Nonchordal and irrational D-optimal designs for the Bradley–Terry model
Year: 2026
Type: preprint
Abstract: Unrefereed research candidate. In one sentence: D-optimal paired-comparison designs for the Bradley–Terry model need not use a chordal set of pairs, and their weights need not be rational, disproving Conjecture 12 of Röttger, Kahle and Schwabe (2025). In a paired-comparison experiment each trial compares two alternatives; the Bradley–Terry model gives alternative i a strength ai and says i beats j with probability ai/(ai+aj). An approximate design assigns each pair a share of the comparison budget, and a locally D-optimal design makes the joint estimate of the strengths as precise as possible. The pairs it uses form a graph. Röttger, Kahle and Schwabe (Journal of Statistical Theory and Practice, 2025, Conjecture 12) conjectured that this graph is always chordal, the property behind their closed rational formulae. With fourteen alternatives and strengths (71 six times, 73, 4260, 4380 six times) the unique optimal design uses 54 of the 91 pairs, its graph contains an induced four-cycle, and every positive weight is a quadratic irrational; all 91 optimality conditions are checked exactly. The phenomenon persists on open sets of strengths, for every number of alternatives from fourteen upwards and for the explicit pairwise-distinct integers a0i(1017+i); along a one-parameter branch the weights are not rational functions of the strengths. Fourteen is the minimum within the symmetric four-level family that produces the example (p > 3+2√2 per outer group), and seventeen within a three-level family; the global minimum is open between five and fourteen. Structure that survives: no biconnected block of an optimal support is a cycle of length at least four; every cut vertex joins exactly two blocks, weaker on one side and stronger on the other; the house and K2,3 are not optimal on five alternatives. This answers Problem 15 of Kahle, Röttger and Schwabe (2021) positively for Bradley–Terry intensities and negatively for unrestricted intensities. The result concerns exact optimal structure and formulae, not efficiency: a reoptimised chordal design reaches D-efficiency about 0.9985 in the example (numerical). The archive contains the manuscript (PDF and LaTeX source), seven exact standard-library certificate programs with a runner that executes each normally and under python -O, their logs, derivation notes, the support-graph figure, the prior-art record, a rating-free internal review summary, CI and SHA-256 sums. Assurance: written proofs for the general theorems; exact finite certificates for the explicit examples and identities; internal AI-assisted review only (not peer review, independent reproduction or formal verification); bounded prior-art search. Paired comparisons only. Creator: Anonymous. Text, figures and data CC0-1.0; code MIT.
Author: Anonymous
Language: English
Field: Decision Sciences
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7225141979