Rollarama World Football Dice Game

Rollarama World Football is a "shootout dice and card game", and so when I saw it on the shelf of a charity shop recently, I decided to give it a try.

The game comes with full instructions on how to play it "properly", but I just liked the idea and the packaging which clearly showed a six-sided die, a 12-sided die and a 20-sided die.

The dice values aren't 1-6, 1-12 and 1-20; instead the values are the number of goals a particular team scores in a game.  Each country (there are 30 in total) has a card which shows if it's assigned the 6, 12 or 20 sided die and you roll that die for that country when it plays a match.  For example, Canada is a D6 team, while Portugal is a D20 team and Mexico is a D12 team.  There are rules included on how to play this as a two-player game, using the dice and the cards, but I found a more interesting game for one player using the same raw materials.

The questions I was most interested in answering are how fair the game is, and if the game could be won by one of the 'weaker' D6 teams.  To do this, I set up my own World Football championship, the 2023-4 Winter World Cup.  This comprises five groups of six countries (the 30 countries in total) who will play in their own mini-league.  The top team from each group goes forward, along with the top three performing runners-up, to give eight teams. These eight will play in quarter-finals, then semi-finals and a final.  The draw for the knockout stages will be entirely random, I'm not seeding any team.

Here's the starting roster and the groups, all drawn randomly from the deck of 30 cards.  As you can see, Brazil got a very good draw as the only D20 country in a group of D6s, while Group A has no D20s.

Group A:
D12: Serbia, Denmark, Mexico
D6: Saudi [Arabia],  USA, Cameroon, 

Group B:
D12: Croatia, Switzerland
D20:  Spain, England, Germany, France

Group C:
D6: Tunisia
D12:  S Korea, Senegal, Japan
D20:  Netherlands, Uruguay

Group D:
D6:  Scotland, Ghana, Canada, Wales, Morocco
D20:  Brazil

Group E:
D6:  Ecuador
D12: Iran, Poland
D20: Portugal, Argentina, Belgium

Each team in the group plays the other teams once, so in a league of six, there are 15 games altogether.  There are three points for a win, one for a draw and none for a loss.  In the event of a tie on points, then goal difference will be taken into account.

Here are the results:

GROUP A

Serbia	1	USA	3
Cameroon	2	Denmark	2
Mexico	2	Saudi	1
Serbia	3	Cameroon	2
USA	3	Mexico	1
Denmark	2	Saudi	1
Serbia	4	Denmark	3
USA	1	Saudi	0
Mexico	3	Cameroon	1
Serbia	4	Mexico	1
USA	1	Denmark	3
Saudi	0	Cameroon	3
Serbia	1	Saudi	1
Cameroon	3	USA	0
Denmark	2	Mexico	4

GROUP B

Spain	3	Croatia	1
Switzerland	3	England	3
Germany	1	France	3
Spain	5	Switzerland	2
Croatia	4	Germany	0
England	5	France	2
Spain	3	England	2
Croatia	0	France	2
Switzerland	4	Germany	3
Spain	1	Germany	1
Croatia	2	England	5
France	5	Switzerland	3
Spain	3	France	5
Croatia	2	Switzerland	1
England	5	Germany	0

GROUP C

South Korea	2	Senegal	3
Japan	4	Netherlands	2
Uruguay	4	Tunisia	3
South Korea	2	Japan	1
Senegal	4	Uruguay	1
Tunisia	0	Netherlands	5
South Korea	1	Netherlands	4
Japan	3	Uruguay	0
Senegal	1	Tunisia	2
South Korea	2	Uruguay	3
Japan	1	Tunisia	0
Netherlands	0	Senegal	2
South Korea	2	Tunisia	3
Senegal	4	Japan	0
Uruguay	2	Netherlands	4

 GROUP D

Brazil	0	Scotland	3
Morocco	2	Canada	2
Wales	1	Ghana	2
Brazil	1	Morocco	0
Scotland	2	Wales	2
Canada	0	Ghana	0
Brazil	2	Canada	2
Scotland	3	Ghana	3
Wales	0	Morocco	2
Brazil	0	Wales	1
Scotland	2	Canada	3
Ghana	2	Morocco	1
Brazil	0	Ghana	0
Scotland	0	Morocco	2
Wales	0	Canada	2

