My 9 year-old daughter’s soccer games are often high-scoring affairs. Double-digit goal totals are not uncommon. So when her team went ahead 2-0 on Saturday someone on the sideline remarked that 2-0 is not the comfortable lead that you usually think it is in soccer.
But that got me thinking. Its more subtle than that. Suppose that the game is 2 minutes old and the score is 2-0. If these were professional teams you would say that 2-0 is a good lead but there are still 88 minutes to play and there is a decent chance that a 2-0 lead can be overcome.
But if these are 9 year old girls and you know only that the score is 2-0 after 2 minutes your most compelling inference is that there must be a huge difference in the quality of these two teams and the team that is leading 2-0 is very likely to be ahead 20-0 by the time the game is over.
The point is that competition at higher levels is different in two ways. First there is less scoring overall which tends to make a 2-0 lead more secure. But second there is also lower variance in team quality. So a 2-0 lead tells you less about the matchup than it does at lower levels.
Ok so a 2-0 lead is a more secure lead for 9 year olds when 95% of the game remains to be played (they play for 40 minutes). But when 5% of the game remains to be played a 2-0 lead is almost insurmountable at the professional level but can easily be upset in a game among 10 year olds.
So where is the flipping point? How much of the game must elapse so that a 2-0 lead leads to exactly the same conditional probability that the 9 year olds hold on to the lead and win as the professionals?
Next question. Let F be the fraction of the game remaining where the 2-0 lead flipping point occurs. Now suppose we have a 3-0 lead with F remaining. Who has the advantage now?
And of course we want to define F(k) to be the flipping point of a k-nil lead and we want to take the infinity-nil limit to find the flipping point F(infinity). Does it converge to zero or one, or does it stay in the interior?
Cheap Talk 
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May 21, 2012 at 9:16 am
Elliot L
I would think F(\infty) should be zero, unless there’s underlying uncertainty about the absolute—not just relative—propensity to score. (“UAPS”)
If there’s no UAPS, then:
– For n large, a lead of n+1 vs. n gives very little additional information about the teams’ relative abilities.
– Thus only the primitive effect of the lead is leftover. [I guess to complete the argument, I need a finite expected number of goals? I’m comfortable with that.]
However, under UAPS:
– The first bullet point above still applies, roughly.
– However, as we learn that the game is more high scoring, a lead takes on less meaning. If the professionals exhibit no UAPS and we have extensive UAPS about the 9-year-olds, then I think we can cook F(\infty) to be anything strictly less than 1.