Tactics in Numbers Games
Posted: Sun Feb 14, 2010 6:18 pm
Following from a discussion in the Thursday 11th February 2010 Recap Thread, Charlie 'sent' me some of his 'data', and I got to work on doing a slightly more thorough look at that incredibly important (that is, not very) question of what you should pick if you're 11+ ahead going into the last numbers game.
So, assuming you are trying to win or draw, the following numbers are an indication of how likely you are to win or draw a numbers round, regardless of rating difference, relative to a 6 small game. A number larger than 0 indicates you are more likely to win that game than a 6 small game, whilst a number smaller than 0 indicates you are less likely to win that game than a 6 small game. The numbers in brackets are 95% confidence intervals, for the more statistically inclined. For the laypeople, these give an indication of how different the estimates might be (if one estimate's confidence interval overlaps with another's, this suggests the two estimates aren't necessarily different).
1 large: 0.386 (0.351,0.421)
2 large: 0.225 (0.184,0.266)
3 large: 0.086 (0.029,0.143)
4 large: 0.018 (-0.029,0.065)
So what these numbers tell us is that by far and away the best choice if you're 11+ ahead is 1 large. With 2 large the next best, and still significantly better than 6 small. 3 large seems a bit better than 6 small, but the confidence interval comes close to 0, which indicates we can't be too confident it is actually a better pick than 6 small, and for 4 large, the confidence interval does include 0 - 4 large, statistically at least, is no better a pick than 6 small.
We can infer, therefore, that 6 small and 4 large are the worst picks if you just need to avoid losing the round, and that 1 large is comfortably the best (and, analogously, if you're 11+ behind you want to pick 6 small or 4 large, and definitely not 1 large).
However, you may not want to ignore the ratings difference between you and your opponent. Supposing you're confident you're much better (or worse) at the numbers, should that change your strategy?
The numbers below indicate how important the ratings difference is, for each selection. The larger a number, the more important the ratings difference between players is.
0 large: 0.116 (0.105,0.127)
1 large: 0.134 (0.129,0.139)
2 large: 0.116 (0.109,0.123)
3 large: 0.133 (0.118,0.148)
4 large: 0.132 (0.121,0.143)
The selections split up into two sets, with 0 large and 2 large seeming to indicate ratings difference plays a less important part than in the other selection. However, the confidence intervals overlap rather a lot, and so these distinctions aren't particularly concrete.
In short, therefore, if you're 11+ ahead, pick 1 large, if you're 11+ behind pick 6 small or 4 large, and ratings don't make much difference across the different selections.
So, assuming you are trying to win or draw, the following numbers are an indication of how likely you are to win or draw a numbers round, regardless of rating difference, relative to a 6 small game. A number larger than 0 indicates you are more likely to win that game than a 6 small game, whilst a number smaller than 0 indicates you are less likely to win that game than a 6 small game. The numbers in brackets are 95% confidence intervals, for the more statistically inclined. For the laypeople, these give an indication of how different the estimates might be (if one estimate's confidence interval overlaps with another's, this suggests the two estimates aren't necessarily different).
1 large: 0.386 (0.351,0.421)
2 large: 0.225 (0.184,0.266)
3 large: 0.086 (0.029,0.143)
4 large: 0.018 (-0.029,0.065)
So what these numbers tell us is that by far and away the best choice if you're 11+ ahead is 1 large. With 2 large the next best, and still significantly better than 6 small. 3 large seems a bit better than 6 small, but the confidence interval comes close to 0, which indicates we can't be too confident it is actually a better pick than 6 small, and for 4 large, the confidence interval does include 0 - 4 large, statistically at least, is no better a pick than 6 small.
We can infer, therefore, that 6 small and 4 large are the worst picks if you just need to avoid losing the round, and that 1 large is comfortably the best (and, analogously, if you're 11+ behind you want to pick 6 small or 4 large, and definitely not 1 large).
However, you may not want to ignore the ratings difference between you and your opponent. Supposing you're confident you're much better (or worse) at the numbers, should that change your strategy?
The numbers below indicate how important the ratings difference is, for each selection. The larger a number, the more important the ratings difference between players is.
0 large: 0.116 (0.105,0.127)
1 large: 0.134 (0.129,0.139)
2 large: 0.116 (0.109,0.123)
3 large: 0.133 (0.118,0.148)
4 large: 0.132 (0.121,0.143)
The selections split up into two sets, with 0 large and 2 large seeming to indicate ratings difference plays a less important part than in the other selection. However, the confidence intervals overlap rather a lot, and so these distinctions aren't particularly concrete.
In short, therefore, if you're 11+ ahead, pick 1 large, if you're 11+ behind pick 6 small or 4 large, and ratings don't make much difference across the different selections.