One of the purest and most fascinating examples of the “wisdom of crowds” in action comes courtesy of a unique online contest called ProbabilitySports run by mathematician Brian Galebach.

In the contest, each participant states how likely she thinks it is that a team will win a particular sporting event. For example, one contestant may give the Steelers a 62% chance of defeating the Seahawks on a given day; another may say that the Steelers have only a 44% chance of winning. Thousands of contestants give probability judgments for hundreds of events: for example, in 2004, 2,231 ProbabilityFootball participants each recorded probabilities for 267 US NFL Football games (15-16 games a week for 17 weeks).

An important aspect of the contest is that participants earn points according to the quadratic scoring rule, a scoring method designed to reward accurate probability judgments (participants maximize their expected score by reporting their best probability judgments). This makes ProbabilitySports one of the largest collections of incentivized¹ probability judgments, an extremely interesting and valuable dataset from a research perspective.

The first striking aspect of this dataset is that most individual participants are very poor predictors. In 2004, the best score was 3747. Yet the average score was an abysmal -944 points, and the median score was -275. In fact, 1,298 out of 2,231 participants scored below zero. To put this in perspective, a hypothetical participant who does no work and always records the default prediction of “50% chance” for every team receives a score of 0. Almost 60% of the participants actually did worse than this by trying to be clever.

Participants are also poorly calibrated. To the right is a histogram dividing participants’ predictions into five regions: 0-20%, 20-40%, 40-60%, 60-80%, and 80-100%. The y-axis shows the actual winning percentages of NFL teams within each region. Calibrated predictions would fall roughly along the x=y diagonal line, shown in red. As you can see, participants tended to voice much more extreme predictions than they should have: teams that they said had a less than 20% chance of winning actually won almost 30% of the time, and teams that they said had a greater than 80% chance of winning actually won only about 60% of the time.

Yet something astonishing happens when we average together all of these participants’ poor and miscalibrated predictions. The “average predictor”, who simply reports the average of everyone else’s predictions as its own prediction, scores 3371 points, good enough to finish in 7th place out of 2,231 participants! (A similar effect can be seen in the 2003 ProbabilityFootball dataset as reported by Chen et al. and Servan-Schreiber et al.)

Even when we average together the very worst participants — those participants who actually scored below zero in the contest — the resulting predictions are amazingly good. This “average of bad predictors” scores an incredible 2717 points (ranking in 62nd place overall), far outstripping any of the individuals contributing to the average (the best of whom finished in 934th place), prompting someone in this audience to call the effect the “wisdom of fools”. The only explanation is that, although all these individuals are clearly prone to error, somehow their errors are roughly independent and so cancel each other out when averaged together.

Daniel Reeves and I follow up with a companion post on Robin Hanson’s OvercomingBias forum with some advice on how predictors can improve their probability judgments by averaging their own estimates with one or more others’ estimates.

In a related paper, Dani et al. search for an aggregation algorithm that reliably outperforms the simple average, with modest success.