Quote:
Originally Posted by grndmstr_c
Okay, Ill take the crow. I think I must have been getting myself crossed on winning possessions instead of winning games, but I just worked out a simple proof and dirno and CD are correct on this. Should've taken the time to look at it systematically first, but for those who are interested, here's an asymptotic argument that illustrates why:
Suppose you have Team A and Team B playing a game that proceeds in discrete fashion, i.e., in possessions, and suppose Team A has constant probability of winning a single possession = p. Now, what we're interested in is what if we extend the game so that it has N possessions, and winning the game is defined as winning a majority of the possessions, i.e., winning (N+1)/2 or more possessions. We can use the normal approximation to the binomial if we assume N is large enough, and doing so it's straightforward to show that the probability of winning (N+1)/2 contests is aproximately equal to the cumulative distribution function (cdf) of the standard normal distribution evaluated at the square root of N times a function of p that is positive if p > .5, and negative if p < .5. The standard normal cdf is strictly increasing from 0 to 1 across the real number line, and it follows that the probability of winning (N+1)/2 contests will go to 0 as N goes to infinity if p is even slightly less than .5, and will go to 1 as N goes to infinity if p is even slightly greater than .5. Obviously, the reverse occurs as N goes from large to small.
My apologies gentlemen, and may this serve as a reminder to all that there is real value in striving even to disagree with civility.
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Yeah, I think that is the way I'm looking at it. For example, let's say that Teams A and B play, and Team A is a 55/45 favorite on each and every possession. If we "pay out" after every single possession, then after enough possessions we would expect the score to be 55/45. But if we wait to "pay out" until we see who is ahead after, say, 100 possessions, then we would expect Team A to win far, far more often than their "per possession" win expectation of 55/45. In fact, after 100 possessions we would expect them to be ahead well better than 90% of the time, right? (I'm just going on gut feel here...100 trials at 55/45 each trial seems like enough to get close to even 100%.)
Am I looking at it right? If so, it would mean that even a 60-win team playing a 20-win team is likely only a small favorite on any one individual possession...hence, the reason that single-possession games would result in all teams moving closer to .500.