What Price Loyalty?

The days before we lock-in a position on Round 1 can drag a little. I've spent some of this idle time in a state that's the mental equivalent of anxious pacing, in the course of which I came to be pondering what we covered in the previous blog.

There we discovered that a competent margin predictor - essentially one whose predictions match the TAB Sportsbet bookmaker's - can be about 50% certain that he or she will be within 4 goals of the final result. Implicitly, this statistic requires that the predicted margin be equal to the true expected value of the final margin or, more loosely, that the predicted margin be an accurate assessment of a team's chances such that, in the long run, teams predicted to win by X points, on average, win by X points.

But what, I wondered, if the margin predictor was biased. Say, for example, that he or she was a supporter of one of the competing teams and so overestimated the true strength of the team that he or she supported. What effect might this have on the 50:50 proposition that an unbiased prediction would deliver?

It turns out that modest biases have only a minor effect, as you can see in the following table:

Consider, for example, a biased supporter who predicts that the Dogs will win by 12 points when an unbiased estimate of the Dogs' chances would have them as just 6-point favourites. In that case, the line of the table that applies to the biased supporter is the one labelled "Off by 6 points" and this tells us that his or her chances of being within 24 points of the actual final margin are about 48%, only a couple of percent smaller than that which could be achieved with an unbiased prediction.

The chart below provides greater detail of the tradeoff between bias and the probability of the prediction being within 4 goals of the eventual result.

As you can see, a predictor doesn't pay much of a penalty for his or her bias until it reaches about 2 goals. At that point the reduction in probability is about 4% points. Biases larger than this have a more substantial effect on the probability of the prediction being within 4 goals of the final margin, so much so that, once the bias has reached 6 goals, the probability has dropped to under one-third.

That's a hefty price to pay for loyalty.