Final Score John D Barrow

The Perils of Averaging Beware of performance league tables

The Weird League 13 teams. Each play the other 12 teams once: 2pts for win 1 pt for draw All Stars win 5 lose 7. All other games are draws. All Stars score 10 points. The teams they beat score 11 points The teams they lose to score 13. All Stars are bottom of the league But……

The Weird League 13 teams. Each play the other 12 teams once: 2pts for win 1 pt for draw All Stars win 5 lose 7. All other games are draws. Rule change after the last game of the season! 3pts for a win 1 pt for a draw Now All Stars score 15 points. The teams they beat score 11 points The teams they lose to score 14. All Stars are top of the league

Squash – Strangely, A Non-Olympic Sport

R = pS S = p + (1-p)R = p +(1-p)pS S = p/(1-p+p 2 ) and R = p 2 /(1-p+p 2 ) An Old Squash Choice – Play to 9 or 10 from 8-8? Receiver’s choice Prob of scoring next pt as receiver Prob of scoring next pt as server p = prob of winning a pt You only score a point if you win it when you are serving

Play to 9 and your chance of winning is R Play to 10 and you could win via 9-8, 10-8 with prob = RS Or via 9-8,9-9,10-9 with prob = R(1-S)R Or via 8-9, 9-9,10-9 with prob = (1-R)RS Prob of winning to 10 = RS + R 2 (1-S) +RS(1-S) So better off playing to 10 if (1-2S)(1-R) < 0  S > ½ This requires p/(1-p+p 2 ) > ½ or p > ½ (3 -5) = Should the receiver play to 9 or 10 from 8-8 ? Weak players might fluke one point but don’t bet on fluking two!

Table Tennis Scoring Luck versus Skill Old rules play games to at least 21 points and best of three games New rules play games to at least 11 points and best of seven games Assume probability of winning a point is p in a series of random trials For evenly matched opponents: p = ½ +s, with 0 < s <<1/2 Want probability of winning n points before you lose n points: Q = ½ + 2s  (n/  ) Marginal probability of winning a game to n points and then m games to win the match is 2  (m/  ) x 2s  (n/  ) = (4s/  ) x  (m x n) {because the marginal advantage for winning m games is 2s’ (m/  ) but s’ = 2s  (n/  ) is the inherited advantage from playing n pts per game} m x n = 21 x 2 = 42 (old rules – and in badminton) m x n = 11 x 4 = 44 (new rules)

Modern Pentathlon Swim 200m Round-robin épée fencing, Show jumping with ‘random’ horse 20 x air pistol shots (now laser) at 10m Run 3000m staggered starting order in run Shooting combined with run now Three lots of shots followed by 1000m run All in one day

Decathlon Running: 100m + 110m hurdles + 400m m Throwing: shot + discus + javelin Jumping: high jump + long jump + pole vault Times and distances  points scores Heptathlon: 200m + LJ + HJ + 100H + 800m + Javelin + Shot put

Power-Law Scoring Run Jump Throw Distance achieved = D Time run = T C>1 ‘progressive’ C = 1 ‘neutral’ C<1 ‘regressive’ IAAF 2001 Tables Fix A, B, C

Heptathlon 200m C = m C = H C = HJ C = LJ C = 1.41 SP C = 1.05 JT C = 1.04 Has a similar pattern for C values

Best Evers and What Ifs If scoring is made highly progressive C = 2 for all events Dvorak (world no 2) becomes new world record holder with 9468 This change dramatically favours the throwers C = 1.1  2 Daley Thompson missed world record by 1 pt in 1984 Olympics Subsequent change in scoring tables gave him the record! Current world decathlon record = 9026 World record in every event = 12,500 Best ever decathlon performances in each event = 10,485 Usain Bolt’s 9.58s 100m record = 1202 Fastest 100m in a decathlon = 10.22s = 1042 ‘Best’ world record is 74.08m discus (Schult) = 1383

All-time top 100 performance patterns What you need to do to Earn 900 points in each event for a 9000 points total Which Events Give the Biggest Pay-off ? World record 9026 points Roman Sebrle in 2001 LJ, 110H, sprints ** 1500m, throws xx

Could avoid points transformations by having B-Total = LJ  HJ  PV  JT  DT  SP_______ T(100)  T(400m)  T(100mH) T(1500m) List all distances in metres and all times in seconds Units of total are m 6 /s 4 A Different Scoring Scheme It has its own biases! Sebrle (9026 pts): B = 2.29 Dvorak (8994 pts): B = 2.40

Goal Average or Goal Difference ? < Goal Average, GA = F/A > Goal Difference, GD = F - A : Arsenal W22 D20 L6 GD = 37 but GA = 73/36 = 2.03 Liverpool W22 D20 L6 GD = 37 but GA = 65/28 = 2.32 The league rule was that with tied GD the winner team with bigger F Arsenal { This rule isn’t now used by the Premier League }

Premier League Table Why bother with points ? Just use goals scored or goal difference

Is the Premier League Random? Random Model uses P(draw) = 1/4 P(home win) = 3/8 P(away win) = 3/8 Random model

The Argentinian Primera Football League Is Mathematically Interesting Relegates the 2 teams with the lowest average points per match over 3 years of performances 2 Championships per year  6 Championships decide (unless haven’t been in the league for those years) The next 2 teams play off against 3 rd and 4 th in the league below Relegation of River Plate after 110 years

A Different Type of Scoring System Traditional league: 2 pts for a win, 1 pt for a draw

If Your Score = Sum of points held by the teams you beat

X = (A,NZ,WI,E,B,SL,I,SA) = (0.729,0.375,0.104,0.151,0.153,0.394,0.071,0.332) First-ranked positive eigenvector New Table

Luck versus Skill in Tennis Prob of winning a point = p Prob of winning game is G 1 way to win 40-0, 4 ways to win 40-15, 10 ways to win 40-30, 20 ways to deuce and the chance of reaching deuce is p 3 (1-p) 3 Prob of winning from deuce = D = p 2 +2p(1-p)D so D = p 2 /{p 2 +(1-p) 2 } If p = ½ + u with u very small (ie evenly matched players) G = ½ + 5u/2 approximately Over 3 sets this grows to ½ + 11u and over 5 sets to ½ + 13u (Ignoring tie-breaks) ‘Skill’ (non-zero u > 0) should win out increasingly in the long run The closer the players are matched the more sets you should play Prob of winning game, G = p 4 + 4p 4 (1-p) +10p 4 (1-p) 2 +20p 3 (1-p) 3 p 2 /{p 2 +(1-p) 2 }

Psychological Momentum Odds on winning first set = Prob you win it/Prob you lose it: O = S/(1-S) Assume winning gives you a psychological ‘boost’, B Odds of winning next set become = O×B But if you lose the odds next time are reduced to O/B In a 3-set match here is how the psychological momentum changes the odds

The Great British Football Team in London 1908