 84): given Axioms 3–5 (HasNeutralEventis Axiom 5) and nondegeneracy, exactly three classes — Luce'sC(1),C(½),C(0)(atLeastThreeClassesis Lemma 11, p. 85).theorem13/theorem14: the two §3.D theorems (pp. 86, 89) behind Luce's proposed experiment;gam_of_factoredis the observable consequence (p. 90) of the suggested local decompositionv(aρb) = w(a,b)·φ(ρ)of §3.D.3 — Luce offers the factoring as a hypothesis, not a theorem, and so does this file.
Formalization notes #
Luce's P and Q are families of probability measures on subsets of size ≤ 3
(Definition 5); here both are total ChoiceFns, which is where the diagonal
bites: binary x x = 1, so Axiom 2 needs its a ≠ b guard
(axiom2_unguarded_false) and Axiom 3 its two x-guards
(complementation_unguarded_false) — for Luce those instances are degenerate
singleton choices. Ratio scales enter as ChoiceFn.BinaryRatioScaleOn, local
to the gamble set under discussion, because Chapter 3 mixes imperfect
discrimination among gambles with perfect discrimination among pure
alternatives (P(a,b) = 1), which a globally positive scale cannot represent.
Axiom 1 lives in the structure (axiom1P/axiom1Q, in
ChoiceFn.HasChoiceAxiom's two-clause form) exactly as Definition 5 demands;
only the §3.C additional axioms and Theorem 10's nondegeneracy enter as
per-theorem hypotheses, as in the book. The three-class theorems are stated
representative-wise — among any four events two are equi-likely
(atMostThreeClasses); three pairwise non-equivalent events to one of which
every event is equivalent (theorem12) — avoiding quotient machinery; a
Setoid-packaged corollary is natural future work. The three-class regime
forces perfect discrimination Q ∈ {0, 1} between extreme classes, which is
exactly what HasChoiceAxiom's deletion clause (and no globally positive
ratio scale) tolerates.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Luce's total alternative set S(A,E) = (A × E × A) ∪ A (p. 78): gambles
together with the pure alternatives.
Equations
- Luce1959.Alternative A E = (Luce1959.Gamble A E ⊕ A)
Instances For
A decomposable preference structure ⟨A, E, P, Q⟩ (Definition 5, p. 78):
choice over gambles and pure alternatives (P), choice over events by
subjective likelihood (Q), coupled by Axiom 2:
P(aρb, aσb) = P(a,b)·Q(ρ,σ) + P(b,a)·Q(σ,ρ).
Deviations from Luce: P and Q are total ChoiceFns rather than
families on ≤3-element subsets (Axiom 1 enters as the axiom1P/axiom1Q
fields, in ChoiceFn.HasChoiceAxiom's two-clause form), and
Axiom 2 carries an a ≠ b guard — at a = b it is unsatisfiable for a
total P (axiom2_unguarded_false), and Luce's own uses all have
P(a,b) ∉ {0, 1} or P(a,b) = 1 with a, b a genuine pair.
- P : Core.ChoiceFn (Alternative A E)
Choice over
S(A,E). - Q : Core.ChoiceFn E
Choice over events by subjective likelihood.
- axiom2 (a b : A) : a ≠ b → ∀ (ρ σ : E), self.P.binary (Sum.inl { win := a, event := ρ, lose := b }) (Sum.inl { win := a, event := σ, lose := b }) = self.P.binary (Sum.inr a) (Sum.inr b) * self.Q.binary ρ σ + self.P.binary (Sum.inr b) (Sum.inr a) * self.Q.binary σ ρ
Axiom 2 (p. 78), in Luce's full mixture form.
- axiom1P : self.P.HasChoiceAxiom
Psatisfies Luce's Axiom 1, per Definition 5. - axiom1Q : self.Q.HasChoiceAxiom
Qsatisfies Luce's Axiom 1, per Definition 5.
