Bayesian Imputation for Unplayed Games in Round-Robin Chess Tournaments: Application to Grand Chess Tour, Bucharest 2026

Abstract: When a player withdraws mid-tournament from a round-robin chess event, organizers face a fundamental problem: how should scores be assigned for games that were never played? Current FIDE guidelines specify annulment if withdrawal occurs before 50% of games are completed, and forfeit (awarding unplayed opponents a full point) thereafter. This dichotomous rule creates arbitrary discontinuities and can substantially distort final standings. We develop a Bayesian framework based on best linear unbiased prediction (BLUP) that optimally combines pre-tournament ratings with observed performance, producing imputed scores that reflect both the withdrawn player's current form and the strength differentials among unplayed opponents. The estimator is consistent, point-conserving, and minimizes mean squared error among linear unbiased predictors. A Monte Carlo simulation study on 180,000 simulated tournaments demonstrates that Bayesian BLUP imputation reduces prediction error by 26% overall compared to FIDE's current rule, with improvements of 41% over forfeit and 12% over annulment. The largest gains occur when the withdrawn player is underperforming, the most common withdrawal scenario. We further show that annulment achieves 15-45% lower RMSE than forfeit across all scenarios. The methodology is applied to GM Alireza Firouzja's withdrawal at Grand Chess Tour, Bucharest 2026, where Bayesian imputation would have awarded unplayed opponents 0.55-0.70 points rather than the 1.0 awarded under forfeit rules. An open-source R Shiny application is provided for tournament organizers. We recommend that FIDE adopt Bayesian imputation for World Championship cycle events, or at minimum replace the current dichotomous rule with uniform annulment.