Andrew Foland provided a solution to this all the way back in January 2011, but it has not been implemented, perhaps because of the complexity of the solution, perhaps simply because nobody has got around to it yet, whichever the case I am here to provide a simple alternative and hope it gets implemented.
The problem is that of combining the prior strength of the teams – the expected difference in performance over a game, let us call this S, with the current state of the game as expressed as a probability, WP. To do this we must consider the total game time, 60 minutes, and the game time remaining, T, in minutes.
Intuitively we can say that S varies such that S(T)=S(60)*T/60, where S(60) is simply the value of S at the start of the game. We can see this is true, intuitively, because if I tell you team A will on average beat team B by ten points over a sixty minute game, then we'd both agree that if the games were cut to thirty minutes then on average A would beat B by five points.
However, we must also consider how the standard deviation of this value varies with time, and the answer happens to be that it varies as the square root of the ratio of time reamining to total time, in other words: Stdev(T)=Stdev(60)*((T/60)^0.5), where Stdev(60) is the standard deviation at the start of the game. As such we reach the equation for the variation of the team strength parameter over the course of the game, S(T)=S(60)*((T/60)^0.5).
How do we get the initial value of S, S(60)? We take it from Brian's win probability for the game:
S(60)=ln(P/(1-P)), where P is Brian's predicted win probability.
So now, lastly, we need the information from Brian's live model, which we get from taking the probability from the model for the current game time, call it WP, and doing as we did to find S(60):
W=ln(WP/(1-WP)), where W is the logit value of WP.
Now, finally, we have everything, and it combines nicely into this neat, simple equation:
And that's it, hardly complicated all in all, maybe if we're very lucky it'll be implemented.