Final ScoreFinal ScoreJohn D John D BarrowBarrow

The Perils of AveragingThe Perils of Averaging

Beware of performance league tables

The Weird LeagueThe Weird League13 teams. Each play the other 12 teams once: 13 teams. Each play the other 12 teams once:

2pts for win 1 pt for draw2pts for win 1 pt for drawAll Stars win 5 lose 7. All other games are draws.All Stars win 5 lose 7. All other games are draws.

All Stars score 10 points. The teams they beat score 11 pointsThe teams they lose to score 13.

All Stars are bottom of the leagueAll Stars are bottom of the league

But……

The Weird LeagueThe Weird League13 teams. Each play the other 12 teams once: 13 teams. Each play the other 12 teams once:

2pts for win 1 pt for draw2pts for win 1 pt for drawAll Stars win 5 lose 7. All other games are draws.All Stars win 5 lose 7. All other games are draws.

Rule change after the last game of the season!Rule change after the last game of the season!3pts for a win 1 pt for a draw3pts for a win 1 pt for a draw

Now All Stars score 15 points. The teams they beat score 11 pointsThe teams they lose to score 14.

All Stars are top of the leagueAll Stars are top of the league

Squash – Strangely, A Non-Olympic SportSquash – Strangely, A Non-Olympic Sport

R = pS

S = p + (1-p)R = p +(1-p)pS

S = p/(1-p+p2) and R = p2/(1-p+p2)

An Old Squash Choice – Play to 9 or 10 from 8-8?An Old Squash Choice – Play to 9 or 10 from 8-8?Receiver’s choiceReceiver’s choice

Prob of scoring Prob of scoring next pt as receivernext pt as receiver Prob of scoring Prob of scoring

next pt as servernext pt as server

p = prob of p = prob of winning a ptwinning a pt

You only scorea point if you

win it when youare serving

Play to 9 and your chance of winning is RPlay to 10 and you could win via 9-8, 10-8 with prob = RSOr via 9-8,9-9,10-9 with prob = R(1-S)ROr via 8-9, 9-9,10-9 with prob = (1-R)RS

Prob of winning to 10 = RS + R2(1-S) +RS(1-S)

So better off playing to 10 if (1-2S)(1-R) < 0 S > ½

This requires p/(1-p+p2) > ½ or

p > ½ (3 -5) = 0.382

Should the receiver play to 9 or 10 from 8-8 ?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 ScoringTable Tennis ScoringLuck versus SkillLuck versus Skill

Old rules play games to at least 21 points and best of three gamesNew 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 trialsAssume probability of winning a point is p in a series of random trials

For evenly matched opponents: p = ½ +s , with 0 < s <<1/2For evenly matched opponents: p = ½ +s , with 0 < s <<1/2

Want probability of winning Want probability of winning nn points before you lose points before you lose nn points: points:

Q = ½ + Q = ½ + 2s2s(n/(n/))

Marginal probability of winning a game to Marginal probability of winning a game to nn points and points and then then mm games to win the match is games to win the match is

22(m/(m/) x 2s) x 2s(n/(n/) = ) = (4s/(4s/) x ) x (m x n)(m x n){because the marginal advantage for winning m games is 2s’ (m/{because the marginal advantage for winning m games is 2s’ (m/) )

but s’ = 2sbut s’ = 2s(n/(n/) is the inherited advantage from playing n pts per game}) 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 = 21 x 2 = 42 (old rules – and in badminton)m x n = 11 x 4 = 44 (new rules)m x n = 11 x 4 = 44 (new rules)

Modern PentathlonModern Pentathlon

Swim 200mRound-robin épée fencing,

Show jumping with ‘random’ horse20 x air pistol shots (now laser) at 10m

Run 3000m staggered starting order in runShooting combined with run now

Three lots of shots followed by 1000m run

All in one day

DecathlonDecathlon

Running: 100m + 110m hurdles + 400m + 1500mRunning: 100m + 110m hurdles + 400m + 1500mThrowing: shot + discus + javelinThrowing: shot + discus + javelin

Jumping: high jump + long jump + pole vaultJumping: high jump + long jump + pole vaultTimes and distances Times and distances points scores points scores

