Arnold Kling  

Soccer and the Law of Large Numbers

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I am not a soccer fan. In my amateur opinion, the rules ought to be changed to foster more scoring. This is not because I want to see more scoring. But having more scoring would reduce the luck factor. It seems as though every game I read about, including Germany-England and Argentina-Mexico yesterday, is decided by a lucky goal or a wrongly-allowed goal or a wrongly-disallowed goal. It makes me think--why doesn't this happen in baseball or football?

There is a lot of luck in baseball or football. There certainly are a lot of bad calls by umpires and referees in baseball and football.

The difference is that with more scoring, the law of large numbers kicks in. The chance that any one lucky bounce or bad call will determine the outcome is low, because scoring one point is not such an overwhelming event.

More scoring would also eliminate the need for a "shootout," which makes a mockery of the whole sport. The equivalent in baseball would be after a certain number of innings to settle the game by bringing in batting-practice pitchers and staging a homerun derby.

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The author at Eli Dourado in a related article titled Meddling with Soccer writes:
    Two of my favorite intellectuals, Arnold Kling and Richard Epstein, have each offered opinions on how to “improve” soccer. I am dismayed and disappointed. Their proposals are not for modest changes (e.g., electronic assistance for the refer... [Tracked on June 28, 2010 10:25 AM]
The author at Fahreunblog in a related article titled Forza Italia! writes:
    Nel senso di "Azzurri". Che pena la Nazionale di calcio italiana ai mondiali in Sudafrica. Ma perchè il nostro calcio è regredito tanto? Anche qui, come nel campo dell' arte, chiederei a Brera più che a Pepe. Ma Brera non c' è più. Quel che è cert [Tracked on June 29, 2010 10:54 AM]
COMMENTS (46 to date)
Zorro writes:

Well, I don't know. Maybe people like luck. Soccer is, after all, way more popular than either baseball or football.

MidCap writes:

First, the game is called football, not soccer. That is because players usually play the ball with the foot, other than in that weird game that is played in combat armor ;)

And the luck factor is exactly what makes football so much more exciting than (e.g.) basketball.
If luck would be less important, most matches would be extremely boring.

Maciano writes:

I read Steve Sailer and he wrote the exact same thing. I'm really glad football doesn't catch on in the US. If you guys were serious about changing the game, it would change unrecognizably and that would be a shame, imho.

Americans should accept that there are certain things that Americans have to adapt to -- or take no part in. The non-American world watches American films, listens to American music, reads about American stars, had to sit out George W. Bush and Sex and the City. The rest of the world offers little back, accept, indeed, maybe football.

The rest of the world is pretty much in agreement that football is fine as it is. Join us or stay out. Don't Americanize the sport and ruin it. I'm almost glad that Sepp Blatter makes it impossible to change any aspect of football; it's a blessing in disguise.

dWj writes:

I thought of the problem as being the single-elimination tournament; in a more extensive group stage (or sequence of group stages, or adaptive tournament of some sort), you would get Large Numbers over several games that would leave large amounts of luck in any given game but wouldn't see bad luck as significant in determining which teams are dropped from the tournament.

Vacslav writes:

Entertainment of the pure skill type is boring, entertainment of the pure luck type is boring, but in a different way. Between these two extremes, there is a space of rules that permit games of mixed skill and luck, each rule being a product appealing to a group of customers.

Henry writes:

How much luck is optimal in sport? I think this depends on how large an edge the best players and teams have over the second best. If it's large, then too little luck makes possible for one player or team to win all the time, which is boring. On the other hand, if it's small and often changing, then we'd likely want it as low as possible. What games do you think have these sorts of skill differentials, and what championship events try to reduce luck? Is there a correlation?

Still, if you really want to know who the best team is, rankings will do that for you. New Zealand's national Rugby Union team has been #1 virtually all the time, though it hasn't won a World Cup since 1987. Should the World Cup rules be changed to reduce variance? Probably not, since it'd likely make the Cups relatively boring.

arne.b writes:

The problem with yesterday's events is that they represent the "wrong" kind of luck -- a result of FIFA's insistence to the right to mess up whenever they want. Without their unexplained* refusal to allow referees to watch instant replays (or introduce one tasked only with this) the small numbers of goals would not be a problem.

