Kenneth ChengMathematical modeling

Editorial project overview

FairChoice Dynamics — When Taking Turns Makes a System Unstable

With fixed cohorts, delayed observations and one update per slot, what is the smallest maximum cyclic update gap that admits a balanced locally stabilizing schedule?

Reproducible studyUpdated 10 Sept 2026
Technical figure from FairChoice Dynamics — When Taking Turns Makes a System Unstable
A reviewed research figure synchronized from the private technical workspace. The original aspect ratio and labels are preserved.

Equal opportunities do not settle the timing question

Four fixed groups choose between two identical options using information that is one update old. A round-robin schedule looks fair: each group updates once every four slots. Yet, in this model, its regularity reinforces delayed overreaction. The balanced eight-slot sequence 1,2,3,1,4,3,2,4 changes the ordering without changing anyone’s long-run frequency.

The exact result is confined to response strengths 3.75–4.25. Across that whole interval, round robin is unstable and the displayed schedule is locally exponentially stable. A separate combinatorial lower bound rules out a smaller stabilizing maximum gap: every schedule with a gap bound of four must repeat a permutation of the groups. Thus the minimal feasible gap is five, not merely the smallest gap found in a numerical search.

What the experiments add

All 2,520 balanced period-8 schedules were evaluated. The period-12 search screened 369,600 balanced candidates and evaluated the 1,488 with maximum gap at most five. The best evaluated gap-5 period-12 schedule contracts faster per slot than the period-8 construction, showing that the smallest fairness relaxation and the fastest convergence are different objectives.

Deterministic initial-state and phase checks and 320 finite-agent runs test specific extensions, not a global theorem. The comparison preserves inconvenient results: independent random updating has smaller fluctuation in the tested populations but cannot promise a fixed maximum update gap; changing the information delay can remove the stabilizing behaviour.

Read When Taking Turns Makes a System Unstable for the model, exact argument, eight research figures, paired comparisons and limits. This is a personal research study using synthetic data. It is not a student contribution record, an empirical service trial or a claim of a new published theorem.

Key findings

  • For four equal fixed cohorts and delay one, the smallest stabilizing maximum gap is five throughout beta in [3.75,4.25]. Round robin is unstable and the displayed gap-5 schedule is locally exponentially stable.
  • At beta=4, the best evaluated gap-5 per-slot radii are 0.97159 at period 8 and 0.90180 at period 12; these are finite-search results, not an unrestricted optimum.
  • IID updates have lower synthetic fluctuation in the tested cases but no deterministic hard update-gap bound. Delay two retains nonzero tail fluctuation for both fixed schedules.

Limitations

  • The theorem concerns local stability in a specified homogeneous model, not global attraction, actual waiting times or human behaviour.
  • Twenty exposed development seeds per policy and population are not a held-out confirmatory sample.
  • Novelty, a journal submission and student authorship are not claimed. The public article is a personal mathematical research explanation.

Technical record

Detailed source, calculations, generated figures, and reproduction instructions are maintained in a private technical workspace. Public articles contain only manually reviewed interpretation and approved figures.

Version history

2026-09-10 — Curated overview reviewed against repository evidence.