Thurstone's Theory of Discriminal Processes @cite{luce-1959}

@cite{luce-1959} §2.D (pp. 53-60): @cite{thurstone-1927}'s Case V model of paired comparison, and the logistic approximation that connects it to the Luce choice rule.

Thurstone Case V

Each stimulus a evokes a random discriminal process — a Gaussian random variable with mean u(a) (the scale value) and standard deviation σ. When a subject compares a and b, they sample one discriminal process for each stimulus. The probability of choosing a over b is the probability that the sample for a exceeds that for b:

P(a,b) = Φ((u(a) - u(b)) / (σ√2))

where Φ is the standard normal CDF. Case V assumes equal variances across all stimuli — the "simplest nontrivial case" in Thurstone's taxonomy.

The Logistic Approximation (pp. 58-59)

Luce observes that the logistic function 1/(1 + exp(-x)) closely approximates the normal CDF Φ(x · π/√3). The maximum absolute deviation between the two is approximately 0.01. This means Thurstone's Case V is approximately a special case of the Luce model:

P(a,b) ≈ 1/(1 + exp(-k(u(a) - u(b))))

for k = π / (σ√6). The logistic approximation is what makes the connection to Luce's ratio-scale framework (§2.A) and hence to softmax (§2).

Strong Stochastic Transitivity

Thurstone Case V satisfies strong stochastic transitivity: if u(a) > u(b) > u(c), then P(a,c) > max(P(a,b), P(b,c)). This is stronger than the weak stochastic transitivity that Luce's axioms alone guarantee.

structure Core.ThurstoneCaseV (Stimulus : Type u_1) : Type u_1

Thurstone's Case V model (@cite{thurstone-1927}; @cite{luce-1959}, §2.D).

Each stimulus has a scale value scale(a) and all stimuli share a common discriminal dispersion sigma > 0. The choice probability is determined by the normal CDF applied to the standardized scale difference.

• scale : Stimulus → ℝ

  The scale value (mean of the discriminal process) for each stimulus.

• sigma : ℝ

  The common discriminal dispersion (standard deviation).

• sigma_pos : 0 < self.sigma

  The dispersion is strictly positive.

noncomputable def Core.ThurstoneCaseV.choiceProb {Stimulus : Type u_1} (m : ThurstoneCaseV Stimulus) (a b : Stimulus) : ℝ

Choice probability under Thurstone Case V: P(a,b) = Φ((u(a) - u(b)) / (σ√2)).

This is the probability that the discriminal process for a exceeds that for b, when both are independent Gaussians with means u(a), u(b) and common variance σ². The difference is Gaussian with mean u(a) - u(b) and variance 2σ², hence standard deviation σ√2.

Equations

• m.choiceProb a b = Core.normalCDF ((m.scale a - m.scale b) / (m.sigma * √2))

theorem Core.ThurstoneCaseV.choiceProb_eq {Stimulus : Type u_1} (m : ThurstoneCaseV Stimulus) (a b : Stimulus) (h : m.scale a = m.scale b) : m.choiceProb a b = 1 / 2

When u(a) = u(b), the choice probability is 1/2 (indifference).

theorem Core.ThurstoneCaseV.choiceProb_complement {Stimulus : Type u_1} (m : ThurstoneCaseV Stimulus) (a b : Stimulus) : m.choiceProb a b + m.choiceProb b a = 1

Complementarity: P(a,b) + P(b,a) = 1.

theorem Core.ThurstoneCaseV.choiceProb_gt_half {Stimulus : Type u_1} (m : ThurstoneCaseV Stimulus) (a b : Stimulus) (h : m.scale b < m.scale a) : 1 / 2 < m.choiceProb a b

If u(a) > u(b), then P(a,b) > 1/2 — the higher-scale stimulus is chosen more often than chance.

theorem Core.ThurstoneCaseV.transitivity_left {Stimulus : Type u_1} (m : ThurstoneCaseV Stimulus) (a b c : Stimulus) (_hab : m.scale b < m.scale a) (hbc : m.scale c < m.scale b) : m.choiceProb a b < m.choiceProb a c

Strong stochastic transitivity (Thurstone Case V).

If u(a) > u(b) > u(c), then P(a,c) > P(a,b) — the "big gap" comparison is easier than either "small gap" comparison.

Proof: u(a) - u(c) > u(a) - u(b), so after dividing by σ√2 > 0, the argument to Φ is larger, and Φ is strictly monotone.

theorem Core.ThurstoneCaseV.transitivity_right {Stimulus : Type u_1} (m : ThurstoneCaseV Stimulus) (a b c : Stimulus) (hab : m.scale b < m.scale a) (_hbc : m.scale c < m.scale b) : m.choiceProb b c < m.choiceProb a c

The right half of strong stochastic transitivity: if u(a) > u(b) > u(c), then P(a,c) > P(b,c).

Thurstone Case V and the Luce Model

Set d = u(a) - u(b) and k = π / (σ · √6). Then the exact identity:

d / (σ√2) = k · d · (√3/π)

rewrites the Thurstone formula as:

P_T(a,b) = Φ(d / (σ√2)) = Φ(k·d · √3/π)

Since Φ(y · √3/π) ≈ logistic(y) numerically (max error ~0.023 with variance matching; see @cite{luce-1959} §2.D.2, Table 3), this gives:

P_T(a,b) ≈ logistic(k·d) = 1/(1 + exp(-k·(u(a) - u(b))))

The constant k = π/(σ√6) arises from matching variances: the standard logistic has variance π²/3, while the Thurstone difference distribution (two i.i.d. N(0,σ²) draws) has variance 2σ². Setting π²β²/3 = 2σ² gives β = σ√6/π, so k = 1/β = π/(σ√6).

