Luce (1959): Individual choice behavior #
This file formalizes Luce's theory of choice built on his choice axiom. Wherever discrimination is imperfect, the axiom makes choice from a finite set of alternatives the conditional probability of a ratio scale on them. The first chapter derives this scale and the semiorder of just noticeable differences it induces. The second applies it to psychophysics, deriving the logistic and power laws of discrimination, its incompatibility with Thurstone's discriminal processes, a response-bias account of signal detection that is exactly a logistic observer, and the probabilities of rankings. The third couples preference among gambles with the subjective likelihood of events and shows that imperfect discrimination among pure alternatives confines the events to at most three classes. The fourth builds learning models on response strengths and derives the asymptotic behavior of the beta model.
Implementation notes #
- A system of choice probabilities is a family of mathlib probability measures indexed by finite
sets, so part i of the choice axiom is conditioning (
ProbabilityTheory.cond). - Luce sets
P(a, a) = 1/2by convention, where hereP(a, a) = 1is the choice from a singleton, so the axioms and theorems of the third chapter carry guards against coincident arguments. - The third chapter's ratio scales are local to a set of gambles, since the chapter mixes imperfect discrimination among gambles with perfect discrimination among pure alternatives.
- The beta model of §4.G is a Markov kernel on the logarithm of the ratio of the response strengths.
- Thurstone's discriminal processes are random variables
U ω aon a probability space, and choice fromTis the random utility model on the processes ofT(ChoiceFn.ofRandomUtility). Luce's independence and densities enter Theorem 7 only through the null ties they imply.
TODO #
- Appendix 1's alternative forms of Axiom 1, stated for the rejection probabilities.
- The special case of §4.G.4 (equations 19–29 and Table 6) and its conjecture.
- Theorem 15 (ii) as printed has
(1 − A(−i))/(1 − B(−i))in its product, where (10) and (11) give(A(−i) − 1)/(1 − B(−i)), which is what the file proves. - The derivation of the gamma form on pp. 105–106 applies the independence-of-unit condition to
gᵢwithout rescaling the boundv_M, which read literally would forceγᵢ = 0, so the formβᵢ·v + γᵢis taken as given.
References #
§1.C–§1.E: The choice axiom and its ratio scale (pp. 5–24) #
Axiom 1 (p. 6) has two parts. Under imperfect discrimination, choice from nested sets composes multiplicatively, and an alternative never chosen over another may be deleted. On a set with imperfect discrimination throughout, the axiom yields a ratio scale unique up to its unit (Theorem 3, p. 23).
Pairwise choice under a ratio scale #
Under a ratio scale v the probability of choosing x over y is v x / (v x + v y). The lemmas
assume positivity only at the alternatives involved, so they apply to scales defined on a local set.
Under the ratio scale v, x is chosen over y with probability v x / (v x + v y).
Equations
- Luce1959.pairwiseProb v x y = v x / (v x + v y)
Instances For
A positive ratio scale is strongly stochastically transitive, P(x, z) ≥ max (P(x, y), P(y, z))
when P(x, y) ≥ 1/2 and P(y, z) ≥ 1/2 (Definition 2, p. 25).
Pairwise choice is the sigmoid of the difference of the log scale values (§2.A.2).
The ratio rule and conditional probability #
A positive scale v gives each alternative a of a finite set T the choice probability
v a / ∑ b ∈ T, v b (Theorem 3, p. 23). Luce remarks that under the choice axiom P_S "acts like a
conditional probability relative to P_T" (p. 24), and for the ratio rule this holds literally.
The ratio rule gives a the probability v a / ∑ b ∈ T, v b of being chosen from T, and 0
if a ∉ T.
Equations
- Luce1959.ratioProb v T a = if a ∈ T then v a / ∑ b ∈ T, v b else 0
Instances For
Within a set the odds of two alternatives are the ratio of their scale values (Lemma 3, p. 9).
The ratio rule does not depend on the unit of the scale (Theorem 3, p. 23).
Under a positive ratio scale choice from T is determined by pairwise choice,
P_T(x) = 1 / ∑ y ∈ T, P(y, x) / P(x, y) (Theorem 1, p. 16).
For a nonnegative scale v, choosing a from T by the ratio rule is conditioning on T the
measure with point masses v (p. 24).
Systems of choice probabilities and the forms of the choice axiom #
For each finite set T the choice probabilities P_T "form an ordinary probability measure on the
subsets of T" (p. 5). Part i of Axiom 1 makes choice from a subset the conditional probability
given it, and a ratio scale is a measure from which every P_T is obtained by conditioning.
