Competitive Variety Show Scoring based on Constrained Bayesian Optimization and Shapley value Decomposition

Abstract

In response to the core question of how to fairly integrate judges' professional ratings and audience voting in Dancing with the Stars (DWTS), this paper proposes a three-stage algorithm framework. In the first stage, a constrained Bayesian optimization model is constructed, mapping the weekly judge scores to Dirichlet priors, and using the elimination results as inequality constraints, the distribution of undisclosed audience votes is inversely estimated by sequential least squares programming (SLSQP). The model achieved a 87.2% agreement rate for elimination prediction over 185 weeks, and the estimated uncertainty decreased from 14.8% in the early stage to 6.2% in the final. In the second stage, the generalized weighted composite score (GWCS) framework and the Controversial Composite Index (CCI) are established, and the system compares the ranking method and the percentage method. Regression analysis (R²=0.996) showed that rank differences contributed 90% of the controversial variance, and 88.1% of the weeks had a balanced effective weight. In the third stage, structural equation model (SEM) and Shapley value decomposition were used to quantify the causal effects of celebrity characteristics and professional dancer quality on the score. The results showed that the number of weeks dominated the change in score (62.0% contribution), there was a significant negative bias in age (β=-0.318), and the partner quality contribution was 11.9%. This framework provides explainable algorithmic support for the scoring design of competitive variety shows.