The Triple Crown
Three races. Three different-looking sets of performance densities. Identical winning probabilities. The reader who agrees with the market on $\{p_i\}$ has agreed on nothing about what the world looks like underneath, and this demo walks through why that matters.
Two of the three races (Preakness and Belmont) sit inside the family where Luce's Choice Axiom holds exactly. The third (Derby) is the Standard Normal Horse Race — the Thurstone Class V alternative. They're tuned to the same winning probabilities; pick whichever you find more believable as a generative story for how runners actually finish.
Set winning probabilities
Enter raw weights for five runners (any positive numbers — they'll be normalised to sum to 1).
Edit the weights then click Recalculate — the lattice calibration for the Derby takes a moment, so we don't rerun it on every keystroke.
The three races
Take a moment with these three panels. The Preakness (exponential) is the Lucian model in its undisguised form — rates $\lambda_i = p_i$ and the win probabilities pop out by force of the memoryless property. The shape is unmistakable: each density peaks at zero and decays. It is not what anyone's mental model of a finishing time looks like.
The Belmont is the same race in disguise. If $Z_i \sim \mathrm{Exp}(\lambda_i)$ then $X_i = Z_i^{1/4}$ has a Weibull (shape 4) distribution — bell-shaped, plausible-looking, almost normal to the eye. But $x \mapsto x^{1/4}$ is monotone, and any rank-determined probability is invariant under a monotone transform of performance. So from the order-statistics point of view, the Preakness and the Belmont are the same race: every quinella, exacta, trifecta, every losing probability, every conditional ranking they assign is identical. They differ only in what units you choose to measure the runners with.
That equivalence is what makes the Belmont's real flaw worth noticing. The flaw is not the shape — the densities look reasonable. The flaw is in the variances. To match the same winning probabilities under Luce's axiom, the Belmont has to inflate the performance variance of the weakest horse: with the canonical $(5,4,3,2,1)$ field, $\sigma_5 / \sigma_1 = (p_1/p_5)^{1/4} = 5^{1/4} \approx 1.50$, and for an extreme favourite it gets close to $2$. The bias-variance plot below shows it directly.
The Kentucky Derby is something genuinely different. It is the Standard Normal Horse Race — Thurstone's Class V model — calibrated by the fast ability transform on a uniform lattice so that the implied winning probabilities match. Each runner has a normal density with the same variance, separated only by translation. That is the natural prior most modellers would write down if they sat down and asked, “how is finishing time distributed around a horse's underlying ability?” It is also not Luce-compliant for $n \ge 3$: its rank-determined probabilities differ from the Preakness/Belmont, even though the marginal winning probabilities agree.
The constraint that Lucian models can't leave
One diamond per horse, both models. Bias on the x-axis, variance ratio on the y, both measured relative to the favourite. Luce's axiom forces you onto a manifold; Thurstone's Normal model spans the plane.
The Derby horses line up along the flat blue line: equal variance by construction, bias free to vary. The Belmont horses climb the green curve: every step away from the favourite in bias also drags the variance ratio upward, and for the weakest horse in an extreme field the variance has to nearly double the favourite's. There is no empirical reason that should be so — Secretariat's 31-length Belmont rebuts it directly — but Lucian arithmetic doesn't leave you a choice. Pick the axiom and you pick the manifold.
This is the Triple Crown argument distilled: the winning probabilities don't adjudicate between sensible generative models, but Luce's axiom — whether you reach it via raw exponentials (Preakness) or a tasteful monotone transform of them (Belmont) — lives on a narrow manifold where bias and variance must move together. The fast ability transform is what lets us walk off that manifold and use whatever performance density actually matches the data.