Probability Of Winning Research Articles

Don’t put all your legs in one basket: Theory and evidence on coopetition in road cycling

We study competition between two groups of individuals with team affiliations, where each group solves a coopetition problem to outperform the other group. Our analysis suggests the presence of a strategic benefit of diversification that goes beyond merely reducing outcome uncertainty. We outline applications in industrial and workplace organization and provide a detailed model of road cycling races, which naturally feature this strategic setup. We find that asymmetric groups tend to cooperate better. Also, having team members in two competing groups is beneficial for a team, as it increases free-riding opportunities in both groups. By analyzing data from over 40 seasons of professional road cycling races, we find empirical evidence in favor of these results. In particular, having a teammate in a group behind positively impacts win probability.

A Nonparametric Global Win Probability Approach to the Analysis and Sizing of Randomized Controlled Trials With Multiple Endpoints of Different Scales and Missing Data: Beyond O'Brien-Wei-Lachin.

Multiple primary endpoints are commonly used in randomized controlled trials to assess treatment effects. When the endpoints are measured on different scales, the O'Brien rank-sum test or the Wei-Lachin test for stochastic ordering may be used for hypothesis testing. However, the O'Brien-Wei-Lachin (OWL) approach is unable to handle missing data and adjust for baseline measurements. We present a nonparametric approach for data analysis that encompasses the OWL approach as a special case. Our approach is based on quantifying an endpoint-specific treatment effect using the probability that a participant in the treatment group has a better score than (or a win over) a participant in the control group. The average of the endpoint-specific win probabilities (WinPs), termed the global win probability (gWinP), is used to quantify the global treatment effect, with the null hypothesis gWinP = 0.50. Our approach involves converting the data for each endpoint to endpoint-specific win fractions, and modeling the win fractions using multivariate linear mixed models to obtain estimates of the endpoint-specific WinPs and the associated variance-covariance matrix. Focusing on confidence interval estimation for the gWinP, we derive sample size formulas for clinical trial design. Simulation results demonstrate that our approach performed well in terms of bias, interval coverage percentage, and assurance of achieving a pre-specified precision for the gWinP. Illustrative code for implementing the methods using SAS PROC RANK and PROC MIXED is provided.

Responsiveness of different disease activity indices in moderate-to-severe ulcerative colitis.

Clinical, endoscopic, histological, and composite instruments are currently used to measure disease activity in patients with ulcerative colitis (UC). We compared the responsiveness of the Mayo Clinic score (MCS), modified MCS (mMS; excluding physician global assessment), partial MCS (pMS; MCS without endoscopic subscore), Robart's Histopathology Index (RHI), and UC-100 score to change after ustekinumab treatment in patients with moderately to severely active UC. Post hoc analysis of the phase 3UNIFI induction trial (ClinicalTrials.gov: NCT02407236) was conducted. Participants with moderately to severely active UC were randomized to receive ustekinumab or placebo. Treatment assignment was the criterion to assess responsiveness, which was quantified using the probability of a treated participant having a larger change in score than a placebo participant, termed the win probability (WinP), and estimated using nonparametric methods. The UC-100 score demonstrated large responsiveness (WinP 0.72 [95% confidence interval: 0.66-0.78]), and the MCS (0.68 [0.62-0.73]), mMS (0.69 [0.63-0.75]), and pMS (0.65 [0.59-0.71]) demonstrated similar effect sizes. Of the component items of the Mayo score, the endoscopic subscore (WinP 0.76 [0.69-0.82]) and the stool frequency subscore (WinP 0.74 [0.69-0.79]) were the most responsive. The Inflammatory Bowel Disease Questionnaire (IBDQ) quality-of-life questionnaire was also responsive (WinP 0.78 [0.72-0.82]). UC disease activity indices are similarly responsive. Depending on the treatment setting, time point of evaluation, and feasibility of measurement, different scores may be used to demonstrate response. These results support the use of mMS as a composite primary endpoint, incorporating both patient-reported and endoscopic outcome measures. The UC-100 score may be more appropriate in settings that also routinely incorporate histological evaluation. There is no funding for this study.