Two measures that give the same choice from T by conditioning agree on T up to a constant
multiple (Theorem 3, p. 23).
A system of choice probabilities assigns to every finite set T a measure P_T concentrated on
T, the distribution of the alternative chosen from T, which is a probability measure when T is
nonempty (pp. 4–5).
- prob : Finset A → MeasureTheory.Measure A
The measure
P_Tof the alternative chosen fromT. - isProbabilityMeasure_prob (T : Finset A) : T.Nonempty → MeasureTheory.IsProbabilityMeasure (self.prob T)
P_Tis a probability measure whenTis nonempty (probability axioms (i) and (ii)). The alternative chosen from
Tlies inT.
Instances For
A system has a ratio scale if a measure μ with positive finite point masses gives every P_T
by conditioning on T (Theorem 3, p. 23).
Equations
- cf.HasRatioScale = ∃ (μ : MeasureTheory.Measure A), (∀ (a : A), μ {a} ≠ 0) ∧ (∀ (a : A), μ {a} ≠ ⊤) ∧ ∀ (T : Finset A), T.Nonempty → cf.prob T = μ[|↑T]
Instances For
A system satisfies the product rule if P_T(R) = P_S(R) P_T(S) whenever R ⊆ S ⊆ T, which is
part i of Axiom 1 without its hypothesis of imperfect discrimination (p. 6).
Equations
- cf.HasProductRule = ∀ (R S T : Finset A), R ⊆ S → S ⊆ T → (cf.prob T) ↑R = (cf.prob S) ↑R * (cf.prob T) ↑S
Instances For
A ratio scale satisfies the product rule, since conditioning on T and then on S ⊆ T is
conditioning on S.
Binary choice P(x, y) is the probability P_{x,y}(x) of choosing x from {x, y} (p. 5).
Instances For
Choice from the singleton {x} gives P(x, x) = 1, where Luce sets P(x, x) = 1/2 by
convention (p. 5).
A system satisfies the constant-ratio rule if every set containing a and b gives them the
odds of the pair, P_T(a) P(b, a) = P_T(b) P(a, b) (Lemma 3, p. 9).
Equations
- cf.HasPairwiseIIA = ∀ (T : Finset A) (a b : A), a ∈ T → b ∈ T → (cf.prob T).real {a} * cf.binary b a = (cf.prob T).real {b} * cf.binary a b
Instances For
A function v, positive on S, is a binary ratio scale on S if choice between distinct
elements of S follows P(x, y) = v x / (v x + v y).
Equations
- cf.BinaryRatioScaleOn S v = ((∀ x ∈ S, 0 < v x) ∧ ∀ x ∈ S, ∀ y ∈ S, x ≠ y → cf.binary x y = Luce1959.pairwiseProb v x y)
Instances For
Discrimination is imperfect throughout T if 0 < P(x, y) < 1 for all distinct x and y in
T (p. 6).
Equations
- cf.ImperfectOn T = ∀ x ∈ T, ∀ y ∈ T, x ≠ y → 0 < cf.binary x y ∧ cf.binary x y < 1
Instances For
A system satisfies Axiom 1 (p. 6) if P_T(R) = P_S(R) P_T(S) for R ⊆ S ⊆ T whenever
discrimination is imperfect throughout T, and P_T(S) = P_{T - {x}}(S - {x}) for S ⊆ T whenever
P(x, y) = 0 for some x and y in T.
- product_rule (T : Finset A) : cf.ImperfectOn T → ∀ (R S : Finset A), R ⊆ S → S ⊆ T → (cf.prob T) ↑R = (cf.prob S) ↑R * (cf.prob T) ↑S
Instances For
A ratio scale satisfies the constant-ratio rule.
A ratio scale is a binary ratio scale on every set, with v a = μ {a}.
A ratio scale satisfies Axiom 1, part ii vacuously since no discrimination is perfect.
When every probability of choice is positive, the constant-ratio rule gives the ratio scale
v x = P(x, x₀) / P(x₀, x) for a fixed alternative x₀ (p. 24).
The ratio form and the constant-ratio rule are equivalent when every probability of choice is positive.
Under Axiom 1 an alternative never chosen over another is never chosen at all (Lemma 1, p. 6).
Under Axiom 1 and imperfect discrimination throughout T, every alternative in T is chosen
from T with positive probability.
Under Axiom 1 and imperfect discrimination throughout T, choice from a nonempty S ⊆ T is
choice from T conditioned on S (p. 24).
Under Axiom 1 and imperfect discrimination throughout T, P_T has positive point masses and
gives choice from every nonempty S ⊆ T by conditioning (Theorem 3, p. 23).
