The geometry of choice

The Thurstone portfolio maximizes ability minus an entropy that knows the correlation.

The Thurstone portfolio is defined procedurally — weights are winning probabilities — but it secretly solves a recognizable program: the weights maximize ⟨θ, p⟩ − Ω(p) over the simplex, where θ is the ability vector and Ω is a convex regularizer determined by the noise law of the race. The noise is decisive. Gumbel noise makes Ω the Shannon entropy and the allocator softmax — a rule that rewards spreading across three names whether or not two of them are the same asset in different wrappers. Gaussian noise with correlation C makes Ω a dependence-aware entropy: it rewards spreading only across genuinely distinct bets.

This page draws that entropy. Three assets; assets 1 and 2 are progressively correlated (the twins), asset 3 stays independent. A fixed grid of ability vectors (radius 3, the "ability budget") is pushed through each allocator, and every attainable portfolio is plotted in the simplex triangle, colored by its diversification reward −Ω(p) — computed live from the race, 8,192 common-seed paths per grid point.

Try it: merge two of the three assets

What to look at

The softmax triangle never moves. Luce's axiom in geometric form: the set of portfolios a given ability budget can reach, and the entropy prices attached to them, are the same whether the twins are independent or identical. Softmax will cheerfully hold (⅓, ⅓, ⅓) of what is really two assets.

The race's triangle collapses to a segment. As ρ₁₂ rises, allocations that distinguish the twins become progressively more expensive in ability terms — the grid's image pinches onto the line w₁ = w₂ — until at ρ₁₂ = 1 the geometry loses a dimension: a three-asset choice becomes a two-asset choice, the redundancy priced into the entropy itself. This is the same near-singularity that makes Σ⁻¹ explode, showing up not as an instability but as a re-shaped penalty.

The equal-ability portfolio drifts. The marker is θ = 0: with no ability information at all, the race's maximum-entropy portfolio moves from (⅓, ⅓, ⅓) to (¼, ¼, ½) — the twins jointly worth one asset, the independent name worth the other. The darkest attainable color also lightens as ρ₁₂ rises: the maximum diversification reward on offer, −Ω at its minimum, shrinks as the third genuine bet disappears.

Honest footnotes: for ρ₁₂ < 1 the race map is still a bijection onto the open triangle — the pinch is about the ability cost of off-median portfolios diverging, not impossibility; at ρ₁₂ = 1 exactly it is impossibility. The two panels use different noise laws (unit-variance Gumbel vs Gaussian), so colors compare within a panel, not across panels. Theory: the implied-objective theorem in the paper (the Williams–Daly–Zachary / perturbed-optimizer construction).