Time Matters: The Science of Winning in Bullet Chess
I recently posted a blog where I attempted to quantify the advantage a player has in a bullet game based on the amount of time they have. To do this, I took the number of moves a player could make and called it δ, defined as
I expanded on this by taking the difference between each players number of moves and dividing it by the total number of moves that can be made:
Essentially, dividing the difference by the total number of moves the players can make tells us who is favored in the sense that the player with more moves available is less likely to lose on time. If black is favored, we will get a negative output. If white is favored, a positive. All we are doing is calculating who has more moves and dividing it by the total number of available moves to see who has the larger amount of moves available to them (and by how much).
From here, I attempted to take into account the engine-based analysis but failed miserably (many thanks to user 'jaldag' for pointing out all of the flaws and for providing the basis for this new method). I have since removed that blog post because it was truly a complete and total mess. Jaldag suggested we take into account the engine based analysis, B, by using a formula similar to
At some point, then, the two advantages should exactly cancel each other out. For example, if white has a winning position but black has an equally comparable time advantage, then the position should be closer to equal. To arrive at a numerical answer for these two weighted constants, I will make the following assumptions:
1. A time advantage of 10 seconds is comparable to one side being up a minor piece, with an average engine evaluation of ± 2.7
2. A time advantage of 30 seconds is comparable to one side being up a queen, with an average engine evaluation of ± 9.9
With these two assumptions in mind, we can arrive at a system of equations to solve for (roughly) what our weighted constants should be.
When white is up 10 seconds on the clock (say, 60 seconds to 50 seconds), we can quantify the time-based advantage as:
Now, suppose in this same game black is up a knight, putting the engine evaluation somewhere around -2.7. Since we are assuming a ten second time advantage is roughly equivalent to being up a piece, the following must be true:
If we repeat this for the situation where white is up 30 seconds on the clock but down a queen (with an engine evaluation of ~-9.9), we arrive at the following:
Now, this system of equations cannot be solved without our two weighted constants being equivalent to zero, so let's look at the case where the position is equal but white has a time advantage. We'll use the same situations as before, where white is up 10 seconds and 30 seconds. Thus, our two equations would be
Now here it is very clear that w_2 has a value of 30. When we look back at our first two equations, this means that w_1 must have a value of 1. Thus, our final equation for taking into account engine-based analysis with our time-based analysis of a game would be
Problems with this:
Obviously, this is very assumption heavy. It is not true that one side being up a queen will always have an advantage of ± 9.9. In fact in most cases, it is probably not true. This number does, however, help us get a more accurate formula for taking time-based advantages into account. It is merely an approximation; a tool that can be used to analyze who is more likely to win a given bullet game.