Compare: Switzerland were lucky to beat Spain 1:0, I think, as Spain had about half a dozen chances to score while SWI had approximantely one, but still it was the Spaniards own fault not to win (my reading of the match - ymmv).

And while I like seeing ENG beaten by GER (nationality bias ;-)), yesterday it did not feel quite right. (Note that neither match was actually decided by the events in question, since both took place well before half-time. Still, we don't know how the matches would have ended without unfair advantage.)

* Does anybody know of an official explanation for this? And I mean a serious one, nothing that can immediately be countered "how come it's working in basketball"?

arnique writes:

It is possible that the reason you seem to think a lot of football games involve bad "luck" (Germany beating England 4-1 isn't due to some random factor—England has a rotten defence—but I digress) is due to self-selection. Matches that get a lot of spotlight in newspapers and air time on American television are necessarily the ones with big controversies (offsides and disallowed goals, mostly referee errors) which pundits love to fight about. You don't hear about a majority of the games in which generally everyone agrees on outcomes.

Penalties (which BTW are only used in knockout rounds) are lots of fun for me to watch—the tension, the drama, the unmissable miss! I wouldn't do without them. It isn't more goals that would solve your problem of luck, it's goal-line technology, which FIFA President Sepp Blatter refuses to use.

What I'd like to know is how would you change the rules to increase scoring? Give the opposing team a ten-second head start at kick-off and keep the back four limited to a twenty-yard area?

Luck's a huge part of enjoying football. Without it, the underdogs would never win and every cup winner could be easily predicted.

Tom Dougherty writes:

Let's make each goal worth 7 points. Germany 28, England 7. Argentina 21, Mexico 7. Or how about the Portugal vs North Korea game - 49 to 0. These now look like football scores.

There is nothing wrong with luck in sports. It can be very exciting - think of the Hail Mary in football. What is not so good is the bad calls. Football has the instant replay and uses video technology. Soccer needs video replay as well to eliminate the bad calls. They can also put a chip in the soccer ball to determine if it crosses the line for a goal. More technology should be used and probably will be used by the next World Cup to eliminate some human errors.

Alex J. writes:

Instead of penalty shots to break ties, they should start removing players from both teams as overtime progresses.

Dave McGowan writes:


The reason FIFA won't use the video ref its to maintain their commitment to having the 'Laws of the Game' (what they call the rules of football/soccer) be the same no matter what level you are playing at.

A video ref would mean that one rule applies in clubs that can afford that necessary infrastructure for a video ref and another for those that cant.

edgar writes:

Yeah, the soccer rules could seem unfair sometimes, however that's part of the trick to keep people talking at any time about the game. I am Mexican but I am not really a big soccer fan, and yet I think the rules are fine just like they are now. There are other options to look at though, like indoors soccer (futbol rapido), where the court's size is smaller and the flooring allows the ball run faster and in general you see more goals in any match. I think is a good alternative for all gringos out there.

Matt writes:

Some unlucky things happened yesterday, but ultimately the better team won in each game. From time to time a team is able to win due to luck, and the same goes for every sport. In soccer, the best team usually does win. At the World Cup in particular, there is never an undeserving champion.

John Jenkins writes:

"It makes me think--why doesn't this happen in baseball or football?"

You don't actually watch a lot of those sports, do you?

Google "Armando Galarraga," just to start.

Arnold Kling writes:

Armando Galarraga proves my point. He did not lose the game because of the bad call. He just lost a particular opportunity to go into the record books.

arne.b writes:

@Dave McGowan:
Considering video refs part of the "Laws of the Game" is a bit of a stretch, isn't it?

"Rules" are things like "The ball must be completely behind the line" or "thou shall not hit your opponent only and not the ball" or something like this, not "if a video screen is present, the guy checking whether you comply to the rules is not allowed to know what everybody else on the field and beyond knows".

I mean, there are probably rules to the extent of "the grass shall be no more than 10 cm and no less than 2 cm high", and still the quality of the field varies, especially in correlation with the "level you are playing at", and the FIFA seems to be able to bear it.

And shouldn't it be an additional motivation for players no matter at what level to know "if you are so good that others want to watch you play on TV, you are also good enough to get the best refereeing possible", not just the putatively best referees?