Under Axiom 1 and imperfect discrimination throughout T, v a = P_T(a) is a binary ratio
scale on T (Theorem 3, p. 23).
Under Axiom 1 and imperfect discrimination on {x, y, z}, a stochastic intransitivity is
exactly as probable as its reverse, P(x, y) P(y, z) P(z, x) = P(x, z) P(z, y) P(y, x) (Theorem 2,
p. 16).
§1.G: Just noticeable differences and the trace (pp. 34–37) #
A cutoff π ∈ (1/2, 1) defines the relations L(π), at least one π-jnd larger, and I(π), at
most one π-jnd apart (Definition 3, p. 34). Under a ratio scale they form a semiorder (Theorem 5),
and the trace, which orders alternatives by how well they fare against every other, is the weak
order of the scale (Theorem 6).
x L(π) y holds when P(x, y) > π, so that x is at least one π-jnd larger than y
(Definition 3, p. 34).
Equations
- Luce1959.jndL v thr x y = (thr < Luce1959.pairwiseProb v x y)
Instances For
x I(π) y holds when 1 - π ≤ P(x, y) ≤ π, so that x and y are at most one π-jnd apart
(Definition 3, p. 34).
Equations
- Luce1959.jndI v thr x y = (1 - thr ≤ Luce1959.pairwiseProb v x y ∧ Luce1959.pairwiseProb v x y ≤ thr)
Instances For
Under a positive ratio scale x L(π) y holds exactly when v x > π / (1 - π) · v y (proof of
Theorem 5, p. 35).
Exactly one of x L y, y L x and x I y holds, which is the first semiorder axiom (Theorem
5, p. 35).
Every alternative bears I to itself, which is the second semiorder axiom (Theorem 5, p. 35).
x L y, y I z and z L w imply x L w, which is the third semiorder axiom (Theorem 5, p.
35).
x L y and y L z exclude x I w together with w I z, which is the fourth semiorder axiom
(Theorem 5, p. 35).
In the trace x ≥ y holds when P(x, z) ≥ P(y, z) for every z (Definition 4, p. 37).
Equations
- Luce1959.traceGe v x y = ∀ (z : A), Luce1959.pairwiseProb v y z ≤ Luce1959.pairwiseProb v x z
Instances For
Under a positive ratio scale the trace is the order of the scale (proof of Theorem 6, p. 37).
Under a positive ratio scale the trace is a weak order (Theorem 6, p. 37).
Under a positive ratio scale x ≥ y in the trace exactly when P(x, y) ≥ 1/2 (corollary to
Theorem 6, p. 37).
§2.A.3: Uniqueness of the logistic curve (pp. 41–42) #
If pairwise choice depends only on differences of a scale u that takes every real value, the ratio
scale is exponential in u and pairwise choice is logistic. Luce reproduces Adams and Messick's
proof, which reduces the claim to Cauchy's functional equation.
If v = g ∘ u for a positive strictly increasing g, u takes every real value, and pairwise
choice is a function of u x - u y, then v x = g 0 · exp (k · u x) for some k > 0 and pairwise
choice is the sigmoid of k (u x - u y) (§2.A.3, pp. 41–42). Luce derives the form v = g ∘ u from
the monotonicity of pairwise choice.
§2.B: The power law (pp. 42–44) #
On prothetic continua Luce assumes the linear generalization of Weber's law, under which
P(x, y) = π exactly when x = (1 + c(π)) y + d(π), and finds the power scale A (x + C)^B (p.
43). He checks by substitution that the power scale solves the equation and cites Luce and Edwards
for uniqueness, which follows here from the uniqueness of the logistic curve in the coordinate
log (x + C).
The power scale is v(x) = A (x + C)^B (p. 43).
Equations
- Luce1959.powerScale A C B x = A * (x + C) ^ B
Instances For
On the power scale P(x, y) = 1 / (1 + ((y + C) / (x + C))^B) (p. 44).
On the power scale P(x, y) = π exactly when x + C = (π / (1 - π))^(1/B) (y + C).
With 1 + c = (π / (1 - π))^(1/B), the power scale satisfies the linear generalization of
Weber's law, P(x, y) = π exactly when x = (1 + c) y + c C, and
B = (log π - log (1 - π)) / log (1 + c) (p. 43).
A scale that is positive and strictly increasing above -C and makes pairwise choice a function
of (x + C) / (y + C) is a power scale, the uniqueness that Luce attributes to Luce and Edwards (p.
43).
On the power scale x L(π) y holds exactly when x + C > (π / (1 - π))^(1/B) (y + C), which is
the linear generalization of Weber's law as an inequality.
