Mathematical Modeling Transfers to Football

Dr. Roger KaufmannJune 17, 2008

Mathematical Modeling Transfers to Football

Part 1 – introduction• Relation football ↔ mathematics• A first glance at the outcome

Part 2 – mathematical approach• Strength of a team• Calculation of probabilities

Part 3 – today's matches• Up-to-date figures for tonight

Part 4 – backtesting and further applications• Backtesting• Outlook

June 17, 2008 2

Football and Mathematics

• Strength of teams can be estimated– Statistics come into play

• Uncertainties play an important role– Probabilities are the key element

• Unexpected events change the initial situation– So-called conditional probabilities need to be

considered

June 17, 2008 3

Wanted: European Champion

June 17, 2008 4

Probabilities

SpainNetherlandsGermanyCroatiaPortugalTurkeyItaly

The favorites:

Mathematical Modeling Transfers to Football

Part 1 – introduction• Relation football ↔ mathematics• A first glance at the outcome

Part 2 – mathematical approach• Strength of a team• Calculation of probabilities

Part 3 – today's matches• Up-to-date figures for tonight

Part 4 – backtesting and further applications• Backtesting• Outlook

June 17, 2008 5

Mathematical Ingredients

Strength of a team• Ranking lists– Matches won, tied, lost– Goals scored, goals received

• FIFA World Ranking– Strength of a team is calculated depending on the

results in each match

• No consideration of single football players (injuries, etc.)– Only measurable information, no personal opinion

June 17, 2008 6

FIFA World Ranking

• Both friendly and qualifying matches considered• Monthly update

June 17, 2008 7

Mathematical Ingredients (cont.)

General football statistics

• Goals scored by home teams• Goals scored by away teams• Frequency of draws• Frequency of favorites underestimating

outsiders

June 17, 2008 8

A Single MatchKnown:• Strength of both teams• Average number of goals in international matchesCalculate:• Expected number of goals for both teams (n1, n2)Account for random effects and their correction:• Use Poisson distributions (with expected values n1, n2)

to model the number of goals scored• Adapt (i.e. increase) probability of drawsOutput:• P[0:0], P[1:0], P[1:1], etc.; and P[win/draw/loss]June 17, 2008 9

Dynamic Sports Analysis – the Output

June 17, 2008 10

Putting the Puzzle together – Calculation of a Championship

The steps for calculating a whole championship (e.g. national championship, world cup, EURO):

• Assess strength of each team• Calculate probability for each match• Simulate a potential result for each match• This yields one potential final ranking list• Repeat the above procedure thousands of times• Calculate probabilities for outcomes of interest

June 17, 2008 11

National Championship vs.World Cup/EURO

National championship:• Many matches• Randomness plays a minor role• Typically the strongest team wins

World cup/EURO (knockout system):• A single bad day can ruin all hopes• Randomness plays an important role• Big chances for outsiders

June 17, 2008 12

Betting Advice

Compare: calculated probability vs. odds[all odds and probabilities as of end April 2008]

Germany 15.0% x 5 = 75.0%Italy 13.4% x 8 = 107.2%Spain 13.2% x 7 = 92.4%Czech Rep. 11.1% x 15 = 166.5%Greece 7.5% x 26 = 195.0%Romania 3.7% x 41 = 151.7%

June 11, 2008 13

Mathematical Modeling Transfers to Football

Part 1 – introduction• Relation football ↔ mathematics• A first glance at the outcome

Part 2 – mathematical approach• Strength of a team• Calculation of probabilities

Part 3 – today's matches• Up-to-date figures for tonight

Part 4 – backtesting and further applications• Backtesting• Outlook

June 17, 2008 14

Today's Matches

Netherlands – Romania49.1% Netherlands wins28.2% draw22.7% Romania wins

June 17, 2008 15

Quarter Finals Semi Finals Final Champion

Netherlands 100.0% 68.4% 33.8% 18.9%

Italy 44.8% 17.7% 8.8% 4.5%

Romania 34.4% 10.8% 5.0% 2.2%

France 20.8% 6.7% 2.9% 1.3%

France – Italy27.0% France wins29.0% draw44.0% Italy wins

European Championevolution of probabilities over time

June 17, 2008 16

Group A 17 June 15 June 11 June 6 JunePortugal 9.9% 15.4% 9.7% 6.8%Turkey 5.1% 1.8% 0.5% 2.6%Czech Rep. --- 7.3% 14.9% 12.2%Switzerland --- --- 0.8% 2.7%

Group B 17 June 15 June 11 June 6 JuneGermany 15.6% 11.5% 17.7% 14.2%Croatia 14.3% 10.7% 6.3% 5.4%Austria --- 0.2% 0.1% 0.7%Poland --- 0.03% 0.6% 2.1%