Do uncertain outcomes matter for ticket prices? Examination of the National Football League's secondary ticket market

PurposeThis study examines the impact of outcome uncertainty on the National Football League (NFL) secondary ticket market prices. As a demand-driven market, it is essential to comprehend how resellers respond to outcome uncertainty, one of the consumer demand factors in sports.Design/methodology/approachUsing real-time ticket prices and money lines as a proxy of the probabilities of winning, this study employs a regression analysis and examines 33,554 price observations from the NFL’s secondary ticket market partner, StubHub.FindingsThe result shows a positive relationship between outcome uncertainty and secondary market ticket prices, indicating that resellers adjust the prices in response to the level of outcome uncertainty and put more value on games with greater uncertainty. This finding confirms the demand-driven nature of the secondary ticket market, as outcome uncertainty is one of the demand factors in sports.Originality/valueThis study links the uncertainty of outcome hypothesis with secondary ticket market pricing and fills a gap in the literature by providing an important perspective on games with uncertainty in the secondary ticket market. Outcome uncertainty has limited understanding in relation to secondary ticket market pricing despite its relationship with consumer demand. The positive relationship between outcome uncertainty and the ticket prices, grounded in real-time price data and win probability from sport betting markets, enhances our understanding of price determinations in the secondary ticket market.

In-game soccer outcome prediction with offline reinforcement learning

Predicting outcomes in soccer is crucial for various stakeholders, including teams, leagues, bettors, the betting industry, media, and fans. With advancements in computer vision, player tracking data has become abundant, leading to the development of sophisticated soccer analytics models. However, existing models often rely solely on spatiotemporal features derived from player tracking data, which may not fully capture the complexities of in-game dynamics. In this paper, we present an end-to-end system that leverages raw event and tracking data to predict both offensive and defensive actions, along with the optimal decision for each game scenario, based solely on historical game data. Our model incorporates the effectiveness of these actions to accurately predict win probabilities at every minute of the game. Experimental results demonstrate the effectiveness of our approach, achieving an accuracy of 87% in predicting offensive and defensive actions. Furthermore, our in-game outcome prediction model exhibits an error rate of 0.1, outperforming counterpart models and bookmakers’ odds.

Sample size planning for estimating the global win probability with precision and assurance

Randomized controlled trials commonly employ multiple endpoints to collectively assess the intended effects of the new intervention on multiple aspects of the disease. Focusing on the estimation of the global win probability (WinP), defined as the (weighted) mean of the WinPs across the endpoints that a treated participant would have a better outcome than a control participant, we propose a closed-form sample size formula incorporating pre-specified precision and assurance, with precision denoted by the lower limit of confidence interval and assurance denoted by the probability of achieving that lower limit. We make use of the equivalence of the WinP and the area under the receiver operating characteristic curve (AUC) and adapt a formula originally developed for the difference between two AUCs to handle the global WinP. Unequal variances between treatment groups are allowed. Simulation results suggest that the method performs very well. We illustrate the proposed formula using a Parkinson's disease clinical trial design example.

Trial Design with Win Statistics for Multiple Time-to-Event Endpoints with Hierarchy

The conventional approach to the analysis of composite endpoints is time-to-first event analysis. However, this approach has been criticized because it ignores the differences in clinical severity and may end up emphasizing the less severe time-to-event. To overcome this limitation, win statistics (win ratio, win odds, or net benefit) have become popular in analysis of hierarchical time to event endpoints. However, design of randomized clinical trials using the win statistics is lagging behind. In this article, we derive formulas for the win statistics and probability of ties under specific assumptions that can be useful in practice. We also address two design issues: the selection of meaningful and justifiable design parameters and power calculations, when the win statistics method is the primary analysis method for multiple time-to-event endpoints for a pre-specified hierarchy. Finally, we identify patterns where the win statistics approach would have greater statistical power than time-to-first event analysis. Several examples are used to illustrate the usefulness of the formula-based power calculations.

Enhancing the Probability of Victor for Top Players via Knockout Tournament Fixture

A tournament fixture is an integral part of the tournament rules. It determines the random pairing of contestants, with several matches played per round. The selection of fixture type that optimizes the top player's winning probability significantly affects the financial aspects for organizers, individuals, and participants while also addressing the interests of millions of fans. In addressing this challenge, this study designed a balanced tournament fixture and employed a labeling system to represent each fixture, utilizing a recursive function. By assigning a strength rating to each player, their rankings were established, leading to varied probabilities of winning. It was decided to represent these abilities with randomly selected integers ranging from 1 to 21, with 1 denoting minimum strength and 21 denoting maximum strength. We explore hierarchical knockout tournament fixtures in competitions to develop optimal tournaments that enhance their attractiveness. In this study, we also performed calculations to determine the probability of each player winning in each round, thereby deducing which tournament fixture minimizes or maximizes the likelihood of the strongest player winning. In cases where the number of players is a power of 2, the first half comprises p/2 matches, where p is the total number of players. However, if the number of players is not a power of 2, k matches are played in the first round, with 𝑝 = 2r + 𝑘, where 0 ≤ k < 2r, followed by implementing a balanced tournament fixture. The findings underscore the effectiveness of employing a balanced tournament fixture to maximize the probability of winning in a single-elimination tournament. Keywords: Single Elimination, Binary Tree, Dummy Players, Election Procedures, Hierarchically.

