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BGonline.org Forums
Simplified Random Walk Model for Chouette $ expectation
Posted By: Rod
Date: Monday, 22 April 2013, at 8:53 a.m.
A number of people on this board are probably favorites in their local chous. I've always understood that my results will be all over the place but will average positive. I've encountered both favorites and dogs who seem to always feel they should be winning more (or losing less). I recently had a few sessions that I didn't feel were anomolous but certainly further piqued my curiosity.
What I found is that you should decrease your expectations for being up at the end of a chou if you feel that since you're better than your chou-mates you should almost "always" be up. Down sessions for the best player in a chou (there's always one) are far more likely than I would have thought.
I created a simplified random walk model that seems to fit with the results (standard deviation-wise) that I've encountered over the past 6 months (probably more than enough sessions to be a meaningful number of data points). The simplifications are: 1) an averaging of the % chance to win each game (as box, captain and team member) and 2) an averaging of the number of points won or lost (absolute value) in each game.
I decided to use 3 points per game as the average win/loss amount. Assuming a 6 man chou, there are all your 1, 2 and 4 (and the occasional 8) point wins/losses as captain and team and your 5, 10 and 20 point wins/losses as box. Throw settlements into the mix and 3 seems to be about right.
The spreadsheet I created has 100,000 data points: 1000 - 100 game sessions. The average and the standard deviations remained in the same general ballpark as I regenerated the spreadsheet for each % chance to win. Here are the results for a 52% favorite, 55% favorite and 60% favorite (unlikely except in extreme circumstances since I have averaged turns as box, captain and team member):
52%: +13 points for the session with a 30 point std dev
55%: +31, 30 std dev
60%: +60, 29 std dev
What these numbers mean is that even for the player who is a 55% favorite on average, 1/3 of the time he will be up between 0 and 30 points, 1/3 of the time up between 30 and 60, 14% down 0 to 30, 14% up 60 to 90, and 2.5% down more than 30 and 2.5% up more than 90. But the interesting thing here is that a full 1/6 of the time the 55% favorite player will have a negative expectation.
And let's say you're playing a heads up session against someone who you are a 60% favorite against. Assuming 2 points as the average amount won or lost per game, the model gives and average of up 40 points after a 100 game session with a 20 point std dev. This means that in a completely mismatched 2 person session, the 60% per game favorite will suffer a loss 2.5% of the time. For the 55% per game favorite, the average up is 20 points with a 20 point std dev, meaning that a 55% fav should have a negative expectation 1/6 of the time.
There was a heads up session a few of us (people in my chou) recently heard about where the 100 game (about) session resulted in a 130 loss by one of the players. A few commented, essentially, that "that shouldn't have happened". Knowing the relative strength of the players I felt that while it might be on the high side that it was well within expected results. Assigning the better player a 65% chance of winning and an average 3 point win/loss per game (cubes started to get passed more readily, as they sometimes do in these types of situations), my model gives an average won of 90 points with a 30 point std dev, meaning that the better player was only a bit lucky, as his result or better would tend to occur about 1/6 of the time. And while he could have lost, that would only be a 1 in 1000 occurrence.
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