§2.C: Interaction of continua (pp. 47–51) #
With an intensity x and a second variable ξ, Weber's law on each continuum makes the scale a
power function of each variable with the other held fixed, and Luce assumes that the two surfaces
coincide (p. 49). He reaches the form (8) by differentiating twice. Since the equation is affine in
log x and log ξ, two of its values suffice here, with no assumption of differentiability.
If A(ξ) x^B(ξ) = A*(x) ξ^B*(x) for all positive x and ξ, then
A(ξ) x^B(ξ) = K ξ^B* x^(B + C log ξ) for constants K > 0, B, B* and C (equation (8), p.
50).
At a fixed ξ the scale of equation (8) is the power scale in x with exponent
B + C log ξ, so P(x, y; ξ) = 1 / (1 + (y / x)^(B + C log ξ)) (p. 50).
§2.D: Discriminal processes (pp. 54–58) #
In Thurstone's model each stimulus a produces a discriminal process U ω a, and the stimulus
whose process is the largest is judged the largest. Choice from T is then the random utility model
on the processes of T (ProbabilityTheory.choiceProb), Luce's integral (9) when the processes are
independent. Judging the smallest is judging the largest of the processes -U ω a, his (10). With
three stimuli the probabilities that x is judged the largest and the smallest differ by
P(x, y) + P(x, z) - 1, and Theorem 7 (p. 57) shows that the two judgments cannot both satisfy
Axiom 1.
The system of choice probabilities of discriminal processes U ω a whose ties have probability
zero. Choice from T is the random utility model on the processes of T, so P_T(a) is the
probability that the process of a is the largest of them, equation (9) for independent processes
(p. 56).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The probability that a is judged the largest from T is the probability that its process
exceeds the others of T.
Theorem 7 (p. 57). Let discriminal processes U with null ties make P_T(a) the
probability that a is judged the largest and P*_T(a) the probability that it is judged the
smallest, the largest of the processes -U. If discrimination is imperfect on {x, y, z} (i) and
P(x, y) + P(x, z) ≠ 1 (ii), then P and P* do not both satisfy Axiom 1 (iii). Luce's
hypothesis (iv), P*(x, y) = P(y, x), holds in every such model. He takes the processes independent
with densities, which makes ties null (Measure.pi_setOf_apply_eq_apply), and the stimuli to be
x, y and z alone, so that Axiom 1 concerns only {x, y, z} and its subsets.
§2.E: Signal detectability theory (pp. 58–64) #
In the Yes-No experiment Luce applies the choice axiom to the two responses, with a signal parameter
α and a response bias v among the scale values (p. 60). He contrasts the response bias with the
criterion of signal detectability theory, yet the resulting model is exactly a logistic observer
with a criterion. The two-alternative forced choice is the Yes-No model with α² in place of α
(p. 62), since the subject makes "in effect, two Yes-No decisions", so it doubles the logistic
sensitivity, while the normal model multiplies its sensitivity by √2.
yesNoProb α v b is the probability of affirming the signal in the axiom-1 Yes-No model with
signal parameter α and response bias v, on signal trials (b = true) and noise trials
(b = false) (p. 61).
Equations
- Luce1959.yesNoProb α v true = α / (α + v)
- Luce1959.yesNoProb α v false = 1 / (1 + v)
Instances For
forcedChoiceProb α v b is the probability of choosing the first interval in the axiom-1
two-alternative forced choice, when the signal is in the first interval (b = true) or the second
(b = false) (p. 62).
Equations
- Luce1959.forcedChoiceProb α v true = α / (α + v)
- Luce1959.forcedChoiceProb α v false = 1 / (1 + α * v)
Instances For
In the axiom-1 Yes-No model the hit rate is α F / ((α - 1) F + 1) at the false-alarm rate F,
which is its receiver operating characteristic (p. 61).
The forced choice is the Yes-No model with signal parameter α² and response bias α v (p.
62).
The axiom-1 forced choice has the receiver operating characteristic of the Yes-No model with
α² in place of α (p. 62).
The axiom-1 Yes-No observer responds as the logistic observer at sensitivity log α and
criterion log v - log α / 2.
The axiom-1 forced choice responds as the logistic observer at sensitivity 2 log α and
criterion log v.
§2.F: Rank orderings (pp. 68–74) #
The ranking postulate (p. 72) ranks first an alternative chosen from the whole set and then ranks the rest by the same rule, so under a ratio scale the probability of a ranking is a product of choices from shrinking sets.
Under the ranking postulate with ratio scale v, a ranking has the probability that its first
alternative is chosen from the whole ranking by the ratio rule, times the probability of the rest
(p. 72).
