Increasing the competition reduces competition

I know the rule is: write what you know. So, I also know I’m going out on a limb here writing something about sport. Specifically, writing about AFL. But, this idea has got into my head, and I think writing it down is the only way to let it out.

Due to utter chance, the weekend newspaper fell open on an article about the AFL competition format, and after glancing at it, I found it to be really interesting. The gist of it was that the AFL has consciously shaped its teams to create a level playing field, through salary caps and priority draft picks for bottom teams, with the result that there is a good rotation of teams winning the grand final, and the sport has achieved enduring popularity. And it doesn’t hurt that they’ve been able to translate that popularity into a pretty decent TV rights deal.

However, it appears that while there is a good rotation of top teams, in recent years, those top teams are typically (and curiously) much better than the rest of the competition. According to the article, since 2000, Essendon, then the Lions, Port Adelaide, Geelong, St Kilda and now Collingwood have had long runs of wins, trouncing all the other teams for a reasonable period of time.

It strikes me that another way that the AFL has shaped their competition may be leading to this very outcome: growing the number of teams. Despite their efforts to limit and reset the strength of the teams each season, there is always going to be a distribution of ability across the set of teams. A spread of ability will exist, even if concentrated, so there can still be outliers. To put it another way, even if the standard distribution is smaller, the probability of a team falling outside a s.d. of the mean is still the same.

According to Wikipedia’s page on the AFL, back in 1982 there were just 12 teams, and this has increased little-by-little over the years to the current 17 teams, with 18 teams are proposed for next year. If we look at the probability of there being an outperformer in the mix (for discussion’s sake, let’s define that as a team with an ability two standard deviations above the mean), it increases from about 1-in-4 for a 12 team competition to about 1-in-3 for an 18 team competition.

# Teams Year P(one or more outperformers)
12 1982 0.241
14 1987 0.275
15 1991 0.292
16 1995 0.308
17 2011 0.324
18 2012 0.339

On one hand, the AFL’s actions are intended to make teams as similar in ability as possible to promote healthy competition. Ironically though, it seems their actions in increasing the size of the competition might be working against this to improve the chance than a season will have a dominant team.

Although from the AFL’s point of view, if the driver of their policies is not competition but popularity, then as long as the new teams are introduced in new areas, the loss of popularity from impacting competition is likely more than made up by the increase due to the additional of new supporters.

How unexpected are 1-in-100 yr events?

I was chatting to someone about once-in-a-century events, and I was reminded of a book I read a while back which pointed out that understanding probability is a pretty recent thing. In fact, there haven’t been that many centuries in which people could talk knowledgeably about 1-in-100 years events.

(I should probably put a disclaimer right here that, just as it seems any public post  about poor grammar is bound to be riddled with grammatical errors, my post about probability is going to be full of  mistakes. So, I promise to fix them, if they are pointed out to me.)

A funny thing about 1-in-100 year events, like 1-in-100 year floods, or storms, or market busts, is that they can appear to come along more frequently than once every hundred years. But it looks like we’re stuck with the name.

A 1-in-100 year event is simply one that has a 1% chance of happening in any particular year, which means that, mathematically, you would “expect” there to be one, on average, every hundred years. But of course, some centuries will have more or less of them.

In fact, (making assumptions about the distribution of events,) you would expect 37% of centuries to never have a particular 1-in-100 year event. You also get exactly one 1-in-100 year event in around 37% of centuries. The rest (26%) have more than one.

While more than a quarter of centuries can see multiple occurrences of a 1-in-100 year event, it’s worth asking: how many times can such an event crop up in a century before you start to wonder if it’s really still a 1-in-100 year event. How many occurrences of 1-in-100 year events should you actually expect?

The answer depends on the threshold for unlikeliness. A reasonable standard might be that anything less than 5% probable is pretty improbable. Back in high-school, playing D&D role-playing games, throwing a 20 sided die and getting a 20 (i.e. a 5% chance) was enough to get you special results. It came up a few times every game, but it was something pretty unlikely indeed. So, let’s use that standard for now. (It’s also common elsewhere.)

So, how many occurrences of a 1-in-100 year event in a single century are needed before we get to a level that’s less than 5% likely (or, you would expect to occur in less than one century out of twenty)?

The answer is 4. It’s only when you get four of a 1-in-100 year event happening in the same century that you might want to start questioning whether something else is going on, because it’s all starting to get a bit improbable. Four or more of such an event should crop up in less than 2% of centuries.

So, in summary, you shouldn’t be shocked to see three of a 1-in-100 year event occur in the last hundred years – it’s perfectly expectable.