Quote:
Originally Posted by Dirkadirkastan
I think assuming the games are independent and have no impact on each other is a reasonable null hypothesis (such is the typical stance of null hypotheses). Some people think wins have a positive impact on future games (momentum) and others think they have a negative impact (letdown). Personally, I find that we see too much of both to believe either one is a genuine force.
Even if you disagree, let me expand on this a bit then I'll address the alternative.
If the games are independent, then their order simply does not matter. Consider the case where you're down 3-1 but still have two home games. Either
A) You have Game Five at home. The good news is you have a good chance to win that game and extend the series. The bad news is the best you can do is force another road game.
OR
B) You have Game Five on the road. The bad news is that you're now more likely to lose that game. But the good news is you have a chance to win, with BOTH remaining games at home. In other words, there's a higher risk but also a higher reward if the game is won.
Mathematically, it all evens out. For the sake of example, let's assume your odds of winning a home game are 60% and your odds of winning a road game are 40%. Then your odds of winning Games 5-7 in scenario A are (.6)(.4)(.6) = 14.4%. In scenario B, the odds are (.4)(.6)(.6) = 14.4%.
What you're proposing is a conditional probability model. That is, the games are not independent; rather, past games impact the win probabilities of future ones. Perhaps in scenario A, winning Game Five increases the likelihood of getting that road win in Game Six to 50%, and a win there in turn builds momentum such that your overall win odds are 70%. Then your odds of winning the series become (.6)(.5)(.7) = 21%. Whereas in the other scenario, perhaps only the odds for Game Seven are increased to 70%, in which case your odds of winning are (.4)(.6)(.7) = 16.8%.
There are two issues I have with this model. One is that I find it too complex to justify the decisive conclusion that you draw from it. Sure, maybe (maybe!) in the specific case you have home field yet trail 3-1, you are better off playing Game Five at home. But to truly evaluate the worth of having this game at home overall, you cannot just analyze this scenario. You have to analyze it under all possible scenarios and weight them accordingly. You may be down 3-1, but you may also be up 3-1, and it could also be 2-2. Mathematically, you have to set up a win probability matrix with each scenario weighted properly. Maybe Game Five is good to have at home when down 3-1, but maybe it's not all that likely you trail 3-1 in the first place. And maybe the consequences of the other scenarios outweigh them. Maybe.
Secondly, and more importantly, in order to abandon the null hypothesis, you have to come up with strong observable evidence that the null hypothesis is false. You can't just feel it in your gut that the guys are more confident and roll with it. You can point to the 2008 World Series and say it was all momentum, but then I'll ask you to explain the 2010 NLCS with the same analysis.
The way I see it, treating the games as independent is as reasonable as any other theory, with the side benefit that it is easier to analyze.
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Question 87 on your SAT test (humor me):
You are the manager of a baseball franchise starting a World Series matchup on the road. You feel that you have the better team and your regular season record indicates as much. Given matchups and the games being on the road, you anticipate a 1-1 series split after the first two games, though there are no guarantees that this will be the outcome. You want to end the series as quickly as possible due to the obvious fatigue that comes with playing a grinding season and long playoff schedule.
That all being said, you have been offered the opportunity to choose whether or not you want to play the series in a 2-3-2 format or a 2-2-1-1-1 format. Your goal is to pick the option that provides the greatest mathematical odds for a series victory for your organization. You decide to ask your team statistician for assistance with making the decision and you inform him of your anticipation for a 1-1 series split after the first two games. He then informs you that there are two fundamental theories that need to be considered, with varying winning percentages associated with each.
The first theory is that baseball teams that win or lose carry no emotional impact from said victory or defeat, and as a result play no better following a win or no worse following a loss. The second theory is that baseball teams that win or lose do carry emotional impact from said victory or defeat, and as a result the win probability for the following game increases. Please note that both theories have had evidence provided that supports that respective stance and both theories have had evidence provided that nullifies that respective stance. Regardless, support exists in the baseball world for both theories so both should be considered as possible.
The statistician goes on to explain that, with either theory, the home team has a 60% chance to win. This means that with the 3-2 scenario along with the independent game theory, the chances of you winning 3 games in a row to quickly end the series is 21.6% (.6 x .6 x .6). The statistician then informs you that with the non-independent game theory, the winning team adds an additional 10% chance to win each following game. This means that with the 3-2 scenario and this specific theory, the chances of you winning 3 games in a row to quickly end the series increases to 33.6% (.6 x .7 x .8).
The second alternative is the 2-1-1-1 format, and with this scenario along with the independent game theory, the chances of you winning 3 games in a row to quickly end the series is 14.4% (.6 x .6 x .4). With this same format and with the non-independent game theory, the chances of you winning 3 games in a row to quickly end the series is 25.2% (.6 x .7 x .6).
While you prefer to end the series quickly, there is also the potential that you can win the series in 6 or 7 games in the event that you don't manage to win 3 games in a row. There is also the potential that the anticipated start won't take place and in reality the series will shift to your home with you either up 2-0 or down 0-2. This all must be calculated and taken into consideration by you, but you still feel that your anticipated results are most likely and you still prefer to end the series as quickly as possible.
Given the above situation and given the feedback from your statistician, as manager of the team, which option would you choose to provide the best possible edge for your team:
1) 2-3-2 format
2) 2-2-1-1-1 format
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Now, that was rushed and all possibilities were not thought out so forgive any mistakes made. That said, a question like this (which was the setup I was suggesting by indicating the 2-3-2 model was beneficial for us IF we could win one of the first two games) points to the 2-3-2 format as the answer I would provide to that question.