Revealing risky mistakes through revisions

We argue that a choice that is modified, absent any informational change, is revealed to have been a mistake. In an experiment, we allow subjects to choose from budgets over binary lotteries. To identify mistakes, which we interpret as deviations from an underlying “true” preference, we allow subjects to revise a subset of their initial choices. The set of revised decisions improves under several standard definitions of optimality. These mistakes are prevalent: subjects modify over 75% of their initial choices when given the chance. Subjects make larger mistakes when inexperienced and when choosing over lotteries with small probabilities of winning.

Some head starts are bigger than others: dynastic legacies and variation in candidate quality in Taiwan's local elections

Abstract Candidates from established political families are more likely to win than their non-dynastic counterparts because their political inheritance often includes significant advantages in resources and reputation. Yet dynastic candidates also bring their own set of experiences and qualities to the game. Is it the case that their individual characteristics can explain their electoral success, or is their success due to their family legacy? Theoretically, how and when are political resources transferred to the new generation? We examine these questions by looking at non-incumbent candidates in city and county council elections in Taiwan from 2009 to 2014, drawing on unique data on politicians' backgrounds. The profile of dynastic candidates differs from that of non-dynastic candidates: they are younger, have less electoral experience and list experience in different sorts of organizations and jobs. However, these differences are not what drives their electoral success; indeed, they tend to win despite these qualities rather than because of them. Even after controlling for candidate quality, dynastic status remains a powerful predictor, conveying roughly a 20 percentage point increase in the probability of winning. This suggests that while dynastic candidates accrue some of their advantage long before they run for office, a large part of the inheritance is transferred during the campaign.

Boldness-Recalibration for Binary Event Predictions

Probability predictions are essential to inform decision making across many fields. Ideally, probability predictions are (i) well calibrated, (ii) accurate, and (iii) bold, that is, spread out enough to be informative for decision making. However, there is a fundamental tension between calibration and boldness, since calibration metrics can be high when predictions are overly cautious, that is, non-bold. The purpose of this work is to develop a Bayesian model selection-based approach to assess calibration, and a strategy for boldness-recalibration that enables practitioners to responsibly embolden predictions subject to their required level of calibration. Specifically, we allow the user to pre-specify their desired posterior probability of calibration, then maximally embolden predictions subject to this constraint. We demonstrate the method with a case study on hockey home team win probabilities and then verify the performance of our procedures via simulation. We find that very slight relaxation of calibration probability (e.g., from 0.99 to 0.95) can often substantially embolden predictions when they are well calibrated and accurate (e.g., widening hockey predictions’ range from 26%–78% to 10%–91%).

ETHNOMATHEMATICS MIJOU DAMAR IN CHILDREN IN DOKA BARAT VILLAGE ARU SELATAN DISTRICT

In everyday life, education and culture are something that cannot be avoided. The purpose of this research is to describe the ethnomathematics contained in the Mijou Damar game for Doka Barat children. The population in this study were all children aged 6-12 years. Incidental sampling technique. Mijou Damar's game has the concepts of operations on integers, measurement, geometry, probability and quantifiers. The concept of integer operations when children calculate Damar. The concept of measurement and quantifier is seen when the child calculates the distance between where the Damar falls and the center of the circle. Making circles, lines and tools is an application of geometric concepts. Meanwhile, the concept of chance is seen when children can predict who will emerge as the winner and calculate the percentage probability of winning. It shows that ethnomatematics is contained in Mijou Damar.