Byron writes:
First, the game is called football, not soccer. That is because players usually play the ball with the foot, other than in that weird game that is played in combat armor ;)

Actually it's called 'football' b/c it was played by peasants on foot, rather than by nobility on horseback. Kicking the ball had nothing to do with the etymology.

The rest of the world is pretty much in agreement that football is fine as it is.

No they're not. England is screaming for video review or goal line refs, Mexicans too. This isn't just a problem Americans are complaining about.

A video ref would mean that one rule applies in clubs that can afford that necessary infrastructure for a video ref and another for those that cant.

It wouldn't mean that at all. The rules would be the exact same, video ref or no, just more accurately enforced.

Kurbla writes:

Injustice (or luck) and the low scores are important part of the game. Highly recommended video:

Matt writes:

I love the venom that is unleashed whenever anyone suggests changing the sacred rules of soccer (which is a game I love by the way).

A no-brainer would be to eliminate offsides. Its a rule that has no beneficial impact to the game that I can see (or that I have heard argued). Its also impossible to enforce effectively with the naked eye at game speed.

TheKm writes:

You mistakenly assume that football is purely an athletic contest. Not so. Football is equal parts athletic and psychological.

The "luck" factor is part of the psychological half. Belief in self, confidence in individual actions, dealing with a call that mistakenly goes wrong - these are all just parts of the game. That aspect is something that fans love. The agony and the ecstacy, if you will.

Not to mention that it is impossible to change the rules such that more scoring. There isn't a single rule that will make keepers less capable or strikers more. The only chance there is is to make it easier to call penalties, which would just create a farcical dive-fest.

Pedro writes:

It seems Americans fundamentally misunderstand the game itself and cannot see past the "low scoring/high thrill" aspect of it. As someone else mentioned above, change the number of points per goal and results could eerily resemble those of American football. And as someone else mentioned, no matter how much "bad luck" the English had, they still conceded 4 goals and the scoreline, not the game, was anything but close in the end. The crucial problem IS the use of technology, which, bizarrely, is already pervasive in the game. The fourth official uses an electronic board to signal substitutions, there's plenty of digital time-keeping going on (which the referee uses to figure out how much time to add on) and, crucially, the referees are allowed to communicate through headsets. Having an electronic beep when the ball crosses the line is, in the abstract, no more than a different signal to be interpreted by the officials (their vision, or lack thereof, being the primary one). So, FIFA's refusal is baffling and insulting while American recommendations to "improve" the sport fundamentally miss the point because they would debase the game completely and not go into the heart of the matter. Finally, the law of large numbers does kick in when in a different kind of competition, like a league, but the appeal of knock out football is precisely the element of fortune that can send the favourites packing - the ultimate embodiment of which being the penalty shoot-outs, which pretty much every football fan enjoys.

Barry Kelly writes:

This post shows that you don't understand one of the key reasons football is so popular: that low scores mean luck has a large part to play, and thus it's possible for anyone to beat anyone.

Luck is a feature, not a flaw, of the game.

Pedro writes:

To Matt above. I'm not sure whether you are a regular player of the game or have played it competitively but as someone who does play often enough and who played at the lower levels in a football deranged country from an early age until the late teens, I can safely say that abolishing the offside rule would have a tremendous impact on the game and, I dare say, that much would be obvious to anyone who, like me, was brought into a game that is culturally omnipresent and part of our very identity. I mentioned the playing then and now because when playing competitively, scores will generally be lower and there will be some organisation to the game. When playing informally, and there are no offsides, it is quite simply chaotic. People go anywhere and everywhere, there are all manner of ridiculous goals being scored and the enjoyment of it is substantially reduced. It turns the game into an absolute farce and that is why people are so reluctant to change that. Surely it would evolve and tactical systems would adapt, but given the length and width of the pitches, you'd see much more distance between lines and the whole face of the game would change. Given that we like it the way it is, why risk turning it into something else?

Martin S writes:

I can't wait until Arnold tell us to improve other games, especially the lottery.

Chris writes:
A no-brainer would be to eliminate offsides. Its a rule that has no beneficial impact to the game that I can see (or that I have heard argued).

Without an offside there would be no need to build up an attack which would mean no passing between players. An attack would consist of the keeper kicking a long ball into the oppositions box while all there team waited for it. The offside rule is probably one of the most important parts of the game, for attack and defence.

hacs writes:

I see soccer as a game of long chains of "if", where shorter chains of "if" are rare, an antipode with relation to other games. Moreover, technology can prevent many of those "strokes of luck".