Instances For
Without its a ≠ b guard, Axiom 2 is unsatisfiable for a total choice
function: at a = b it forces P(aρa, aσa) = Q(ρ,σ) + Q(σ,ρ) = 1 in both
argument orders, contradicting binary complementarity. In Luce's system
P(a, a) is the degenerate singleton choice.
Luce's P(a, b) for pure alternatives.
Instances For
Luce's P(g, h) for gambles.
Instances For
The reduced decomposition under perfect discrimination, as used inside the
proofs of Theorems 13–14 (p. 87): if P(a,b) = 1 then
P(aρb, aσb) = Q(ρ, σ).
Definition 6: the subjective likelihood order #
Definition 6 (p. 79): ρ ≿ σ iff Q(ρ, σ) ≥ ½ — ρ is deemed at least
as likely as σ.
Instances For
Subjective equi-likelihood ρ ∼ σ: the symmetric part of Definition 6's
≿. On distinct events this is Q(ρ, σ) = ½ (eventIndiff_iff_eq_half);
on the diagonal it holds since Q(ρ, ρ) = 1.
Equations
- dp.EventIndiff ρ σ = (dp.EventPref ρ σ ∧ dp.EventPref σ ρ)
Instances For
Totality of ≿ in strict form: a non-equi-likely pair is strictly
ordered one way or the other.
The nondegeneracy hypothesis of Theorem 10 (p. 80): some genuine pair
of alternatives is discriminated imperfectly and asymmetrically,
P(a, b) ∉ {0, ½, 1}.
Equations
- dp.Nondegenerate = ∃ (a : A) (b : A), a ≠ b ∧ dp.alt a b ≠ 0 ∧ dp.alt a b ≠ 1 / 2 ∧ dp.alt a b ≠ 1
Instances For
The three-class theorems (§3.B.2) #
Lemma 5 (p. 80), in denominator-cleared form: Luce's identity
(K+1){2[Q(ρ,σ)+Q(σ,τ)+Q(τ,ρ)] − 3} + K²[Q(ρ,σ)Q(σ,τ)Q(τ,ρ) − Q(ρ,τ)Q(τ,σ)Q(σ,ρ)] = 0, K = P(a,b)/P(b,a) − 1, multiplied through by
P(b,a)². From Axiom 2 and Theorem 2 for the gamble triple
{aρb, aσb, aτb}, whose pairwise discrimination is imperfect whenever
P(a,b) ∉ {0, 1}.
Lemma 6 (p. 80): ≿ is transitive (with gt_half_or_of_not_eventIndiff
totality, a weak ordering of E).
∼ is transitive: with eventIndiff_refl and EventIndiff.symm, an
equivalence relation (the content of Theorem 10's first clause).
Lemma 7 (p. 81): three distinct events, pairwise imperfectly
discriminated, cannot lie in three distinct ∼-classes.
Lemma 8 (p. 81) / the partition clause of Theorem 10 (p. 80): in
a nondegenerate decomposable preference structure, ∼ partitions the
events into at most three classes — among any four events, two are
subjectively equi-likely.
Theorem 11 (p. 82): Q is constant across ∼-classes — if ρ ∼ ρ'
and σ ∼ σ' then Q(ρ, σ) = Q(ρ', σ'). Both comparisons must be genuine
pairs: at ρ' = σ' the total-ChoiceFn diagonal Q(ρ', ρ') = 1 breaks
the unguarded claim, which Luce's P(x, x) = ½ convention (p. 5) hides.
Theorem 13 (p. 86): if P(a,b) = P(c,d) = 1 and "all pairwise
discriminations in the set T = {aρb, aσb, cρd, cσd} are imperfect",
then P(aρb, cρd) = P(aσb, cσd) — the step-function prediction of §3.D.
The local ratio scale of Luce's proof is supplied by Theorem 3
(ChoiceFn.HasChoiceAxiom.binaryRatioScaleOn).
Axiom 3 (p. 83): "P(aρb, x) = P(bρ̄a, x), where ρ̄ denotes the
complement of ρ" — aρb and bρ̄a are the same prospect relabeled.
