Additive representation of separable preferences over infinite products
Bibliographic record
Abstract
Let X be a set of states, and let I be an infinite indexing set. Our first main result states that any separable, permutation-invariant preference order (>) on X^I admits an additive representation. That is: there exists a linearly ordered abelian group A and a `utility function' u:X-->A such that, for any x, y in X^I which differ in only finitely many coordinates, we have x>y if and only if the sum of [u(x_i)-u(y_i)] over all i in I is positive. Our second result states: If (>) also satisfies a weak continuity condition, then, for any x, y in X^I, we have x>y if and only if the `hypersum' of [u(x_i)-u(y_i)] over all i in I is positive. The `hypersum' is an infinite summation operator defined using methods from nonstandard analysis. Like an integration operator or series summation operator, it allows us to define the sum of an infinite set of values. However, unlike these operations, the hypersum does not depend on some form of convergence (recall: A has no topology) ---it is always well-defined. Also, unlike an integral, the hypersum does not depend upon a sigma-algebra or measure on the indexing set I. The hypersum takes values in a linearly ordered abelian group A*, which is an ultrapower extension of A. These results are applicable to infinite-horizon intertemporal choice, choice under uncertainty, and variable-population social choice.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".