Sports Analytics Weekly by kubeia.io - 22/2025

Explore Bill Benter’s iconic horse betting strategy, enhanced with modern code and contrasted against three decades of evolving data.

Abstract:The construction of confidence intervals for the mean of a bounded random variable is a classical problem in statistics with numerous applications in machine learning and virtually all scientific fields. In particular, obtaining the tightest possible confidence intervals is vital every time the sampling of the random variables is expensive. The current state-of-the-art method to construct confidence intervals is by using betting algorithms. This is a very successful approach for deriving optimal confidence sequences, even matching the rate of law of iterated logarithms. However, in the fixed horizon setting, these approaches are either sub-optimal or based on heuristic solutions with strong empirical performance but without a finite-time guarantee. Hence, no betting-based algorithm guaranteeing the optimal \mathcal{O}(\sqrt{\frac{\sigma^2\log\frac1\delta}{n}}) width of the confidence intervals are known. This work bridges this gap. We propose a betting-based algorithm to compute confidence intervals that empirically outperforms the competitors. Our betting strategy uses the optimal strategy in every step (in a certain sense), whereas the standard betting methods choose a constant strategy in advance. Leveraging this fact results in strict improvements even for classical concentration inequalities, such as the ones of Hoeffding or Bernstein. Moreover, we also prove that the width of our confidence intervals is optimal up to an 1 o(1) factor diminishing with n. The code is available on~this https URL.

What xGOT measures, how it fixes xG’s blind spots, how analysts rely on it, and a Python shortcut to bulk-download the data

Abstract:During football matches, a variety of different parties (e.g., companies) each collect (possibly overlapping) data about the match ranging from basic information (e.g., starting players) to detailed positional data. This data is provided to clubs, federations, and other organizations who are increasingly interested in leveraging this data to inform their decision making. Unfortunately, analyzing such data pose significant barriers because each provider may (1) collect different data, (2) use different specifications even within the same category of data, (3) represent the data differently, and (4) delivers the data in a different manner (e.g., file format, protocol). Consequently, working with these data requires a significant investment of time and money. The goal of this work is to propose a uniform and standardized format for football data called the Common Data Format (CDF). The CDF specifies a minimal schema for five types of match data: match sheet data, video footage, event data, tracking data, and match meta data. It aims to ensure that the provided data is clear, sufficiently contextualized (e.g., its provenance is clear), and complete such that it enables common downstream analysis tasks. Concretely, this paper will detail the technical specifications of the CDF, the representational choices that were made to help ensure the clarity of the provided data, and a concrete approach for delivering data in the CDF.

Abstract:In tennis tournaments, momentum, a critical yet elusive phenomenon, reflects the dynamic shifts in performance of athletes that can decisively influence match outcomes. Despite its significance, momentum in terms of effective modeling and multi-granularity analysis across points, games, sets, and matches in tennis tournaments remains underexplored. In this study, we define a novel Momentum Score (MS) metric to quantify a player's momentum level in multi-granularity tennis tournaments, and design HydraNet, a momentum-driven state-space duality-based framework, to model MS by integrating thirty-two heterogeneous dimensions of athletes performance in serve, return, psychology and fatigue. HydraNet integrates a Hydra module, which builds upon a state-space duality (SSD) framework, capturing explicit momentum with a sliding-window mechanism and implicit momentum through cross-game state propagation. It also introduces a novel Versus Learning method to better enhance the adversarial nature of momentum between the two athletes at a macro level, along with a Collaborative-Adversarial Attention Mechanism (CAAM) for capturing and integrating intra-player and inter-player dynamic momentum at a micro level. Additionally, we construct a million-level tennis cross-tournament dataset spanning from 2012-2023 Wimbledon and 2013-2023 US Open, and validate the multi-granularity modeling capability of HydraNet for the MS metric on this dataset. Extensive experimental evaluations demonstrate that the MS metric constructed by the HydraNet framework provides actionable insights into how momentum impacts outcomes at different granularities, establishing a new foundation for momentum modeling and sports analysis. To the best of our knowledge, this is the first work to explore and effectively model momentum across multiple granularities in professional tennis tournaments.