Football Momentum Analysis based on Logistic Regression

In tennis, momentum is pivotal and can be quantified using metrics like Consecutive Win Rate (CWR), Unforced Error Rate (UER), Break Point Save Rate (BPSR), and Fatigue Factor (FF). Each metric provides insight into a player's performance and state during a match. CWR is a clear momentum indicator, reflecting a player's game dominance, while UER highlights potential lapses in concentration or physical condition. BPSR evaluates a player's clutch performance in critical situations, and FF gauges physical exertion. Utilizing logistic regression, we can predict a player's probability to win at any scoring point, incorporating these metrics as variables. The coefficients obtained from MATLAB analysis (e.g., p1_cwr at 22.73 and p2_ff at -3.26) reveal the positive or negative correlation of these factors with a player's winning chances. In the case of the "2023-wimbledon-1301" match, the logistic model's predictions showed a symmetrical distribution of win probabilities between players, suggesting a balance in momentum swings throughout the match. Initial volatility in Player 1's success rate indicated a strong start, which diminished over time, possibly due to fatigue or the opponent's improving performance. Despite the fluctuations and a period of deadlock, Player 1's consistent performance and superior win rate for most of the game secured the victory. This outcome underscores the importance of maintaining momentum and physical resilience in tennis.

Modeling and prediction of tennis matches at Grand Slam tournaments

In this manuscript, different approaches for modeling and prediction of tennis matches in Grand Slam tournaments are proposed. The data used here contain information on 5,013 matches in men’s Grand Slam tournaments from the years 2011–2022. All regarded approaches are based on regression models, modeling the probability of the first-named player winning. Several potential covariates are considered including the players’ age, the ATP ranking and points, odds, elo rating as well as two additional age variables, which take into account that the optimal age of a tennis player is between 28 and 32 years. We compare the different regression model approaches with respect to three performance measures, namely classification rate, predictive Bernoulli likelihood, and Brier score in a 43-fold cross-validation-type approach for the matches of the years 2011 to 2021. The top five optimal models with highest average ranks are then selected. In order to predict and compare the results of the tournaments in 2022 with the actual results, a comparison over a continuously updating data set via a “rolling window” strategy is used. Also, again the previously mentioned performance measures are calculated. Additionally, we examine whether the assumption of non-linear effects or additional court- and player-specific abilities is reasonable.

Modelling tactical changes in association football using a Markov game

Abstract Accepted by: Phil Scarf We model tactical changes in association football as a Markov game. The pitch is discretized into nine zones and the states of the Markov game are defined according to the zone in which the ball is located in-play, the team in possession and the score. We first model tactical changes in a Markov decision process framework, wherein one team maximizes their probability of winning. Then, we model tactical changes in a two-person zero-sum Markov game framework, wherein both teams maximize their probability of winning. Fundamental to our modelling is the notion that tactical changes impact upon transition rates. We verify the models using data from matches in a season of the Japan Professional Football League. We define a change in transition rates that can be realized by changes in tactics, and illustrate an example of optimal tactical changes when both teams can vary their tactics. The models we develop in the paper can support managers who are considering important decisions about substitutions and changes to formation, for example, when a match is in-play.

P280 Comparative responsiveness of disease activity indices in moderately-to-severely active Ulcerative Colitis: a post-hoc analysis of the UNIFI trial

Abstract Background Clinical, endoscopic, histological, and composite instruments are used to measure disease activity in ulcerative colitis (UC). We compared responsiveness of the Mayo Clinic Score (MCS), modified Mayo Clinic Score (mMS), partial Mayo Clinic Score (pMS), Robarts Histopathology Index (RHI), and UC-100 score after treatment with ustekinumab in patients with moderately-to-severely active UC. Methods Publicly available data from the phase 3 UNIFI induction trial (NCT02407236) comparing the efficacy and safety of ustekinumab versus placebo in patients with moderate-to-severe UC were obtained from Vivli (Vivli ID #7549) and Yale University Open Data Access Project (#2021-4848). Post-hoc analyses were performed on patient level data to ascertain outcomes of interest. Treatment assignment was the criterion for change in evaluation of responsiveness, which was quantified using the probability of a treated patient having a larger change than a placebo patient (win probability [WinP]) and estimated using nonparametric methods. Extent of responsiveness was interpreted according to Cohen’s benchmarks as very small (WinP<0.56); small (0.56≤ WinP<0.64); medium (0.64≤ WinP<0.71); and large (WinP≥0.71). Results Data from 961 patients with moderately-to-severely active UC randomised in UNIFI were included. At baseline, 319 patients received placebo and 624 patients received ustekinumab. A total of 912 (94.9%) patients completed the induction trial. Overall mean baseline index scores were: MCS 8.9 (±1.6), mMS 6.6 (±1.3), pMS 6.2 (±1.4), RHI 16.6 (±8.1), and UC-100 score 71.4 (±16.5) (Table). The UC-100 demonstrated a large degree of responsiveness (WinP 0.72 [95% CI: 0.66, 0.78]). The MCS (0.68 [0.62, 0.73]), mMS (0.69 [0.63, 0.75]), and pMS (0.65 [0.59, 0.71]) demonstrated similar, medium effect sizes (all comparisons p>0.05) (Figure). The RHI alone had a relatively small effect size (WinP 0.63 [0.56-0.70]). Among individual UC-100 components (Mayo Clinic stool frequency subscore, Mayo Clinic endoscopic subscore, and RHI), the endoscopic score was most responsive (WinP 0.59 [0.36-0.82]). The combination of endoscopy and RHI was associated with a medium degree of responsiveness (WinP 0.69 ([0.34, 1.00]). Conclusion UC disease activity indices are similarly responsive to change. Depending on trial design and measurement feasibility, different instruments can be considered optimal for evaluation of efficacy. The mMS is similarly responsive to the MCS without including the physician global assessment and may be more appropriate for determining trial eligibility and defining primary endpoint, whereas the UC-100 may be more appropriate for measuring treatment effect when an objective continuous measure is needed.

