1 Oliver Hatfield 1 (Supervisor) 1 Lancaster University September 2, 2016
Contents 1 Introduction and Motivation 2 Simulations of Matches Markov Chains Improvements & Conclusions 3 Detecting Changes in Performance Likelihood Ratio Test Finding an Appropriate Threshold Improvements & Conclusions 4 Probabilities of Winning Matches Motivation Methodology 5 Modified ELO Rating System Why need a new rating system? Why Modified ELO? How does ELO work? Further Improvements 6 Overall Conclusions Richard Ings, Former ATP executive vice president.
Introduction and Motivation January 2016, documents released revealing widespread accusations of match-fixing. Reports of players throwing matches in return for large sums of money. Article on Buzzfeed The Tennis Racket 1 1 https://www.buzzfeed.com/heidiblake/the-tennis-racket
Introduction and Motivation Most publicised match-fixing accusation between Davydenko and Arguello. Davydenko was ranked 4th and Arguello 87th in the world. Davydenko won the first set. Surely the betting odds would have Davydenko as favourite? Figure: Davydenko was involved in the most highly publicised match-fixing accusation. Figure: Pilgrim Tennis Club where Davydenko trained.
Introduction and Motivation However the betting data had Davydenko as the underdog. Large sums of money placed on Arguello to win. Davydenko lost the match. So how do we detect match-fixing?
Simulations of Matches Simulate tennis matches to gain data to test upon. Probability of winning a point on serve is p = S ± D/2. S = 0.645. D is the difference in ability between the players. Random numbers, X i, drawn from a binomial distribution X i Bin(p, n). Random numbers, X i, determine who wins the points.
Example Random binomial numbers, 1 0 1 1 0 1, would result in: 1 15-0 1 40-15 0-0 0 1 30-15 0 1 W1 15-15 40-30
Markov Chains Markov Chains Problems with deuce and at end of tiebreaks. Absorbing Markov chains can be used. 40 40, A1 and A2 are transient states. W1 and W2 are absorbing states. p p 1-p 1-p W1 A1 40-40 A2 W2 1-p p
Markov Chains Absorbing Markov Chains Canonical form used for the transition matrix ( Q R ) 0 I Q is for the transient states. R is for the absorbing states. We use equations 1 and 2 to obtain probabilities of reaching the absorbing states. N = (I Q) 1 (1) B = NR (2)
Improvements & Conclusions Simulations of Matches Improvements: Introduce return abilities, creating better estimations of p. Track individual points within deuce and end of tiebreaks. Conclusions Programme that simulates tennis matches based on player s abilities. Creating data that can tested upon.
Likelihood Ratio Test Detecting Changes in Performance Changepoint methods used to detect changes in player s performance. Assume points are i.i.d. Likelihood ratio test used to detect single changepoints. Produces test statistic, λ, which is tested against a threshold, c.
Finding an Appropriate Threshold Finding an Appropriate Threshold Computed many simulations on simulated data with no changepoints. Found the 95% quantile of the test statistics. c = 3.76. Figure: Histogram of test statistics over multiple simulations of matches with 95% quantile marked
Improvements & Conclusions Detecting Changes in Performance Improvements: Different probabilities dependant on who s serving. Online changepoint methods. Conclusions: Detect potential changes in player s performance. Assumption of i.i.d data is unrealistic.
Motivation Probabilities of Winning Matches Need ability to calculate probability of winning matches from different scores. Then compare to the betting odds. If large discrepancies, potential case of match fixing.
Methodology Working out the Probabilities Using the Law of Total Probability: Pr(Win Match) = Pr(Win Match Win Set)Pr(Win Set) + Pr(Win Match Lose Set)Pr(Lose Set). Pr(Win Set) = Pr(Win Set Win Game)Pr(Win Game) + Pr(Win Set Lose Game)Pr(Lose Game). Probabilities calculated using absorbing Markov chains.
Why need a new rating system? Modified ELO Rating System Why do we need a new rating system? Current system only includes previous year s results. Doesn t incorporate strength of the opposition. Long term injuries result in rapid ranking declines. Figure: Del Potro has had many injuries and his rating has been severely affected in ATP system.
Why Modified ELO? Why use Modified ELO? Ratings are accumulated over all previous matches. Strength of opponent accounted for. Inactive players ratings declines slower. Expectation, E, of winning the match is calculated. E used to calculate an estimate for D. Figure: How ratings decay for an inactive player over 2 years.
How does ELO work? How does ELO work? Each player has a ratings R i,t. i is the unique player number. t is time. Expectation for each player to win the game is calculated. E i = Ratings are updated by: 1 1 + 10 (R j,t R i,t )/400. R i,t+1 = R i,t + K(S i E i ), S i is 1 or 0 depending who won the match. K is the maximum possible adjustment. K optimised to be 15.6.
How does ELO work? The Modifications Increased K by 50% for Grand Slam so more points can be gained. Exponential decay in rating points for players who are inactive for 8 weeks or more. Ratings only start to decay when completed a tournament.
How does ELO work? Initial Ratings 100 log(atp Ranking Points+1) Scales ratings to below 1000. +1 to allow for players entering system with 0 rating. Ran data, using ELO, for all matches from 2005-2016. Player ATP ELO Federer 6525 878 Roddick 3655 820 Safin 3360 812 Moya 2520 783 Coria 2400 778 Henman 2360 777 Agassi 2100 765 Nalbandian 1945 757 Table: Rankings in 2005 for ATP and new ratings for ELO.
How does ELO work? Player Ratings Figure: Graph showing changes in players ranking points. Things to note: Ratings don t decay until played first tournament When players retire, ratings decay to zero. Players with low ratings can increase rapidly.
How does ELO work? Up to Date Rankings ELO ATP Player Rating Player Rating Djokovic 920 Djokovic 16790 Federer 749 Murray 8945 Murray 694 Federer 8165 Nadal 595 Wawrinka 6865 Wawrinka 575 Nadal 5230 Nishikori 553 Berdych 4560 Berdych 550 Nishikori 4235 Ferrer 526 Ferrer 4145 Table: Table showing the top 8 players in the ELO and ATP rating systems as of 01-18-2016
Further Improvements Further Improvements Optimize increase in K for Grand Slams. Introduce different K s for different standard tournaments. Stop average ratings from decreasing due to exponential decay.
Conclusion Simulations used to generate data. Compare the probability of the player winning the match to betting odds to check for differences. If discrepancies are detected, check for changes in performance. Estimate the difference in abilities of the players using the ELO rating system s expectations of winning the matches.
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