Minimax regret against an adversarial experiment: certified values for three to eight binary signals and the two-point support of Nature's optimal mixture
Abstract
A decision-maker who will observe n i.i.d. binary signals of unknown precision π ∈ [1/2,1] commits in advance to a belief aₖ for every count k of high signals; an adversarial Nature mixes over π; the payoff is the mean-squared-error regret against an oracle who knows π, under a uniform prior on the binary state. Che, Li and Luo (arXiv:2602.15246) solve this game for n ≤ 3 — Nature mixes the two endpoint experiments {1/2,1} for n ≤ 2 and {1/2,π^*} with an interior π^* at n=3 — state that a closed-form characterisation for n>3 is intractable by their approach, conjecture the two-point structure {1/2,πₙ^*} for every n, and plot the values that the conjecture predicts for 3 ≤ n ≤ 18. We prove, without assuming anything about the structure of Nature's strategy, rational brackets Lₙ ≤ Vₙ ≤ Uₙ on the minimax regret for n=3,…,8, of width at most 2.1 · 10⁻¹¹: Lₙ is what an explicit rational two-point mixture guarantees against every belief rule, and Uₙ is what an explicit rational belief rule guarantees against every precision, proved by a Bernstein subdivision certificate. The numbers agree with the source's conjecture-based numerics; what is new is that they are bounds. A second certificate, with a margin vanishing exactly at the two conjectured atoms, shows that every finitely supported mixture guaranteeing Lₙ — in particular every finitely supported optimal one — satisfies Σᵢ wᵢ (2pᵢ-1)²(2pᵢ-1-hat dₙ)² ≤ 8.4 · 10⁻¹¹, which localises Nature's optimal support at the two conjectured atoms with an explicit error bar; at n=4 at most 5.25 · 10⁻⁴ of the mass lies further than 1/100 from both. Alongside, the game in which Nature is confined to {1/2,1} has the closed-form value 1/(4(1+2^((n-1)/2))²) for every n, which reproduces the source's n ≤ 2 strategies and is strictly worse for Nature from n=3 on, by a factor that grows from below 2 to above 21; and V₃>V₄>…>V₈. Every statement is verified in Lean 4 against Mathlib.
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Claim ledger
Stated results
NM1routine2026-09-07
The game in which Nature is restricted to mixtures of the two endpoint experiments 1/2, 1 has value exactly 1/(4(1+2^((n-1)/2))²) for every n >= 1, attained by Nature's weight 1/(1+2^((n-1)/2)) on pi = 1 against the equalising rule a₀ = 1/(2(1+2^((n-1)/2))), aₙ = 1 - a₀, aₖ = 1/2 otherwise
NM2known data2026-09-07
Model-fidelity controls: the general restricted-value formula reproduces the source's Proposition 1 (n = 1: weight 1/2, rule (1/4, 3/4)) and Theorem 1(i) (n = 2: weight sqrt2 - 1); and the DM's action space may be taken to be all of Rⁿ⁺¹ rather than the source's [0,1]ⁿ⁺¹, since truncating a rule into [0,1] never raises the regret
NM3candidate2026-09-07
The minimax regret of the source's MSE / binary-signal game at n = 3 satisfies 0.039156935533 <= V₃ <= 0.039156935541 (a bracket of width 8e-12), by an exact rational two-point Nature mixture and a Bernstein-certified upper bound for an explicit rational belief rule
NM4candidate2026-09-07
The minimax regret of the source's MSE / binary-signal game at n = 4 satisfies 0.036580028069 <= V₄ <= 0.036580028090 (a bracket of width 2.1e-11), by an exact rational two-point Nature mixture and a Bernstein-certified upper bound for an explicit rational belief rule
NM5candidate2026-09-07
The minimax regret of the source's MSE / binary-signal game at n = 5 satisfies 0.036399353288 <= V₅ <= 0.036399353295 (a bracket of width 7e-12), by an exact rational two-point Nature mixture and a Bernstein-certified upper bound for an explicit rational belief rule
NM6candidate2026-09-07
The minimax regret of the source's MSE / binary-signal game at n = 6 satisfies 0.035570811186 <= V₆ <= 0.035570811191 (a bracket of width 5e-12), by an exact rational two-point Nature mixture and a Bernstein-certified upper bound for an explicit rational belief rule
NM7candidate2026-09-07
The minimax regret of the source's MSE / binary-signal game at n = 7 satisfies 0.035382000376 <= V₇ <= 0.035382000378 (a bracket of width 2e-12), by an exact rational two-point Nature mixture and a Bernstein-certified upper bound for an explicit rational belief rule
NM8candidate2026-09-07
The minimax regret of the source's MSE / binary-signal game at n = 8 satisfies 0.035026812023 <= V₈ <= 0.0350268120238 (a bracket of width 8e-13), by an exact rational two-point Nature mixture and a Bernstein-certified upper bound for an explicit rational belief rule
NM9routine2026-09-07
Nature's n <= 2 support 1/2, 1 is strictly suboptimal from n = 3 on: Vʳestrₙ < Vₙ for n = 3..8, with the price rising from a factor below 2 at n = 3 to a factor above 21 at n = 8; and against the certified rule at n = 4 the fully revealing experiment is not even a best reply, R(a₄, 1) < R(a₄, 1/2)
NM10routine2026-09-07
Negative controls: for each n = 3..8 an explicit precision at which the certified rule aₙ pays more than a threshold within 10⁻11 of Uₙ, so no materially smaller upper bound holds for that rule; V₄ is refuted both below 0.036580028068 and above 0.036580028091; and no hypothesis is vacuous (pi-hat strictly interior, w-hat a strict mixture, the regret strictly positive)
NM11known data2026-09-07
The minimax regret is strictly decreasing in the sample size over the certified range: V₃ > V₄ > V₅ > V₆ > V₇ > V₈, because the six brackets are pairwise disjoint and ordered (Uₙ₊₁ < Lₙ for every n = 3..7)
NM12candidate2026-09-07
Support localisation, unconditional: for n = 3..8 there is an explicit rational gₙ > 0 with R(aₙ, pi) <= Uₙ - gₙ (2pi-1)² (2pi-1-d-hat)² on all of [1/2,1], whence every finitely supported Nature mixture that guarantees at least Lₙ – in particular every optimal one – satisfies sumᵢ wᵢ (2pᵢ-1)² (2pᵢ-1-d-hat)² <= (Uₙ - Lₙ)/gₙ, a number of order 10⁻11; at n = 4 at most 5.25e-4 of Nature's mass lies at precisions further than 1/100 from both 1/2 and pi-hat₄
Provenance
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- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source: arXiv:2602.15246v1, Yeon-Koo Che, Longjian Li, Tianling Luo, *Learning Against Nature: Minimax Regret and the Price of Robustness*, econ.TH, submitted 16 Feb 2026 (v1 only; the live abs page was checked on 2026-09-07). Everything below refers to Section 3 of the compiled e-print, *Special Case: MSE with Binary Signals* (p. 7); theorem and page numbers are read off the.aux and the PDF produced by tectonic from the arXiv e-print, never counted from the raw LaTeX.
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- 2026-09-07 03:53 UTC
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