The Gumbel-Luce model (GumbelLuce.lean) gives exactly logistic(d/β) by McFadden's theorem — no approximation. The Thurstone model gives exactly Φ(d/(σ√2)). They agree up to Φ ≈ logistic which is a purely numerical fact (~0.023 max error with variance matching, ~0.009 with the optimal constant 1.702).

noncomputable def Core.thurstoneLuceK (sigma : ℝ) : ℝ

The scaling constant connecting Thurstone and Luce: k = π / (σ · √6) so that (u(a)-u(b))/(σ√2) = k·(u(a)-u(b))·(√3/π).

Equations

• Core.thurstoneLuceK sigma = Real.pi / (sigma * √6)

theorem Core.thurstone_luce_identity {Stimulus : Type u_1} (m : ThurstoneCaseV Stimulus) (a b : Stimulus) : m.choiceProb a b = normalCDF (thurstoneLuceK m.sigma * (m.scale a - m.scale b) * (√3 / Real.pi))

Thurstone–Luce identity (@cite{luce-1959}, §2.D): the Thurstone choice probability equals normalCDF evaluated at the variance-matched Luce argument scaled by √3/π.

P_T(a,b) = Φ(d/(σ√2)) = Φ(k·d·√3/π)

where k = π/(σ√6) and d = u(a) - u(b). Since Φ(y·√3/π) ≈ logistic(y) numerically, this gives P_T(a,b) ≈ logistic(k·d) — the Luce model.

The approximation Φ(y·√3/π) ≈ logistic(y) has max error ~0.023 (variance matching) and is a numerical fact without analytical proof.

Theorem 7: Luce and Thurstone Diverge for Three or More Alternatives

@cite{luce-1959} Theorem 7 (§2.D.3): for pairwise comparisons (n = 2), the Luce and Thurstone models are approximately equivalent (thurstone_luce_identity). For n ≥ 3 alternatives, they are fundamentally incompatible: no independent Thurstone discriminal processes can generate both the "choose best" and "choose worst" probabilities predicted by the Luce model.

The proof has two steps:

1. Thurstone integral identity: For independent discriminal processes, P_best(x) - P_worst(x) = P(x,y) + P(x,z) - 1 (expanding the product of CDFs).

2. Algebraic contradiction: Under axiom 1, P_best(x) = v(x)/Σv and P_worst(x) = (1/v(x))/Σ(1/v). Setting the axiom 1 difference equal to P(x,y) + P(x,z) - 1 forces P(x,y)·P(y,x)·P(z,x) = 0, contradicting non-degeneracy.

The algebraic core (step 2) is formalized below.

theorem Core.luce_thurstone7 (v : Fin 3 → ℝ) (hv : ∀ (i : Fin 3), 0 < v i) (h_nd : v 0 / (v 0 + v 1) + v 0 / (v 0 + v 2) ≠ 1) (h : v 0 / (v 0 + v 1 + v 2) - v 1 * v 2 / (v 0 * v 1 + v 0 * v 2 + v 1 * v 2) = v 0 / (v 0 + v 1) + v 0 / (v 0 + v 2) - 1) : False

Luce–Thurstone incompatibility (@cite{luce-1959}, Theorem 7): for three alternatives with positive Luce scales, the axiom 1 "best-worst difference" does NOT equal P(x,y) + P(x,z) - 1 (the value predicted by independent Thurstone processes).

Specifically, axiom 1 gives:

• P_best(0) = v₀ / (v₀ + v₁ + v₂)
• P_worst(0) = v₁v₂ / (v₀v₁ + v₀v₂ + v₁v₂)
• P(0,1) + P(0,2) - 1 = v₀/(v₀+v₁) + v₀/(v₀+v₂) - 1

If these are equal (as the Thurstone integral identity requires), and P(0,1) + P(0,2) ≠ 1 (Luce's hypothesis ii, equivalent to v₀² ≠ v₁v₂), then v₀v₁v₂ = 0, contradicting positivity.

The proof clears denominators, factors out (v₀² - v₁v₂), and shows the remaining factor is -v₀v₁v₂ ≠ 0.

theorem Core.luce_thurstone_incompatible {n : ℕ} (v : Fin n → ℝ) (hv : ∀ (i : Fin n), 0 < v i) (i j k : Fin n) (_hij : i ≠ j) (_hik : i ≠ k) (_hjk : j ≠ k) (h_nd : v i / (v i + v j) + v i / (v i + v k) ≠ 1) (h_int : v i / (v i + v j + v k) - v j * v k / (v i * v j + v i * v k + v j * v k) = v i / (v i + v j) + v i / (v i + v k) - 1) : False

Luce–Thurstone incompatibility (general): the n = 3 result extends to any n ≥ 3 alternatives by restricting to a 3-element subset.

By IIA (Luce's axiom 1), the pairwise probabilities P(i,j) = vᵢ/(vᵢ+vⱼ) are the same regardless of choice set. The Thurstone integral identity only needs to hold for one triple {i, j, k} to derive a contradiction. So the incompatibility between Luce and independent Thurstone processes holds whenever n ≥ 3 and any non-degenerate triple exists.

This is why Luce states Theorem 7 for |T| = 3 — the general case follows immediately from IIA + the base case.