GROUP E

Iran	3	Portugal	3
Argentina	2	Belgium	5
Ecuador	1	Poland	4
Iran	3	Argentina	5
Portugal	3	Ecuador	1
Belgium	2	Poland	3
Iran	4	Belgium	2
Portugal	4	Poland	1
Ecuador	2	Argentina	1
Iran	3	Ecuador	0
Portugal	4	Belgium	3
Poland	2	Argentina	2
Iran	1	Poland	4
Portugal	5	Argentina	2
Ecuador	0	Belgium	3

Final tables (this game is a statistician's or an analyst's dream!)

GROUP A
Team	P	W	D	L	F	A	GD	Pts
Serbia	5	3	1	1	13	10	3	10
Mexico	5	3	0	2	11	11	0	9
USA	5	3	0	2	8	8	0	9
Cameroon	5	2	1	2	11	8	3	7
Denmark	5	2	1	2	12	12	0	7
Saudi	5	0	1	4	3	9	-6	1

GROUP B
Team	P	W	D	L	F	A	GD	Pts
France	5	4	0	1	17	12	5	12
England	5	3	1	1	20	10	10	10
Spain	5	3	1	1	15	11	4	10
Croatia	5	2	0	3	9	11	-2	6
Switzerland	5	1	1	3	13	18	-5	4
Germany	5	0	1	4	5	17	-12	1

GROUP C
Team	P	W	D	L	F	A	GD	Pts
Senegal	5	4	0	1	14	5	9	12
Netherlands	5	3	0	2	15	9	6	9
Japan	5	3	0	2	9	8	1	9
Tunisia	5	2	0	3	8	13	-5	6
Uruguay	5	2	0	3	10	16	-6	6
South Korea	5	1	0	4	9	14	-5	3

GROUP D
Team	P	W	D	L	F	A	GD	Pts
Canada	5	2	3	0	9	6	3	9
Ghana	5	2	3	0	7	5	2	9
Morocco	5	2	1	2	7	5	2	7
Scotland	5	1	2	2	10	10	0	5
Brazil	5	1	2	2	3	6	-3	5
Wales	5	1	1	3	4	8	-4	4

GROUP E
Team	P	W	D	L	F	A	GD	Pts
Portugal	5	4	1	0	19	10	9	13
Poland	5	3	1	1	14	10	4	10
Iran	5	2	1	2	14	14	0	7
Belgium	5	2	0	3	15	13	2	6
Argentina	5	1	1	3	12	17	-5	4
Ecuador	5	1	0	4	4	14	-10	3

Notes:

Serbia and Mexico were first and second in Group A, as would be expected as D12s.

Group B was won by France, and there were no major upsets.

Senegal won Group C, ahead of the two D20 teams Uruguay and Netherlands.  Senegal won four of their five games, including a 4-1 win over Uruguay and a 2-0 win against the Netherlands.

In a significant shock, Brazil game fifth in Group D.  As a D20 country playing D6s, Brazil were expected to come first, but only won one of their games.  It was very tight among the other teams, as they were all D6 teams and therefore had a high probability of drawing each other.  Six of the 15 games were drawn.

Portugal topped Group E easily, winning four of their five games and remaining unbeaten throughout.  Ecuador, the only D6 team in the group, came last, winning only one game.

The group winners were:
Serbia (D12), France (D20), Sengal (D6), Canada (D6) and Portugal (D20).

The best-performing runners-up were Mexico (A), England (B) and Netherlands (D).

The Quarter Finals, drawn from the eight qualifying teams:

Serbia vs Netherlands
France vs Senegal
Canada vs Portugal
Mexico vs Ghana

Serbia       2     Netherlands     2 (Netherlands 5-2 on penalties)
France       2    Senegal              1
Canada     1    Portugal             2
Mexico     1     Ghana                1 (Mexico 11-10 on penalties)

The Semi Finals, drawn from the four winners

Netherlands vs France
Portugal vs Mexico (who have done surprisingly well)

Netherlands 4 - France 5
Portugal 4 - Mexico 4 (Portugal win 3-2 on penalties)

The Final was unsurprisingly between two D20 teams, France and Portugal.