European Championevolution of probabilities over time

June 17, 2008 17

Group C 17 June 15 June 11 June 6 JuneNetherlands 18.9% 18.3% 11.1% 4.5%Italy 4.5% 4.3% 5.7% 15.6%Romania 2.2% 2.1% 3.0% 3.6%France 1.3% 1.2% 4.3% 5.9%

Group D 17 June 15 June 11 June 6 JuneSpain 24.6% 23.9% 19.5% 13.2%Sweden 2.3% 2.1% 3.5% 1.3%Russia 1.3% 1.3% 0.5% 1.7%Greece --- --- 1.6% 7.5%

Comparison with UBS, DeKaBank and University of Vienna

June 17, 2008 18

Quarter Finals Semi Finals Final Champion

UBS CZE, GER, ITA, SPA, SUI, CRO, NED, GRI CZE, SUI, ITA, NED CZE, ITA CZE

DeKaBankCZE, GER, ITA, SPA, ???, ???,

FRA, ???CZE, GER, ITA, SPA GER, ITA GER

University Vienna POR, GER, ITA, SPA, CZE, CRO, FRA, GRI POR, GER, ITA, SPA GER, SPA GER

Roger Kaufmann CZE, GER, ITA, SPA, POR, CRO, FRA, GRI CZE, GER, ITA, SPA GER, ITA ITA

• Other researchers and risk managers performed calculations on the most probable outcome of the EURO 2008 as well.

• Although based on different data sources, most results resemble each other.

Mathematical Modeling Transfers to Football

Part 1 – introduction• Relation football ↔ mathematics• A first glance at the outcome

Part 2 – mathematical approach• Strength of a team• Calculation of probabilities

Part 3 – today's matches• Up-to-date figures for tonight

Part 4 – backtesting and further applications• Backtesting• Outlook

June 17, 2008 19

BacktestingOnline betting pools• About 60 participations. Always among first 1/3

• Several 1st ranks, won many prizesSwiss lottery• Several times 12 correct results out of 13 • Return more than twice the expected oneMathematical backtesting• Backtesting possible for accumulation of predictions;

not for a single match• e.g. 20 events with a probability of 80% each => expect

14 to 18 occurrences

June 17, 2008 20

Outlook on Further ApplicationsLive calculations during a match• Impact of:– Goals scored– Red cards given– Penalties given– Time evolved

• Help manager to decide:– New forward in order to score a further goal– New defender in order to keep the current result– How much risk to take at a given moment

June 17, 2008 21

June 11, 2008 22

Assumed results CZE – POR 2:1 CZE – POR 1:1 CZE – POR 1:2

SUI – TUR 2:1

CZE 100.0%POR 72.5%SUI 27.5%TUR 0.0%

CZE 96.6%POR 78.4%SUI 25.0%TUR 0.0%

CZE 75.1%POR 95.6%SUI 22.6%TUR 6.7%

SUI – TUR 1:1

CZE 100.0%POR 76.5%SUI 23.0%TUR 0.5%

CZE 83.8%POR 81.9%SUI 19.8%

TUR 14.5%

CZE 80.7%POR 100.0%

SUI 2.8%TUR 16.5%

SUI – TUR 1:2

CZE 97.1%POR 81.1%

SUI 4.5%TUR 17.3%

CZE 79.7%POR 99.6%

SUI 0.0%TUR 20.6%

CZE 76.8%POR 100.0%

SUI 0.0%TUR 23.2%

Example of a Manager DecisionQualification for Quarter Finals

Assumed results CZE – POR 2:1 CZE – POR 1:1 CZE – POR 1:3

SUI – TUR 2:1

CZE 100.0%POR 72.5%SUI 27.5%TUR 0.0%

CZE 96.6%POR 78.4%SUI 25.0%TUR 0.0%

CZE 64.9%POR 99.8%SUI 28.1%TUR 7.3%

SUI – TUR 1:1

CZE 100.0%POR 76.5%SUI 23.0%TUR 0.5%

CZE 83.8%POR 81.9%SUI 19.8%

TUR 14.5%

CZE 79.9%POR 100.0%

SUI 2.8%TUR 17.3%

SUI – TUR 1:2

CZE 97.1%POR 81.1%

SUI 4.5%TUR 17.3%

CZE 79.7%POR 99.6%

SUI 0.0%TUR 20.6%

CZE 61.9%POR 100.0%

SUI 0.0%TUR 38.1%

Thank you…

…for your attention!

• Questions?

Enjoy tonight’s match!

June 17, 2008 23