A new mathematical preprint gives an explicit formula for the dimension of a geometric object built from every possible ranking and its best-and-worst selections. The result answers a question about how this dimension changes with the number of alternatives.
The construction is entirely theoretical. It uses all linear rankings of a set of alternatives and the deterministic best-worst choice vectors produced by those rankings. In other words, the analysis considers every allowed ordering and records which option is treated as best and which as worst under the model.
A formula that scales with the number of options
For every n at least 2, the theorem states that the dimension is dim BW_n = n(n - 3)2^(n - 2) + C(n,2) - C(n,3) + n. The expression provides a direct calculation from the number of alternatives.
The paper's introduction places the result alongside the reported case of four alternatives, whose polytope had 24 vertices and dimension 22. The new formula is presented as the general answer to the dimension question, rather than as a calculation for one particular number of options.
In this setting, dimension is a count of the independent directions in the mathematical object. It is a structural property of the specified construction.
Breaking a large rank problem into local pieces
To obtain the formula, the proof first turns the dimension question into a matrix-rank problem. It then studies a filtration, meaning a nested sequence of subspaces generated by choice functions on subsets of the alternatives. This lets the authors examine the model in successive layers instead of handling the full construction in one step.
Each increase in that filtration is identified with a direct sum of local quotient spaces, one for every subset of the corresponding size. A quotient here records the new directions that remain after contributions from smaller subsets have been removed. The decomposition is the bridge between the global polytope and the smaller pieces used in the calculation.
For a local subset with k at least 4, the remaining quotient has dimension k(k - 2). The relevant lower-order intersection is generated by best-worst coordinates associated with proper subsets. These local dimensions are then combined across subsets to recover the overall rank.
The two smallest nontrivial cases are handled separately: the local quotient dimension is 1 for subsets of size 2 and 2 for subsets of size 3. These small pieces provide the starting contributions before the general k(k - 2) pattern applies.
What the result does and does not establish
Although representation theory motivates the proof, the argument presented in the preprint uses elementary linear algebra. The final polytope dimension is obtained from the vector-space rank after subtracting 1 for the normalization condition.
The result applies to the specified linear-ranking construction and its generated polytope. It does not provide evidence about human choice behavior, estimate choice probabilities from observed data, or compare behavioral models empirically. No empirical uncertainty estimates, model-fit measures, or validation data are reported.
That boundary matters because a closed formula for a model's dimension is not evidence that the model describes real-world decisions. The paper leaves open how the result relates to empirical adequacy, what a fuller representation-theoretic analysis might add, and whether similar formulas hold for other choice-polytope constructions.
The supplied record identifies the work as arXiv version 1 dated 26 August 2026. The acknowledgements state that it was supported by the German Research Foundation under grant MA 6503/1-1.
Paper data and sources
Original title: On the Dimension of the Best-Worst Choice Polytope
Authors: Keivan Mallahi-Karai, Majid Salamat
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text