The two guards exclude x ∈ {aρb, bρ̄a}: for Luce those instances are
degenerate singleton choices, and over a total ChoiceFn the unguarded
axiom is unsatisfiable (complementation_unguarded_false).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Without its guards, Axiom 3 is unsatisfiable for a total choice function:
x := bρ̄a forces P(aρb, bρ̄a) = 1, and symmetrically
P(bρ̄a, aρb) = 1, contradicting binary complementarity.
Axiom 4 (p. 83): some pair of alternatives and some pair of events are
discriminated away from ½. Distinctness is explicit: Luce's P(a*, b*)
presupposes a genuine pair (P(a, a) = 1 ≠ ½ would satisfy the inequality
degenerately and break the determinant step of Lemma 9).
Equations
- dp.NontrivialPreference = ((∃ (a : A) (b : A), a ≠ b ∧ dp.alt a b ≠ 1 / 2) ∧ ∃ (ρ : E) (σ : E), ρ ≠ σ ∧ dp.Q.binary ρ σ ≠ 1 / 2)
Instances For
Lemma 9 (p. 84): under Axioms 3–4, Q(ρ, σ) = Q(σ̄, ρ̄).
Under a global binary ratio scale for Q, Lemma 9 pins v(ρ)·v(ρ̄) to a
constant — the source of the φ(ρ)φ(ρ̄) = constant clause of the §3.D.3
decomposition (p. 89).
An event indifferent to its own complement: membership in Luce's class
C(½) (Lemma 11, p. 85).
Instances For
An event deemed more likely than its complement: Luce's class C(1).
Instances For
An event deemed less likely than its complement: Luce's class C(0).
Equations
- dp.Unfavorable ρ = (dp.Q.binary ρ ρᶜ < 1 / 2)
Instances For
Every event is unfavorable, neutral, or favorable.
An event is neutral iff its complement is.
An event is favorable iff its complement is unfavorable.
Two distinct neutral events are indifferent — the first clause of
Lemma 10 (p. 84), "if ρ ∼ ρ̄ and σ ∼ σ̄, then ρ ∼ σ", via Theorem 11
and Lemma 9. Distinctness is required: Q(ρ, ρ) = 1.
Favorable events are preferred to unfavorable ones: the between-class
ordering C(1) > C(0) of the three-class picture (§3.C.2, p. 85),
under a global ratio scale for Q.
Theorem 14 (p. 89): with Axiom 3, P(a,b) = P(d,c) = 1, and a local
ratio scale over the six gambles involved, the scale satisfies
v(aρb)·v(dρ̄c) = v(aσb)·v(dσ̄c). Together with Theorem 13 this is what
"suggests that v may be of the form v(aρb) = w(a,b)·φ(ρ)" (§3.D.3).
Axiom 5 (p. 84): some event is subjectively as likely as its complement.
Equations
- dp.HasNeutralEvent = ∃ (ε : E), dp.Q.binary ε εᶜ = 1 / 2
Instances For
The second clause of Lemma 10 (p. 84): anything equi-likely with a
neutral event is itself neutral — C(½) is exactly the neutral class.
Lemma 11 (p. 85): with Axioms 3–5 and nondegeneracy there are at least three classes — a neutral event, a non-neutral event, and its complement are pairwise non-equivalent.
Theorem 12 (p. 84): given Axioms 3–5 and a nondegenerately
discriminated pair of alternatives, ∼ partitions the events into
exactly three classes: three pairwise non-equivalent events to one of
which every event is equivalent.
The observable content of the §3.D.3 suggested factoring
v(aρb) = w(a,b)·φ(ρ) (pp. 89–90): between gambles on the same event
the event weight cancels, so binary choice follows the Luce rule on the
outcome weights alone — "the step function described in theorem 13 can
have only one step intermediate between 0 and 1" (p. 90). Luce offers the
factoring as a hypothesis consistent with Theorems 13–14, not a theorem;
accordingly it enters here as a hypothesis.