France 2 - Portugal 1

The dice:

Although I've described the teams as D6, D12 and D20, the dice aren't normal dice with values 1-6, 1-12 or 1-20.  Most football matches don't end with scores like Liverpool 18 - Man Utd 11.  The dice have duplicated numbers - here are the values for their sides:

D6:  0, 1, 1, 2, 2, 3 --> mean = 1.5
D12: 0, 1, 1, 1, 2, 2, 2, 3, 3, 4 --> mean = 1.58 (19/12)
D20: 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5 --> mean = 2.7 (54/20)

In conclusion - this is a great game.  I didn't play it according to the instructions, but took the raw materials from the game and made it my own - and thoroughly enjoyed it.  I didn't analyse all the maths (what's the probability of a D6 team drawing against a D12 team? Or winning?) but rolled the dice, recorded the scores, and built and Excel spreadsheet to solve the league maths for me!  Was it fair?  Yes - the D6 teams have a chance of winning (and the multiple upsets during the championship show this) and the D20s have a chance of being knocked out (Brazil in a league of D6 teams).  The values on the dice make sure that every team has a chance of winning, even if it's slim.

In a future post, I'll play a cricket-by-dice game, and compare the results!

Other articles I've written looking at data and football

Checkout Conversion:  A Penalty Shootout
When should you switch off an A/B test?
The Importance of Being Earnest with your KPIs
Should Chelsea sack Jose Mourinho? (It was a relevant question at the time, and I looked at what the data said)
How Exciting is the English Premier League?  what does the data say about goals per game?

Statistics: Type 1 and Type 2 Errors

 In statistics (and by extension, in testing), a Type I error is a false positive conclusion (we think a test recipe won when it didn't), while a Type II error is a false negative conclusion (we think the test recipe lost, when it didn't).  

Making a statistical decision always involves uncertainties, because we're sampling instead of looking at the whole population.  This means the risks of making these errors are unavoidable in hypothesis testing - we don't know everything because we can't measure everything.  However, that doesn't mean we don't know anything - it just means we need to understand what we do and don't know.

The probability of making a Type I error is the significance level, or alpha (α), while the probability of making a Type II error is beta (β).  Incidentally, the statistical power of a test is measured by 1- β.  I'll be looking at the statistical power of a test in a future blog.

These risks can be minimized through careful planning in your test design.

To reduce Type 1 errors, which mean falsely rejecting the null hypothesis - and calling a winner when the results were flat - it is crucial to choose an appropriate significance level and stick to it. Being cautious when interpreting results and also considering what the findings mean may also help mitigate Type 1 errors.  Different companies have different significance levels that they use when testing, depending on how cautious or ambitious they want to be with their testing program.  If there are millions of dollars at risk per year, or developing a new site or design will cost months of work, then adopting a higher significance level (90% or higher) may be the order of the day.  Conversely, if you're a smaller operator with less traffic, or a change that can be easily unpicked if things don't go as expected, then you could use a lower significance level (80% or higher).

It's worth saying at this point that human beings are lousy at understanding and interpreting probabilities, and that's generally.  Confidence levels and probabilities are related but are not directly interchangeable.  The difference in confidence between 90% and 80% is not the same as between 80% and 70%.  It becomes more and more 'difficult' to increase a confidence level as you approach 100% confidence.  After all, can you really say something is 100% certain to happen when you've only taken a sample (even if it's a really large sample)?  On the other hand, it's easy to the point of inevitable that a small sample can give you a 50% confidence level.  What did you prove?  That a coin is equally likely to give you heads or tails?

 Type 2 errors can be minimised by using high levels of statistical significance, or (unsurprisingly) by using a larger sample size.  The sample size determines the degree of sampling error, which in turn sets the ability to detect the differences in a hypothesis test. A larger sample size increases the chances to capture the differences in the statistical tests, and also increases a test's power. 

Practically speaking, Type 1 and Type 2 errors (false positives and false negatives) are an inherent feature of A/B testing, and the best ways to minimize them is to have a pre-agreed minimum sample size, and a pre-determined confidence level that everyone (business teams, marketing, testing team) are all agreed on.  Otherwise, there'll be discussions and debates afterwards about what's a winner, what's confident, what's significant and what's actually a winner.