The Reconstruction of Unobserved Audience Voting and the Robustness of Evaluation Mechanisms Based on Constrained Optimization Models and Sensitivity Analysis

Abstract

To address the black-box problem caused by unobserved audience voting data in television competition systems, this study uses panel data from 34 seasons of Dancing with the Stars to construct a computational framework that combines constrained vote reconstruction with robustness analysis of evaluation rules. In the audience-vote estimation phase, a season-specific constrained optimization model is developed using known expert scores and recorded elimination outcomes. The Python-Gurobi solver is used to obtain representative vote distributions under non-negativity, total-vote, and elimination-logic constraints, with a variance-minimization objective used to avoid extreme allocations. The reconstructed votes reproduce nearly all single-stage elimination outcomes and achieve an 83.9% tie-aware at-risk-set agreement in modern two-stage seasons. These results support the use of inferred votes for downstream rule comparison, while also showing that absolute vote magnitudes remain interval-uncertain in some weeks. In the mechanism evaluation phase, the ranking method and the percentage method are compared under different levels of competition intensity, and a weekly stability score is introduced to identify rule-sensitive weeks. The findings suggest that disputes mainly arise near elimination thresholds, where small vote fluctuations may alter at-risk membership. By quantifying otherwise unobserved decision variables, this study provides a reproducible constrained-optimization approach for fairness and robustness audits in hybrid expert-audience evaluation systems.