Equations
- Luce1959.rankProb v [] = 1
- Luce1959.rankProb v (a :: rest) = Luce1959.ratioProb v (a :: rest).toFinset a * Luce1959.rankProb v rest
Instances For
The rankings of T are the lists without repetition whose elements are those of T.
Equations
- Luce1959.allRankings T = T.val.toList.permutations.toFinset
Instances For
The distribution of the rankings of T under the ranking postulate puts the mass rankProb v r
on each ranking r of T.
Equations
- Luce1959.rankMeasure v T = ∑ r ∈ Luce1959.allRankings T, ENNReal.ofReal (Luce1959.rankProb v r) • MeasureTheory.Measure.dirac r
Instances For
Under a positive scale the rankings of T form a probability distribution.
Under the ranking postulate and a positive ratio scale, a ranking of T places x above y
with probability P(x, y) (Theorem 9, p. 72).
Let P* choose the worst alternative under Axiom 1 with P*(x, y) = P(y, x), and so with the
reciprocal scale. Ranking {x, y, z} from the top, P_T(x) P(y, z), and from the bottom,
P*_T(z) P(x, y), give x > y > z the same probability exactly when P(x, y) = P(y, z) (Theorem
8, p. 69).
§3.B–D: Decomposable preference structures (pp. 78–90) #
A gamble aρb has the outcome a if the event ρ occurs and b otherwise. A decomposable
preference structure couples preference among gambles and pure alternatives with the subjective
likelihood of events through the decomposition axiom (p. 78). If some pure alternatives are
discriminated imperfectly, the events fall into at most three classes of equal likelihood (Theorem
10, p. 80).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
The alternatives S(A, E) = (A × E × A) ∪ A are the gambles and the pure alternatives (p. 78).
Equations
- Luce1959.Alternative A E = (Luce1959.Gamble A E ⊕ A)
Instances For
A decomposable preference structure couples choice P among the alternatives with choice Q
among events by subjective likelihood, each satisfying Axiom 1, through Axiom 2 (Definition 5, p.
78).
- P : ChoiceFn (Alternative A E)
Choice among gambles and pure alternatives by preference.
- Q : ChoiceFn E
Choice among 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(aρb, aσb) = P(a, b) Q(ρ, σ) + P(b, a) Q(σ, ρ)fora ≠ b(p. 78). - axiom1P : self.P.HasChoiceAxiom
Psatisfies Axiom 1. - axiom1Q : self.Q.HasChoiceAxiom
Qsatisfies Axiom 1.
Instances For
Without its guard a ≠ b, Axiom 2 would force P(aρa, aσa) = P(aσa, aρa) = 1, since
P(a, a) = 1.
dp.alt a b is the choice probability P(a, b) between pure alternatives.
Instances For
dp.gam g h is the choice probability P(g, h) between gambles.
Instances For
If P(a, b) = 1 then P(aρb, aσb) = Q(ρ, σ), a step in the proofs of Theorems 13 and 14
(pp. 87, 89).
ρ ≿ σ holds when Q(ρ, σ) ≥ 1/2, so that ρ is deemed at least as likely as σ (Definition
6, p. 79).
Instances For
Events ρ and σ are equi-likely, ρ ∼ σ, when ρ ≿ σ and σ ≿ ρ.
Equations
- dp.EventIndiff ρ σ = (dp.EventPref ρ σ ∧ dp.EventPref σ ρ)
Instances For
Two events that are not equi-likely are strictly ordered.
A structure is nondegenerate if P(a, b) ≠ 0, 1/2, 1 for some distinct a and b, which is
the hypothesis of Theorem 10 (p. 80).
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
With K = P(a, b)/P(b, a) - 1,
(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)]
vanishes, here multiplied through by P(b, a)² (Lemma 5, p. 80).
In a nondegenerate structure ≿ is a weak order (Lemma 6, p. 80).
In a nondegenerate structure, of three distinct events among which discrimination is imperfect, two are equi-likely (Lemma 7, p. 81).
In a nondegenerate structure ∼ has at most three classes, so of any four events two are
equi-likely (Lemma 8, p. 81).
If some P(a, b) ≠ 0, 1/2, 1, equi-likelihood is an equivalence relation with at most three
classes (Theorem 10, p. 80).
In a nondegenerate structure ρ ∼ ρ' and σ ∼ σ' give Q(ρ, σ) = Q(ρ', σ') for ρ ≠ σ and
ρ' ≠ σ' (Theorem 11, p. 82).