Abstract:We use the fertile ground of betting markets to study the anticipation of major news in financial markets. While there is a considerable body of literature on the accuracy and efficiency of betting markets after important in-match events, there are no studies dealing with the anticipation of such events. This paper tracks bookmaker odds and betting stakes to provide insights into the movement of both prior to goals. Utilising high-resolution (1 Hz) data from a leading European bookmaker for a full season of the top German football league, we analyse whether market participants anticipate major news. In particular, we consider the case of the first goal scored within a match, with its strong impact on the match outcome. Using regression models and state-space models (SSMs) accounting for an underlying market activity level, we investigate whether the bookmaker adjusts odds and bettors tend to place higher stakes on the scoring team right before the first goal is scored. Our results indicate that neither side of the market anticipates goals by significantly adjusting their behaviour.

Machine learning models have become increasingly sophisticated, but this complexity often comes at the cost of interpretability. You can build an XGBoost model that achieves excellent performance on your housing dataset, but when stakeholders ask “why did the model predict this specific price?” or “which features drive our predictions?” you’re often left with limited answers beyond feature importance rankings.SHAP (SHapley Additive exPlanations) bridges this gap by providing a principled way to explain individual predictions and understand model behavior. Unlike traditional feature importance measures that only tell you which features are generally important, SHAP shows you exactly how each feature contributes to every single prediction your model makes.

Ergodicity economics is an umbrella term for addressing issues in economics research by carefully considering the ergodicity problem — that’s the problem that the expected value of something may be different from its time average. One key finding which illustrates the field’s significance within the history of economics is the mapping between utility functions and ergodicity transformations. This mapping provides a physical (as opposed to psychological) interpretation of where utility functions really come from. It also highlights a weakness of expected-utility theory: agents who follow the prescriptions of expected-utility theory do not maximize utility in the long run; agents who follow the prescriptions of ergodicity economics, on the other hand, do maximize utility in the long run.

Abstract:Traditional interpretations of probability, whether frequentist or subjective, make no reference to the concept of energy. In this paper, we propose that assigning hypothetical energy levels to the outcomes of a random variable can yield improved probability estimates. We apply this Boltzmann-informed approach to the context of sports betting and analyze five seasons of the English Premier League data. It was found that when used to compute the Kelly criterion, Boltzmann-informed probabilities consistently outperform probabilities derived from the original betting odds. These findings demonstrate the value of integrating energy-informed probabilities into studying complex social systems.

Abstract:Reinforcement learning with verifiable rewards (RLVR) has boosted math and coding in large language models, yet there has been little effort to extend RLVR into messier, real-world domains like forecasting. One sticking point is that outcome-based reinforcement learning for forecasting must learn from binary, delayed, and noisy rewards, a regime where standard fine-tuning is brittle. We show that outcome-only online RL on a 14B model can match frontier-scale accuracy and surpass it in calibration and hypothetical prediction market betting by adapting two leading algorithms, Group-Relative Policy Optimisation (GRPO) and ReMax, to the forecasting setting. Our adaptations remove per-question variance scaling in GRPO, apply baseline-subtracted advantages in ReMax, hydrate training with 100k temporally consistent synthetic questions, and introduce lightweight guard-rails that penalise gibberish, non-English responses and missing rationales, enabling a single stable pass over 110k events. Scaling ReMax to 110k questions and ensembling seven predictions yields a 14B model that matches frontier baseline o1 on accuracy on our holdout set (Brier = 0.193, p = 0.23) while beating it in calibration (ECE = 0.042, p < 0.001). A simple trading rule turns this calibration edge into $127 of hypothetical profit versus $92 for o1 (p = 0.037). This demonstrates that refined RLVR methods can convert small-scale LLMs into potentially economically valuable forecasting tools, with implications for scaling this to larger models.

While the Premier League continues to set new records for revenue, thus cementing its enormous advantage over the leagues in other countries, the question remains whether it can ever be profitable, given its large losses in recent years. This is a particularly important topic for rational investors, who would surely like to see a return on their capital at some stage.