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Jun 24, 2016
Jun 24, 2016

The Paradox of Skill

The Paradox of Skill
From finances to sports, betting is a mixture of skill and luck. The key to long-term success is not to pick winners, but to find value and make consistent profits. Betting tipsters often promise high strike rates, but can they perform consistently in the long run? The answer lies in what is called the Paradox of Skill.

The variability in your betting outcomes, and therefore your profits, is a combination of the variance in your skill and the variance in luck. Although sporting outcomes are subject to inherent randomness, on average if your assessment of an outcome probability (in other words the odds) is better than those published by the bookmaker, you will have secured positive expectation. 

What do the odds represent?

The bookmaker’s odds, however, are to a significant extent a reflection of the market’s opinion about the outcome probability, discounting the influence of the margin. What is the market? It’s simply the collection of bettors expressing opinions about the likelihood of the outcome with their money.

In other words, the betting odds merely represent a weighted average opinion of players engaged in a prediction contest, in which the key to success is to have better forecasting skills than your opponents. If you do, you are likely to find value and make a profit over the long term. Sounds easy enough. Why, then, is finding consistent value so difficult? The answer lies in what is called the Paradox of Skill. 

If your assessment of an outcome probability (odds) is on average better than those published by the bookmaker, you will have secured positive expectation (long-term profits).

Lessons from baseball

In 1941, Ted Williams, a Major League Baseball player for the Boston Red Sox, had a batting average of 0.406. Considering that typical batting averages have remained largely unchanged since the origins of the professional game in the 1870s (around 0.25 to 0.28), this was a remarkable achievement, and something that has not been repeated since.

Arguably, however, Williams would not score anything like that average in today’s league, given the improvements in training, fitness, diet and general professionalism. So what’s going on? Firstly, the batting average is simply a measure of relative skill, between the pitcher on the one hand and the batter on the other.

As Major League Baseball has become more professionalised, batters have individually become more skilled at hitting. At the same time, however, pitchers have become more skilled at pitching. Michael Mauboussin, author of The Success Equation, describes this as a kind of arms race: absolute skills improve across the board, but relative skills, on average, remain more or less the same.

Secondly, whilst overall skill levels have improved, the difference between the best and worst hitters (and pitchers) has shrunk. We can explain this by imagining there to be a ‘wall’ of human ability. In the early years of professional baseball, a few players were already approaching the ‘wall’ but most were still quite some way away. Over time, progressively better hitters (and pitchers) were replacing the weaker ones, and as a consequence the difference between best and worst has narrowed.

The harder I work, the luckier I get

Since the observed variance in outcomes is the variance in skill plus the variance in luck, a decrease in the variance of skill (with the variance in luck remaining the same) should reveal itself in the observed variation in batting averages, quantified by means of the standard deviation.

As the variance in skill diminishes the variance in luck, luck will assume an ever-increasing importance in the calculation of outcomes, and therefore profits.

Sure enough, during the first years of US professional baseball (1870s), standard deviation in batting averages was around 0.05, meaning around two-thirds of all batting averages were roughly in the range 0.2 to 0.3, with about 95% between 0.15 and 0.35.

Today, the standard deviation is about half of what it was. Consequently, extreme outliers have become less common. In the 19th century we might have expected a batting average of 0.40 to appear once in every 1,000 hitters; today that might be more like 1 in a million.

Mauboussin also reminds us that as the variance in skill diminishes the variance in luck, luck will assume an ever-increasing importance in the calculation of outcomes. “If everyone gets better at something, luck plays a more important role in determining who wins.”

The case of betting tipsters

Like baseball, betting is really a relative skills contest between backers and layers, and in the financial world, between buyers and sellers too. If one side is more skilled than the other at forecasting (the outcome of a game or the intrinsic value of an asset), the potential exists to secure long-term positive expectation, after good and bad luck has cancelled out.

However, the Paradox of Skill has revealed that as more and more players adopt ever more sophisticated methods for predicting the future, their range of abilities will narrow and chance will assume an ever-increasing influence on betting outcomes.

Referring to the performance of institutional professional investors, Mauboussin says that “the more everyone’s level of skill looks the same, the more you’d expect the range of excess returns for money managers to shrink.” Sure enough, that’s exactly what’s happened.

As more and more people took up the challenge of beating the market from 2007 to 2014, the harder it became to do so, with sharper forecasters converging towards a ‘wall of truth’.

Between 1960 and 1997 the standard deviation in excess returns for Morningstar mutual funds trended downwards from about 13% to 8%. Just as in baseball, the big hitters were disappearing; not because they were less skilled at forecasting returns, but because they were competing against more of the same. 

Similarly, as a verifier of betting tipsters from 2001 to 2015, I observed a decline in the standard deviation in running 5-year aggregated yields from about 2.5% in the period 2002-2007 to just 1% by 2009-2014.

Arguably, this period witnessed the biggest growth in online sports betting, but as more and more took up the challenge of beating the market the harder it became to do so, with sharper forecasters converging towards a ‘wall of truth’.

Players may have become sharper in absolute terms, but with that profitable returns are now harder to come by. As the efficiency of the market is increasing the ‘0.400 hitters’ of betting tipsters may be disappearing. 

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Joseph is a betting analyst who manages the website www.Football-Data.co.uk, providing historical results, match statistics and betting odds data. He is also the author of Fixed Odds Sports Betting: Statistical Forecasting & Risk Management (2003), How to Find a Black Cat in a Coal Cellar: The Truth about Sports Tipsters (2013) and Squares & Sharps, Suckers & Sharks: The Science, Psychology & Philosophy of Gambling (2016).

By Joseph Buchdahl

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