Heptathlon: 200m + LJ + HJ + 100H + 800m + Javelin + Shot put

Power-Law ScoringPower-Law Scoring

Run

JumpThrow

Distance achieved = DDistance achieved = D

Time run = TTime run = T

C>1 ‘progressive’C>1 ‘progressive’C = 1 ‘neutral’C = 1 ‘neutral’C<1 ‘regressive’C<1 ‘regressive’

IAAF 2001IAAF 2001TablesTables

Fix A, B, CFix A, B, C

HeptathlonHeptathlon

200m C = 1.81800m C = 1.88100H C = 1.835HJ C = 1.348LJ C = 1.41SP C = 1.05JT C = 1.04

Has a similar pattern for C values

Best Evers and What IfsBest Evers and What IfsIf scoring is made highly progressive C = 2 for all eventsIf scoring is made highly progressive C = 2 for all events

Dvorak (world no 2) becomes new world record holder with 9468Dvorak (world no 2) becomes new world record holder with 9468

This change dramatically favours the throwers C = 1.1 This change dramatically favours the throwers C = 1.1 2 2

Daley Thompson missed world record by 1 pt in 1984 OlympicsDaley Thompson missed world record by 1 pt in 1984 OlympicsSubsequent change in scoring tables gave him the record!Subsequent change in scoring tables gave him the record!

Current world decathlon record = 9026World record in every event = 12,500

Best ever decathlon performances in each event = 10,485

Usain Bolt’s 9.58s 100m record = 1202Usain Bolt’s 9.58s 100m record = 1202

Fastest 100m in a decathlon = 10.22s = 1042Fastest 100m in a decathlon = 10.22s = 1042

‘‘Best’ world record is 74.08m discus (Schult) = 1383Best’ world record is 74.08m discus (Schult) = 1383

All-time top 100 performance patternsWhat you need to do toEarn 900 points in each event

for a 9000 points total

Which Events Give the Biggest Pay-off ?Which Events Give the Biggest Pay-off ?

World record 9026 points World record 9026 points Roman Sebrle in 2001Roman 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 secondsUnits of total are m6/s4

A Different Scoring SchemeA Different Scoring Scheme

It has its own biases!

Sebrle (9026 pts): B = 2.29Sebrle (9026 pts): B = 2.29Dvorak (8994 pts): B = 2.40Dvorak (8994 pts): B = 2.40

Goal Average or Goal Difference ?Goal Average or Goal Difference ?

<1976-7 Goal Average, GA = F/A

> 1976-7 Goal Difference, GD = F - A

1988-9: Arsenal W22 D20 L6 GD 73-36 = 37 but GA = 73/36 = 2.03 Liverpool W22 D20 L6 GD 65-28 = 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 }

2009-10 Premier League Table2009-10 Premier League Table

Why bother with points ?Why bother with points ?

Just use goals scoredJust use goals scoredor or

goal differencegoal difference

Is the Premier League Random?Is the Premier League Random?

Random Modeluses

P(draw) = 1/4 P(home win) = 3/8P(away win) = 3/8

Randommodel

The Argentinian Primera Football LeagueThe Argentinian Primera Football LeagueIs Mathematically InterestingIs Mathematically Interesting

Relegates the 2 teams with the lowest average points per matchover 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 3rd and 4th in the league below

Relegation of River Plate after 110 years

A Different Type of Scoring SystemA 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 TennisLuck versus Skill in Tennis

Prob of winning a point = pProb 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 andthe chance of reaching deuce is p3(1-p)3

Prob of winning from deuce = D = p2 +2p(1-p)D so D = p2/{p2+(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 runSkill’ (non-zero u > 0) should win out increasingly in the long runThe closer the players are matched the more sets you should playThe closer the players are matched the more sets you should play

Prob of winning game, G = pProb of winning game, G = p44 + 4p + 4p44(1-p) +10p(1-p) +10p44(1-p)(1-p)22 +20p +20p33(1-p)(1-p)3 3 pp22/{p/{p22+(1-p)+(1-p)22}}

Psychological MomentumPsychological MomentumOdds on winning first set = Prob you win it/Prob you lose it: O = S/(1-S)

Assume winning gives you a psychological ‘boost’, BOdds of winning next set become = O×BBut 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 oddsIn a 3-set match here is how the psychological momentum changes the odds

The Great British Football Team in London 1908The Great British Football Team in London 1908