If distinct events have Q(ρ, σ) = Q(σ, τ) = q and Q(ρ, τ) = 1, then any
P(a, b) ≠ 0, 1/2, 1 forces 3/4 < q < 1 and P(a, b) = (1 ± √(4q - 3) / (2q - 1)) / 2 (§3.C.2,
p. 85).
§3.C: Additional axioms (pp. 83–86) #
Axiom 3 identifies aρb with bρ̄a, Axiom 4 asks that neither all alternatives nor all events be
indifferent, and Axiom 5 posits an event as likely as its complement. Under them the classes are
exactly three, and the imperfect discriminations among pure alternatives share one probability.
Axiom 3 asks that P(aρb, x) = P(bρ̄a, x) for every x other than the two gambles (p. 83).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Without its guards Axiom 3 would force P(aρb, bρ̄a) = P(bρ̄a, aρb) = 1.
Axiom 4 asks for distinct a and b with P(a, b) ≠ 1/2 and distinct ρ and σ with
Q(ρ, σ) ≠ 1/2 (p. 83).
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
Under Axioms 3 and 4, Q(ρ, σ) = Q(σ̄, ρ̄) (Lemma 9, p. 84).
An event ρ is neutral if Q(ρ, ρ̄) = 1/2, and by Lemma 10 the neutral events form Luce's
class C(1/2) (p. 85).
Instances For
Distinct neutral events are equi-likely (Lemma 10, p. 84).
Axiom 5 posits an event as likely as its complement (p. 84).
Equations
- dp.HasNeutralEvent = ∃ (ε : E), dp.Q.binary ε εᶜ = 1 / 2
Instances For
An event equi-likely with a neutral event is neutral (Lemma 10, p. 84).
Under Axioms 3–5 a neutral event ε, a non-neutral event ρ and ρ̄ lie in three distinct
classes (Lemma 11, p. 85).
In a nondegenerate structure satisfying Axioms 3–5, ∼ has exactly three classes (Theorem 12,
p. 84).
Under the hypotheses of Theorem 12 the three classes have representatives ρ ≻ σ ≻ τ with
Q(ρ, σ) = Q(σ, τ) > 1/2 and Q(ρ, τ) = 1 (§3.C.2, p. 85).
Under the hypotheses of Theorem 12, every P(a, b) ≠ 0, 1/2, 1 equals
(1 ± √(4q - 3) / (2q - 1)) / 2 for one q with 3/4 < q < 1 (§3.C.2, p. 85).
§3.D: A proposed experiment (pp. 86–90) #
Luce's prediction concerns gambles whose outcomes are perfectly discriminated. When two gambles pay their preferred outcomes on the same event, choice between them is a step function of the event (Figure 7), and when they pay them on complementary events, the ratio scale obeys a product rule that suggests factoring it into outcome and event weights.
If P(a, b) = P(c, d) = 1 and discrimination among aρb, aσb, cρd and cσd is imperfect
throughout, then P(aρb, cρd) = P(aσb, cσd) (Theorem 13, p. 86).
Under Axiom 3, if P(a, b) = P(d, c) = 1, a ratio scale v on the gambles involved has
v(aρb) v(dρ̄c) = v(aσb) v(dσ̄c) (Theorem 14, p. 89). Luce takes the scale on aρb, aσb, cρd
and cσd from Theorem 4 and extends it by Axiom 3, where here it is given on all six gambles.
If the ratio scale factors as v(aρb) = w(a, b) φ(ρ), as Theorems 13 and 14 suggest, then
P(aρb, cρd) = w(a, b) / (w(a, b) + w(c, d)), so the step function of Theorem 13 has at most one
step strictly between 0 and 1 (pp. 89–90).
§4: Response-strength operators (pp. 93–106) #
A learning event transforms the vector of response strengths, and the choice probabilities follow from the strengths by the ratio rule. The alpha model takes the transformation to be a nonnegative matrix, the form Luce derives from the unboundedness, superposition and independence-of-unit conditions, the beta model multiplies each strength by a constant, and the gamma model adds a constant as well.
§4.C: The alpha model #
An operator M satisfies the proportional change assumption (p. 97) with constant a if it
multiplies the total strength of every positive vector by a.
Equations
- Luce1959.ProportionalChange M a = ∀ (v : A → ℝ), (∀ (i : A), 0 < v i) → ∑ i : A, M.mulVec v i = a * ∑ i : A, v i
Instances For
The proportional change assumption holds exactly when every column of M sums to a
(equation (1), p. 98).
Under the proportional change assumption the probability operator of the alpha model is
linear, with matrix aᵢⱼ / a (p. 97).
