We all have our guilty pleasures, and one of mine is Daniel Negreanu’s Twitter polls. As a fan of statistics, I should be resistant to unscientific methodologies, and it’s hard to get less scientific than an online poll conducted through social media by a celebrity in a field with demographics that bear almost no resemblance to those of larger society. Nonetheless, the questions themselves often offer some food for thought, and sometimes knowing what his followers think is itself interesting, even if it’s unlikely to be reflective of the general public’s sentiment.
With a couple of recent polls, Kid Poker has been trying to find the correct Over/Under line for the age of this year’s World Series of Poker (WSOP) Main Event champion. His first guess was 30, but that poll had 65% of respondents answering that they would expect the champion to be younger. He then tried again, setting the line at 28, and came much closer, although a slim majority (54%) were still taking the Under.
It was a typo in the latter poll which initially caught my attention – he incorrectly said that 65% had taken the Over in the first poll, which caused me to wonder both why he’d responded by lowering his figure and how that could cause a reversal in results. The reality is a bit less bizarre, but it still strikes me as somewhat surprising that lowering the bar by two years was enough to change 11% of people’s minds. The odds of a champion who is exactly 29 or 30 seem fairly slim to me, so one would have to imagine that the first poll was very close to an even money proposition for this adjustment to flip one’s bet from Under to Over.
QOTD: how old will the winner of this years WSOP main event be? 65% had 30 or over, how about 28 or over?
— Daniel Negreanu (@RealKidPoker) May 13, 2016
A question of distribution
To figure out whether either of Negreanu’s lines are approximately correct, and whether there’s much of a difference between 28 and 30, the first thing we need to ask ourselves is what we would expect the age distribution of champions to look like if the Main Event were a thing we could repeat many times.
The rate at which players of various ages would win can be broken down as the product of two separate curves; the overall demographic curve of Main Event entrants, and the statistical edge you’d expect to be enjoyed by an average player of a given age.
Fortunately, the WSOP releases demographic information about the Main Event, so we can answer the first question exactly. The average age last year was just a hair under 40, but the largest five-year age bracket was 26-30, with players in that range making up nearly a quarter of the field. No median is specified, but it would fall somewhere towards the low end of the 36-40 bracket. In other words, we have a skewed distribution with a large number of players in their late 20s and early 30s, but a long tail of older people pulling the mean and median upwards.
If all players were of equal skill and style, then, you’d expect the line to be something like 37, but historical data suggests that this would be a serious overestimate. If we define the “modern era” of the Main Event as beginning in 2003, the year Chris Moneymaker won, then there has never been a champion older than 40, and only four who’ve been older than 30. The last and oldest of these was Jerry Yang, who won in 2007, at age 40; since him, every champion has been in his early to mid-20s. The oldest champion since Yang, as crazy as it sounds to say, is Martin Jacobson, who won at the ripe old age of 26.
Of course, there have only been 12 “modern era” WSOP Main Events to date, so we’re talking about a small sample size, susceptible to random variance. Still, a glance at the Player of the Year races in any live tournament circuit confirms what you probably knew to begin with, that men in their 20s win tournaments at a disproportionate rate. Looking at recent champions, you’d guess that the peak age is in fact very young indeed; the most common age of winners is 22, shared by Peter Eastgate, Jonathan Duhamel, Pius Heinz and Ryan Riess. Given that one has to be 21 to gamble legally in the United States, then, it seems that the best age to be is as young as possible, and players’ chances of winning decrease monotonically with age.
Multiplying a positively skewed curve by a monotonically decreasing function should generally just increase the amount of skewness. That is, our hypothetical distribution curve for Main Event winners under repeated trials should look qualitatively the same as the overall demographic curve, but peaking even earlier and with an even longer tail. That theory is borne out by recent results, with a huge majority of winners in their early 20s and then a smattering of other champions in their late 20s and 30s.
The median’s the thing
Heavily skewed distributions have a tendency to mess with people’s expectations. We’re used to looking at symmetric (aka “normal”) distributions, where the mean, median and mode are all one and the same. If you’re not familiar with those terms, the nutshell explanations are that the mean is what we usually think of as “the average,” or expectation value, the median is the number at the midpoint of the list if you put the values in order, and mode is the most common value.