As well as nobody was born enjoying baseball and football (handball with scarce kicks and frequent crashes), soccer (football) is enjoyed by billions because is played by them since they were 3 years old (of course, there are late fans).

philipp writes:

The great thing about football is that even the underdogs on a good day have a chance to beat the big guys.
Thats a part of what makes the worldcup so facinating sometimes.
And for that you need the luck factor.

And there is nothing compared to the drama of a shootout.

John Thacker writes:
And there is nothing compared to the drama of a shootout.

A Test match that has been even for several days coming down to one team having 9 outs but within a small numbers of runs is the most absurd drama ever. Either side could win, and both a draw and a tie are still available.

First, the game is called football, not soccer.

The game is one of seven codes of football all derived from the same lineage. All of them have and are called "football" when they are the most popular sport in the area. All of them need an alternate name when referring to them in an area where another code is more popular.

I am able to accept that, in different parts of the world, rugby union, rugby league, Gaelic football, Canadian football, Australian rules football, American football, and Association football may all be called simply football by adherents. There's no reason for Association football fans to be so chauvinistic.

John Thacker writes:
As well as nobody was born enjoying baseball and football (handball with scarce kicks and frequent crashes), soccer (football) is enjoyed by billions because is played by them since they were 3 years old (of course, there are late fans).

Strange, the infants and toddlers I've known have all preferred using their hands to do things. I believe it takes training to get them to avoid doing so.

eccdogg writes:

As someone else mentioned above, change the number of points per goal and results could eerily resemble those of American football.

As a person who is a huge American football fan and has been trying to get into and excited about the World Cup (with some success), this just doesn't seem to be the case to me.

American football is almost never zero zero and it is very seldom that an American football team "keeps a clean sheet"

Average scores in the NFL are 21 pts per game per team or 3 scores per team. As far as I can tell looking around the internet the average goals scored per team in the world cup is about 1 per team today having fallen from about 2 per team in the 60's.

Hume writes:

The low-scoring aspect of soccer ("football," futbol, whatever) is what makes it so exciting, especially in a tournament setting. Because it is so difficult to score, every single possession is extremely exciting: there is the possibility of a goal every time down the field. Combine this with the importance of every goal, there is extreme tension every time the ball passes midfield.

Also, I think your statements about luck (1) are not backed up by the facts, and (2) your insights are misguided because of the tournament setting.

Regarding the facts: there have only been 7 countries to win the World Cup. I guess these are just really lucky teams. Also, in the major professional leagues, the same teams compete year after year. If luck was such a huge aspect of the game, you would expect to see more diversity in success.

Regarding the tournament setting: if your only familiarity with the NFL was 3 games plus a knockout tournament, you would hear far more calls that pure luck influences outcomes (e.g., that lucky fumble recovery or unlucky int.). Soccer seasons are longer than NFL seasons, yet your only basis is a short tournament played once every 4 years.

Thomas Schminke writes:

A couple observations ...

1) American football and baseball are terrible for luck. Almost every year some team that shouldn't even have made the playoffs makes the finals against better teams. The World Cup usually comes down to a Brazil-France or Germany-Argentina final. If the point of more scoring is to balance out luck, it seems to fail badly in American sports.

2) Tournament soccer is really itself an not true to this spirit. European leagues give the championship to the team that plays best over the course of the whole season ... no play-offs. This is the best way to control against luck.

Pedro writes:

To eccdogg: Fair enough, but the point was simply to outline that the low scoring can be deceptive. Furthermore, touchdowns are not the only way to get points in American Football, just as there are other ways to score other than tries in rugby. I think this is actually the major difference between the sports in terms of how scoring opportunities affect the outcome of the match.
Being Portuguese, football is almost in the bloodstream but even for us there wouldn't be much resistance to changes in the rules if we felt they would actually benefit the game. Most of those proposed by Americans seem to focus too much on the issue of low scoring, which I think is one of the main features of the sport. There are certainly plenty of things on the disciplinary side which could and should change, along with a thought out use of technology and we could use some fresh ideas, but people calling for the end of the offside rule or ideas like Epstein's simply lack a basic understanding of what people enjoy about the sport and thus can only be dismissed as ultimately unnecessary contributions.