With two alternatives, indexed 0 and 1 for Luce's 1 and 2, the alpha model is the linear
operator P' = α P + (1 − α) λ of [BM55], with α = (a₁₁ − a₁₂)/a and
λ = a₁₂/(a₁₂ + a₂₁) (p. 99).
§4.D: The beta model #
An operator on a response strength that is independent of the unit is multiplication by its
value at 1 (p. 100).
In probability terms the beta operator multiplies each probability by its βᵢ and
renormalizes, P'(i) = βᵢ·P(i) / ∑ⱼ βⱼ·P(j) (p. 101).
The beta operators commute on the probabilities, as they do on the strengths (p. 101).
In the simple beta model only the chosen alternative i changes strength, βᵢ = β and
βⱼ = 1 for j ≠ i, so P'(i) = β·P(i) / (1 + (β − 1)·P(i)) and
P'(j) = P(j) / (1 + (β − 1)·P(i)) (p. 101).
With two alternatives the general beta model is the simple one with β = β₁/β₂
(p. 102).
§4.E: The gamma model #
The additive constant of the gamma model vᵢ ↦ βᵢ·vᵢ + γᵢ keeps its probability operator from
being a function of the probabilities, so path independence fails at the level of the
probabilities (p. 106).
§4.F: Partial reinforcement (pp. 107–110) #
In a partial reinforcement experiment the organism chooses between aρb and aρ̄b, with a a
reward and b nothing, and Theorem 14 makes the product of their strengths independent of the
event. An alpha or gamma operator that keeps this product fixed confines the strengths to a few
values unless it does not learn or reduces to a beta operator, and a beta operator needs only
β₁ β₂ = 1.
If P(a, b) = 1, the product of the strengths of aρb and aρ̄b is the same for every event,
by Theorem 14 with c = b and d = a (p. 108).
An alpha-model operator that keeps the product K of the two strengths fixed either confines
v(aρb) to at most four values, the roots of a quartic, or is the identity or the swap of the
two strengths, neither of which learns (pp. 108–109).
A beta-model operator that keeps the product of the two strengths fixed has
β₁·β₂ = 1, and is then the simple beta model with β = β₁² (p. 109).
A gamma-model operator vᵢ ↦ βᵢ·vᵢ + γᵢ that keeps the product K of the two strengths
fixed either confines v(aρb) to at most two values, the roots of a quadratic, or holds both
strengths constant, or is a beta-model operator with β₁ = 1/β₂ (p. 110).
§4.G: Asymptotic properties of the beta model (pp. 111–120) #
With two alternatives and two outcomes, the ratio vₙ of the response strengths is a Markov chain
that moves by one of four steps on the log scale. Three equations govern its expectations, the
moment recursion (10), the identity (11), and the drift (16) of the expected log ratio, and from
them Theorems 15–18 relate the limit of E(Pₙ) to the moments of vₙ and 1/vₙ. Each second part
is the first part for the model with the alternatives exchanged.
The beta model for two alternatives and two outcomes multiplies the ratio v = v(1)/v(2) of the
response strengths by β₁ⱼ when alternative 1 is chosen and outcome j occurs, and divides it by
β₂ⱼ when alternative 2 is chosen, where outcome 1 follows the choice of i with probability πᵢ
(p. 112).
- β₁₁ : ℝ
The multiplier of
vafter choice 1 and outcome 1. - β₁₂ : ℝ
The multiplier of
vafter choice 1 and outcome 2. - β₂₁ : ℝ
The divisor of
vafter choice 2 and outcome 1. - β₂₂ : ℝ
The divisor of
vafter choice 2 and outcome 2. - π₁ : ℝ
The probability of outcome 1 after choice 1.
- π₂ : ℝ
The probability of outcome 1 after choice 2.
- β₁₁_pos : 0 < self.β₁₁
- β₁₂_pos : 0 < self.β₁₂
- β₂₁_pos : 0 < self.β₂₁
- β₂₂_pos : 0 < self.β₂₂
- π₁_nonneg : 0 ≤ self.π₁
- π₁_le_one : self.π₁ ≤ 1
- π₂_nonneg : 0 ≤ self.π₂
- π₂_le_one : self.π₂ ≤ 1
Instances For
Each row of the kernel is the four-point distribution of (7).
The conditional expectation given the log ratio u averages over the four events of (7).
m.law μ₀ n is the distribution of the log ratio on trial n from the initial distribution
μ₀.
Instances For
When the translates of f are integrable, the expectation of f on trial n + 1 is the
expectation on trial n of its conditional expectation.
Finite moments E(v₀ᵗ) of every real order persist on every trial.
The constant A(k) of (9) is π₁β₁₁ᵏ + (1 − π₁)β₁₂ᵏ.