For the ages of “modern” Main Event champions, the mode is, as I said above, 22, which is shockingly young, given that under-25s only make up about 8% of the field. The median is only slightly higher, or 24, while the mean is 28. When trying to make predictions, many people’s instincts lead them to look at either the mean or the mode, but when you’re looking to set an Over/Under line, the relevant type of average is, by definition, the median.
Based on that, then, if you were going solely on recent results, you’d say the line has to be set at 24; since 2003, there have been 7 champions aged 24 and up, and 6 who were 23 or younger.
Is twenty-two the new thirty-nine?
That said, 24 seems like an insane place to set the line, and it likely is too low. The principle of “regression towards the mean” tells us that the young ages of recent champions are likely due in part to random variance and the actual statistical distribution of ages, if we could do lots of trials, would tend to regress back towards the overall demographic curve at least a little bit. Just how much, though, is anyone’s guess, and there’s even a chance that the reverse could be true.
What makes things complicated is that poker has changed a lot since 2003 and it would almost certainly be incorrect to assume that the skill-versus-age curve has remained consistent over the past 13 years. Poker in general, and internet poker in particular peaked around 2010, shortly before Black Friday, and it was the internet poker boom which has led to the prevalence of extremely talented young players.
It’s probably no coincidence, then, that Yang, in 2007, was the last time we saw an older player win the Main, and that it’s been dominated by guys in their early 20s ever since. If you drop Yang and everyone before him from the calculation, you could make a case that even 24 is too high, rather than too low. Post-Yang, four of the eight champions have been exactly 22, and then there’s Joe Cada (21) and Joe McKeehen (23), so if you set the line at 24, Under would have taken it 75% of the time since 2008.
Things are still changing
If we believe that the recent trend towards extremely young champions is a real thing and not just variance, then the final question we need to ask is whether we expect the trend to continue or start to reverse itself. I would argue the latter, and the reason I would give is the drying up of the online economy and the industry’s shift away from pros, towards recreational players.
The recent dominance of young players in the live arena is only partially due to the fact that online play makes it easier to gain experience and master the game. The other part of the equation is that, as online play becomes more and more competitive and hard to profit at, these same young players are being driven out of the online ecosystem and into the world of live events. For most of the past decade, players would start off their careers online – either playing illegally under-age or in places where the legal age is lower – and then start transitioning into live events at 21, with several years and millions of hands of experience under their belts.
What’s changed now is that it’s more or less impossible to make substantial profits at the lower stakes online, due to tough fields and high rake. The days of people being able to grind up a bankroll from scratch online are gone, which means that players are switching to live games earlier on, and therefore not racking up as much experience playing high volumes online. Those 22 year-olds who were the best in the game four or five years ago may still be the best now, as the ladder has been pulled up somewhat for those who’ve followed. I would expect, then, that the median age of winners will start to creep back up in coming years.
You realize almost no one @RealKidPoker builds a bankroll online anymore right. Rake is impossible to beat lower and there aren't good games
— Doug Polk (@DougPolkPoker) May 22, 2016
Is there an answer to the question?
So far, this article has largely consisted of me vacillating on the issue. Negreanu’s guesses of 28 and 30 are substantially lower than the mean and median age of the overall field, yet also much higher than the demographics of recent winners. Those numbers would suggest that the line be set at 24, yet I’ve argued that the real number could be either higher or lower than this. It may seem, then, that I’m doing everything in my power to avoid giving a concrete opinion, but in fact I’ve just been laying out all my reasoning up front.
My conclusion, weighing all the above observations, is that there’s probably not that much difference between 28 and 30, certainly not enough to justify an 11% shift in opinion among Negreanu’s followers. The Main Event’s structure places such emphasis on skill that the weaker recreational players, many of whom are older, have essentially zero chance of winning, and so, despite the numerical advantage the over-30s have, I think that whether the line is set at 28 or 30, it’s a slam dunk to take the Under.
Based on my expectations that we should both see a regression towards the mean and a real trend towards slightly higher ages of winners, however, I think that 24 would be too low. I’m going to guess that someone along the lines of a Martin Jacobson will win this year, and therefore go with his age (at the time of his win) and say that I’d set my line at 26 – that is, I believe it’s about equally likely that we’ll see yet another 21-25 year old champion as one who is 26 or older – but with the expectation that I’d get considerably more takers on the Over.
Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Montreal, Quebec, Canada.