Solution of a functional equation arising in an axiomatization of the utility of binary gambles
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Bibliographic record
Abstract
For a new axiomatization, with fewer and weaker assumptions, of binary rank-dependent expected utility of gambles the solution of the functional equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis z slash p right-parenthesis gamma Superscript negative 1 Baseline left-bracket z gamma left-parenthesis p right-parenthesis right-bracket equals phi Superscript negative 1 Baseline left-bracket phi left-parenthesis z right-parenthesis psi left-parenthesis p right-parenthesis right-bracket left-parenthesis z comma p element-of right-bracket 0 comma 1 left-bracket right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi> γ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">[</mml:mo> <mml:mi>z</mml:mi> <mml:mi> γ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi> φ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">[</mml:mo> <mml:mi> φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi> ψ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mspace width="2em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">[</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(z/p)\gamma ^{-1}[z\gamma (p)] = \varphi ^{-1}[\varphi (z)\psi (p)] \qquad (z,p\in ]0,1[)</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> is needed under some monotonicity and surjectivity conditions. We furnish the general such solution and also the solutions under weaker suppositions. In the course of the solution we also determine all sign preserving solutions of the related general equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis u right-parenthesis left-bracket g left-parenthesis u plus v right-parenthesis minus g left-parenthesis v right-parenthesis right-bracket equals f left-parenthesis v right-parenthesis g left-parenthesis u plus v right-parenthesis left-parenthesis u element-of double-struck upper R Subscript plus Baseline comma v element-of double-struck upper R right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="2em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>v</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h(u)[g(u+v)-g(v)]=f(v)g(u+v)\qquad (u\in \mathbb {R}_+,\ v\in \mathbb {R}).</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula>
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it