Instances For
The constant B(k) of (9) is π₂/β₂₁ᵏ + (1 − π₂)/β₂₂ᵏ.
Instances For
Given vₙ, the expectation of vₙ₊₁ᵏ is (A(k) − B(k))Pₙvₙᵏ + B(k)vₙᵏ (8).
The choice probability satisfies Pₙvₙ = vₙ − Pₙ (11).
m.swap is the model with the two alternatives exchanged, whose ratio of strengths is 1/v.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Exchanging the alternatives turns A(k) into B(−k).
Exchanging the alternatives turns B(k) into A(−k).
The transition of the exchanged model is the reflection of the original one.
The log ratio of the exchanged model is the negated log ratio.
The exponent σ₁ of (14) solves β₁₁^σ₁ = 1/β₁₂, so it counts the occurrences of E₁₁ that
undo one occurrence of E₁₂.
Instances For
The exponent σ₂ of (14) solves β₂₁^σ₂ = 1/β₂₂.
Instances For
The probability P* of (15) is the limit of E(Pₙ) under the conditions of Theorem 17.
Equations
Instances For
Expectations under the exchanged model are expectations of the reflected function.
The exchanged model chooses its first alternative with probability 1 − E(Pₙ).
Taking expectations in (8) gives E(vₙ₊₁ᵏ) = (A(k) − B(k))E(Pₙvₙᵏ) + B(k)E(vₙᵏ) (10).
When β₁₁ ≠ 1, β₂₁ ≠ 1 and the coefficient of (16) is nonzero, the expected log ratio moves
by that coefficient times E(Pₙ) − P* (16).
Multiplying (11) by vₙᵗ and integrating gives E(Pₙvₙᵗ⁺¹) = E(vₙᵗ⁺¹) − E(Pₙvₙᵗ).
By (10) and (11) the first moment satisfies E(vₙ₊₁) = A(1)E(vₙ) − (A(1) − B(1))E(Pₙ).
If E(vₙⁱ) converges to L i and A(i) ≠ B(i), A(i) ≠ 1 for i = 1, …, k, then E(Pₙ)
converges to some p, and L k is (A(k) − B(k))/(A(k) − 1) times the product of
(1 − B(i))/(A(i) − 1) over 1 ≤ i < k times p (Theorem 15 (i), p. 114).
The limits of the moments of 1/vₙ satisfy Theorem 15 (ii) with the factors
(A(−i) − 1)/(1 − B(−i)), where the book prints (1 − A(−i))/(1 − B(−i)) (p. 114).
If E(vₙ) converges and A(1) ≠ B(1), or E(1/vₙ) converges and B(−1) ≠ A(−1), then E(Pₙ)
converges (the corollary to Theorem 15, p. 115).
Suppose E(vₙ) converges to L. If L = 0 then E(Pₙ) → 0 and E(1/vₙ) → ∞, and if
E(Pₙ) → 0 and A(1) ≠ 1 then L = 0 (Theorem 16 (i), p. 115).
Suppose E(1/vₙ) converges to L. If L = 0 then E(Pₙ) → 1 and E(vₙ) → ∞, and if
E(Pₙ) → 1 and B(−1) ≠ 1 then L = 0 (Theorem 16 (ii), p. 116).
If E(Pₙ) converges to p and the coefficient of (16) is nonzero, then p = P*, or
E(vₙ) → ∞, or E(1/vₙ) → ∞ (Theorem 17, p. 116).
If E(vₙ) and E(1/vₙ) both converge, A(1) ≠ B(1) or B(−1) ≠ A(−1), and the coefficient
of (16) is nonzero, then E(Pₙ) → P* (the corollary to Theorem 17, p. 117).
If A(1), B(1) > 1 then E(vₙ) → ∞, if A(1), B(1) < 1 then E(vₙ) → 0, and if
E(vₙ) → ∞ then A(1) ≥ 1 (Theorem 18 (i), p. 117).
If A(−1), B(−1) > 1 then E(1/vₙ) → ∞, if A(−1), B(−1) < 1 then E(1/vₙ) → 0, and if
E(1/vₙ) → ∞ then B(−1) ≥ 1 (Theorem 18 (ii), p. 117).
For x, y > 1, log x/(x − 1) < y log y/(y − 1) (Lemma 12, p. 119).
For x = βᵢ₁ > 1 and y = βᵢ₂ ∈ (0, 1), the ratio σᵢ/(σᵢ + 1) with σᵢ = −log y/log x from
(14) lies strictly between (1 − y)/(x − y) and x(1 − y)/(x − y) (Theorem 19, p. 119).