Opponent choice in tournaments: winning and shirking

Abstract We propose an alternative design for tournaments that use a preliminary stage, followed by several rounds of single elimination play. The conventional “bracket” design of these tournaments suffers from several deficiencies. Specifically, various reasonable performance criteria for the tournament are not satisfied, there is an unnecessary element of luck in the matchups of players, and there are situations where players have an incentive to shirk. To address all these issues, we allow higher ranked players at the single elimination stage to choose their next opponent sequentially at each round. We allow each player’s ranking either to remain static, or to improve by beating a higher ranked player (Guyon, J. 2022. “Choose your opponent”: a new knockout design for hybrid tournaments. J. Sports Anal. 8: 9–29). Using data from 2215 men’s professional tennis tournaments from 1991 to 2017, we demonstrate the reasonableness of the results obtained. We also perform sensitivity analysis for the effect of increasing irregularity in the pairwise win probability matrix on three traditional performance measures. Finally, we consider strategic shirking behavior at both the individual and group levels, and show how our opponent choice design can mitigate such behavior. Overall, the opponent choice design provides higher probabilities that the best player wins and also that the two best players meet, and reduces shirking, compared to the conventional bracket design.

An efficient and straightforward online vector quantization method for a data stream through remove-birth updating.

The growth of network-connected devices has led to an exponential increase in data generation, creating significant challenges for efficient data analysis. This data is generated continuously, creating a dynamic flow known as a data stream. The characteristics of a data stream may change dynamically, and this change is known as concept drift. Consequently, a method for handling data streams must efficiently reduce their volume while dynamically adapting to these changing characteristics. This article proposes a simple online vector quantization method for concept drift. The proposed method identifies and replaces units with low win probability through remove-birth updating, thus achieving a rapid adaptation to concept drift. Furthermore, the results of this study show that the proposed method can generate minimal dead units even in the presence of concept drift. This study also suggests that some metrics calculated from the proposed method will be helpful for drift detection.

Using Conformal Win Probability to Predict the Winners of the Canceled 2020 NCAA Basketball Tournaments

The COVID-19 pandemic was responsible for the cancellation of both the men’s and women’s 2020 National Collegiate Athletic Association (NCAA) Division I basketball tournaments. Starting from the point when the Division I tournaments and unfinished conference tournaments were canceled, we deliver closed-form probabilities for each team of making the Division I tournaments, had they not been canceled, under a simplified method for tournament selection. We also determine probabilities of a team winning March Madness, given a tournament bracket. Our calculations make use of conformal win probabilities derived from conformal predictive distributions. We compare these conformal win probabilities to those generated through linear and logistic regression on college basketball data spanning the 2011–2012 and 2022–2023 seasons, as well as to other publicly available win probability methods. Conformal win probabilities are shown to be well calibrated, while requiring fewer distributional assumptions than most alternative methods.

A generalized model of the Colonel Blotto stochastic game

Abstract A generalized stochastic modification of the Colonel Blotto game, also known as the game of gladiators, is considered. In the original model, each of two players has a set of gladiators with given strengths. The battle of gladiator teams takes place through individual gladiator battles. In each fight, the probability of gladiator winning is proportional to its strength. Kaminsky et al. in 1984 had obtained a formula for the probability of winning in terms of weighted sums of exponential random variables. Here we provide an interpretation of this result from the Markov chains with continuous time point of view, and a more general statement of the problem is considered, for which a similar expression is obtained.