Incuhed writes:

Eliminating offsides would require defenders to play man to man and open the game up. I'm all for trying it out. Shootouts need to go - dramatic, but an insulting way to end a game. the team that's last standing in field play should win.

The idea that some 'purists' here and elsewhere are in support of Sepp Blatter's archaic, insolent and wildly conceited dissmisals for technology in the game is disturbing. I love American Football and Basketball, but soccer is my favorite sport. Those two sports impliment replay and get the calls right. Justice. Injustice and inaccuracy at the expense for purity? Galileo might have a beef with you. Oh, BTW, refs do use wireless radio to communicate - technology, Sepp, it's a good thing.

I am in full favor of review for all goals and straight red cards, possibly even yellows. All it takes is a quick replay evaluation from the fourth referee to confirm whether a goal is valid, and whether a player truly deserves to be ejected. The incentives are aligned such that it's WAY too easy to convince the referee on the fly that a player has been fouled harshly. People wonder why soccer players roll around like they've been shot when they're nudged, or sometimes never even touched -although it will appear from a ref's perspective that he could have been fouled. The fact that players do this, and flop all over the place to kill time, is embarrassing to the sport (see Ghana, what a shameful display towards the end of the US game)

How can anyone say with a straight face that having this controversy (i.e. the England no goal) is good for the sport because it gets people talking???!!!! Murder gets people talking too. Unbelievable hubris from some soccer snobs, no more so than Mr. Blatter himself.

Even with all its problems, there's nothing like the excitment and tension of soccer.

John V writes:


This isn't about what Americans want vs. what the rest of the world wants. Believe it or not, many soccer/football fans from other countries would agree about making changes. The opinion is not USA vs. the world.

I have watched the sport all my life and the topic of change came up while I was visiting friends in France a few days ago. They were in total agreement about using technology to improve the officiating and would also like subtle rule changes to improve scoring. The lack of those changes isn't going to make them like soccer less but they would still like it.

Personally, I am 100% for developing simple replay technology for certain applicable situations in the world game. Most people are. Also, I think rules changes that made scoring go from 0-2 goals on average to 4-5 on average would help the game and help better teams win more than not.

It's not about being American or not American.

John V writes:


One change that would encourage more scoring would be to make the goal mouth slightly larger. People on average are taller than they used to be. Goalkeepers are like basketball players these days. Taking that advantage away y adding about 8-12 inches to the goal width and maybe 4-6 inches to the goal height would improve scoring.

Matt Flipago writes:

Many people are open to using technology to see if they scored. Using technology on every foul is just ridiculous, almost foul is a possible yellow, it's just a matter of how reckless they are. And you almost never get red cards unless you are blatantly reckless, with not need for review. Goal scoring review opens up the gave for linesmen, as you don't have to juggle between watching offside and whether the ball crosses the line, and rarely would need to be used. Abandoning offsides is a bad idea, it would result in much less movement, and

Also the Germany-England and Argentina-Mexico, neither were affected by the bad calls. England looked pathetic the whole game, never really seemed like they were going to win, and Mexico let in two goals, neither of which had anything to do with the first goal Argintina scored. The second goal by Mexico was about a terrible pass, nothing to do with putting to much pressure up. And the other goal was off a set piece. I can't see how either team who suffered bad luck from those calls could have won. The emphasis on a bad call is because of sensationalist journalism.

Incuhed writes:

Red cards should be reviewed. What about that oscar-worthy performance of the Ivory Coast player that got Kaka sent off? What a joke. Or that dive a Chilean took against the Swiss player after he swung his arm around to shield his body? His arm came close to the Chilean's face - no intent to elbow - and the Chilean goes down like someone sprayed sulphuric acid in his eyes, rolls around in 'agony'. Result? Swiss guy ejected. Replays show it's a yellow offense at worst. The Swiss are forced to play even more defensive (if that's possible), and are broken down. If players know that their antics will be reviewed they will be less likely to fake injury.

The journalistic emphasis is not on bad calls, it's the griping that FIFA or its refs don't give a damn and have no accountability for their horrible mistakes. They hide behind the 'human' element and purity arguments that have been outdated for years. Get it right, or at least admit you're judgement was incorrect, and nobody will complain.

