Tactics in Numbers Games

[Michael Wallace]

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.

[Kirk Bevins]

James Robinson eat your heart out.

[Charlie Reams]

Nice work Raccoon, good to get that discussion settled.

[Gavin Chipper]

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)

But what are these numbers?

[Michael Wallace]

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)

But what are these numbers?

The reason I dumbed it down was because I doubted any/many people would understand/care about what they really are. They're the coefficients associated with each variable (relative to a 6 small selection, in this case) of a logistic regression model (with a binary outcome of 0 = loss, 1 = win/draw, and each selection as a categorical variable and ratings difference as explanatory variables).

[Gavin Chipper]

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)

But what are these numbers?

The reason I dumbed it down was because I doubted any/many people would understand/care about what they really are. They're the coefficients associated with each variable (relative to a 6 small selection, in this case) of a logistic regression model (with a binary outcome of 0 = loss, 1 = win/draw, and each selection as a categorical variable and ratings difference as explanatory variables).

Is there not a simpler way to explain what these numbers mean, and if not, is it actually the case that this gives a better indication of what you are trying to say than just the probability itself, for example?

[Michael Wallace]

Is there not a simpler way to explain what these numbers mean, and if not, is it actually the case that this gives a better indication of what you are trying to say than just the probability itself, for example?

"The probability itself"?

[Andrew Feist]

Is there not a simpler way to explain what these numbers mean, and if not, is it actually the case that this gives a better indication of what you are trying to say than just the probability itself, for example?

"The probability itself"?

I think he means "the probability of winning or drawing the numbers game".

My quibble is with the "11 points behind strategy is the opposite of 11 points ahead strategy", because I don't think it is. You seem to be claiming "if you are 11 points behind, you should pick 6 small, since that's least likely to give you any points". I would think that for the people who are behind, you want to look at just winning, not winning and drawing. (EDIT: It might be clear, from what you've done, that the winning-only numbers will have to be in the same order as the winning-and-drawing numbers. It's not clear to me, just from the description, that it is so.)

[Gavin Chipper]

Is there not a simpler way to explain what these numbers mean, and if not, is it actually the case that this gives a better indication of what you are trying to say than just the probability itself, for example?

"The probability itself"?

It seems I'm not at all clear what you've done then. I thought you'd worked out the probability of someone at random winning or drawing a numbers game against someone else picked at random.

Maybe it's more advanced than that, but take the 0.386 as the 1 large statistic. Is this that you are 1.386 times as likely to get the job done with 1 large as with 0 large, or that p(getting the job done with 0 large) + 0.386 = p(getting the job done with 1 large) or something else entirely?

[Michael Wallace]

You seem to be claiming "if you are 11 points behind, you should pick 6 small, since that's least likely to give you any points". I would think that for the people who are behind, you want to look at just winning, not winning and drawing.

That's not what I'm claiming, I'm saying if you're 11 points behind you should pick 6 small because that's most likely to gain you points. Winning and drawing is the opposite of just losing here, so the outcome that is most likely to win or draw is the least likely to lose you points.

[craig]

If you're 11-20 points behind then you want to win the numbers game. 6 small gives more chance of your opponent missing the game which means you have a chance at a crucial conundrum. Basically what Michael beat me to saying.

[Michael Wallace]

It seems I'm not at all clear what you've done then. I thought you'd worked out the probability of someone at random winning or drawing a numbers game against someone else picked at random.

Maybe it's more advanced than that, but take the 0.386 as the 1 large statistic. Is this that you are 1.386 times as likely to get the job done with 1 large as with 0 large, or that p(getting the job done with 0 large) + 0.386 = p(getting the job done with 1 large) or something else entirely?

You could try reading this, it'll explain it about as quickly as I could.

[Andrew Feist]

You seem to be claiming "if you are 11 points behind, you should pick 6 small, since that's least likely to give you any points". I would think that for the people who are behind, you want to look at just winning, not winning and drawing.

That's not what I'm claiming, I'm saying if you're 11 points behind you should pick 6 small because that's most likely to gain you points. Winning and drawing is the opposite of just losing here, so the outcome that is most likely to win or draw is the least likely to lose you points.

I think I'm with you now. Let me summarize what I'm thinking and we'll go from there: there are only two outcomes: win/loss and draw (I was originally counting 0/0 as a separate thing, but that's really just a draw). If the W&D percentage comes out to p, what that really means is that 2(1-p) of the games were outright won [i.e., if 60/100 games were wins, and therefore 40/100 were draws, then the win+draw percentage would be calculated as 60W+80D=140/200 people for 70%, leaving 30% behind as half of the win/loss games]. Hence the lower p is, the more outright wins there are, and if you're 11+ points down, only an outright win does you any good.

[Clive Brooker]

Presumably every player has an individual profile of success rates for each selection, and I would imagine this would become quite stable once sufficient games have been played. Intuitively I would expect players' profiles to differ considerably, but perhaps they don't.

The clear conclusion so far has been that if you need a result from a numbers game, you should pick 6 small. If players are markedly different in their success profiles, then when player A plays player B, there should be an optimum strategy for each and this won't necessarily be 6 small. If, on the other hand, players are pretty similar - generally better or worse at each selection type in roughly equal measure - then 6 small will turn out to be right for everyone.

I'd still put my money on there being considerable differences, which would mean that it's important to play the player.