Ryan writes:

a lucky goal or a wrongly-allowed goal or a wrongly-disallowed goal are not symptoms of the same problem. In fact, the first, "luck", is not a problem at all. The latter two surely can be debated regarding a solution. The real law of large numbers (LLN) is evident in Hume's and Thomas Schminke's posts -- that when the experiment (the game) is performed many times the best teams seem to consistently come out on top. Otherwise, we'd see much more variance, if your "luck" argument had legs. Practically speaking, it is much easier to modify the tournament format or **tournament rules to lessen the impact of bad refereeing.

** We all know it would be nearly impractical to expect instant replay to be available at all levels of the game. The real challenge, if implemented, would be should one expect it in qualifying for such a tournament.

letterofcreditguy writes:

This worldcup has been dismal in regards to referees; however, one bad case does not make good policy.

Luck is part of the attraction of soccer, since statistically you can better predict baseball and football compared to soccer outcomes. The better team does win less often in soccer.
Hence, soccer can be a sport of the underdog where the most expensive team might NOT win.

39389 writes:

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John V writes:

Agreed Incuhed about reviewing cards.

The best way would to have that useless 4th official actually do something and quickly review such calls while everyone is arguing with the referee about it and buzz the right call down to ref on the field. This is an excellent deterrent to play-acting. You know, I'm always amazed at how players can take 70mph+ direct kick on their body and play on unfazed upon impact or take a header shot on goal from a speeding cross without a problem. YET, a finger scraps their face at worst and role in agony. I wonder if they would fall down and role around if someone slapped them in a bar or in the street. Probably not.

The next best way would be to have a review board examine games during the between games with the power to overturn cards and hand out fines and suspensions for play-acting and missed fouls away from the action.

Both are easy and neither interrupts the flow of play because action is stopped.

John V writes:

That's all true Ryan.

But tournaments like the WC are not long seasons were the cream always rises to the top.

They are short knock-out affairs where a ref's mistake is the difference between a team reaching the the next round or going home. There's no "next week" or tomorrow.

Luís writes:

Eliminating the offside rule would be terrible; the strikers would just stay upfront waiting for the ball, and so would the defenders marking them. So, instead of the game moving up and down the field, with defenders attacking and full backs almost playing as wingers, we would see a bunch of players in each penalty area and not much in the middle, I'm afraid. Soccer is interesting because of the way plays are build, pass by pass from defense to attack, until a goal is scored. And the reason I really like football is that great moment when a goal is scored and you know it is a game-winning goal, it is a very important moment. I can't enjoy basketball because of the feeling that everytime a team scores, it is just another one, most likely not game-changing. Shootouts should stay; they are not that common, make for memorable moments and only 14 players can play in a game for each team; 120 minutes is more than enough for 8 of those men, unlike other sports where subs com in and out without limitations.

Dave writes:

As usual, I'm late to the party, but Rick Bookstaber wrote a post with a similar theme.

James writes:

I am a Scotsman, and I would like to point out that England did not lose because of a disallowed goal. They were in fact hammered 4-1 by a superior German team. Just to clear that point up.

But seriously, football used to have higher scoring matches than exist nowadays. Back in the old times, formations committed more men up the field attacking, but nowadays tactics developed in Europe often see 1 striker taking on 4 defenders as a rule, and when in defensive formation, 9 or 10 men behind the ball. Under these circumstances space is scarce and attacks frequently break down before a goal scoring opportunity presents itself. It is a fearful, negative attitude which I believe comes from coaches who concentrate on league football where 'not losing' is deemed as more important than going for victory at the risk of losing. Knock out football however, is very different, and you see the reason major competitions like the World Cup is so exciting is that teams must come out and play for a win. It is no good putting 10 men on the edge of your own penalty box to stop the opposition scoring.

This is one of the reasons England failed I believe. Coach Capello is undoubtedly one of the most successful coaches to ever grace the game. But he had success in European leagues playing defensive football. That isn't good enough to win the World Cup. You must be more attack minded. Part of Brazil's perennial success relative to the rest of the world (if not to their own fans' galactic expectations) is that until recently, Brazil had no national league system. Instead they had a playoff system similar to US sports, which encouraged more open, attacking play. Even though they now do have a European style national league, they also have the regional state championships which are big affairs and continue with the playoff format in some states.

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