So I've been trying to apply more mathematical models to look at balance in the mod. However, a key notion to do that is the concept of Yield Discounting.
We all know that 1 science on Turn 1 is worth more than 1 science on Turn 200 due to the snowball nature of the game. The key question is.... how much?
A true model would be extraordinarily complicated, but we can simplify it using some actual game experience and get a ballpark.
To start, I am going to assume a SV victory, where techs are gathered over 375 turns (the last 15 is for building a spaceship, which is 400 turns total). I think this is too slow for higher difficulties but its probably about in the ballpark for lower difficulties. So ultimately we care about getting science for 375 turns, after which science value becomes effectively 0.
So the idea is, each era (technically each row of the tech tree) has an increase in science costs. In other words, it takes more science to achieve the same effect (getting a tech). Another way to put this is that the value of a single unit of science is going down (aka a discount). Since we know exactly how much science it takes to get a tech by looking at the tech tree, this gives us a feeling for how science discounts over time.
So calculating the total science for a given era is easy. But now the real question, how long do we spend in each era? This is a tricky question, often we don't complete an era fully before going to the next one, and its not exact at all parts of the game. But as a ballpark we can make some assumptions, and I could really use some feedback on this part to get the right feel.
Turn Times per Era (FEEDBACK REQUESTED)
So this is the Era, and the number of turns a player spends researching that's Eras tech (whether I do it initially, or backpedal and fill in the tree later, doesn't matter). The total is 375.
Ancient 60
Class 60
Medieval 55
Ren 55
Ind 50
Modern 40
Atomic 35
Info 20
My general feeling is that eras go a little quicker each time as the game goes on (and Information blitzes by when you go SV), but again I could really use some thoughts on this table, its the crux of the whole model.
Average SPT at Different Points in the Game
So with these numbers and the totals from the tech tree we can now calculate roughly how much science per turn (SPT) we need to have at various turn numbers in order to stay competitive.
0 15.75
60 35.33333
120 112.6
175 410.4545
230 871.5
280 1981.875
320 2955
355 9161.25
Our Discount Formula
With these numbers in hand, we can now do an Excel regression to give us a formula, telling us the expected SPT at any given turn.
= 14.595e0.0176T
T = Turn Number
To keep the numbers easy, I am going to normalize the numbers by diving by 14.595. This means our Turn 0 number is now 1 SPT, and therefore its easy to compare at different turns.
= 14.595e0.0176T / 14.595
Some Example Numbers
So with our formula, the question is....how much SPT is "equal" to 1 SPT at the very beginning of the game. Aka how much more science does it take to move the needle?
0 1.0
50 2.4
100 5.8
150 14.0
200 33.8
250 81.5
300 196.4
350 473.4
374 722.3
So to show how you can use these numbers. If we look at Turn 100, the value is 2.46 (so roughly 2.5) times higher than the number at 50. This means if you could find a way to gain an extra 10 science on Turn 50, it would be equivalent to getting an extra 25 science on Turn 100.
Another example, if you built a council (1 SPT) on Turn 50, you would have to build roughly 197 councils on Turn 350 to have the same overall impact (not including the growth science, just the base +1).
One More Chart
So this chart shows the speed at which yields lose value.
50 0.83%
100 0.41%
150 0.28%
200 0.21%
250 0.17%
300 0.14%
So what this means is that on Turn 50, every turn that passes your yield loses .83% of its current value. On Turn 200, its .21%. This is the snowball, the earlier you can get yields the more they are worth. This means that even a small different in the first 100 turns makes a lot more difference than the same difference in the next 100 turns.
The Failures of the Model
To be clear, this is a rough model, as doing a true in depth model would take much more work than I am willing to do. This model looks at each era as a block, a superior model would look at what turn times we get each and every tech.
The model also doesn't consider discounts for getting the same tech as some one else, spy techs, or TR rubber bands. All of that washes out across the whole model, but it also means that each data point is going to be a little off. Those numbers are pretty small compared to the main numbers so the effect should be small but it is present.
And lastly, the model greatly depends on the amount of turns spent in each era, so once again I implore anyone looking at these numbers to give me feedback there if you think I'm off, that is the most critical data for the model to work well.
We all know that 1 science on Turn 1 is worth more than 1 science on Turn 200 due to the snowball nature of the game. The key question is.... how much?
A true model would be extraordinarily complicated, but we can simplify it using some actual game experience and get a ballpark.
To start, I am going to assume a SV victory, where techs are gathered over 375 turns (the last 15 is for building a spaceship, which is 400 turns total). I think this is too slow for higher difficulties but its probably about in the ballpark for lower difficulties. So ultimately we care about getting science for 375 turns, after which science value becomes effectively 0.
So the idea is, each era (technically each row of the tech tree) has an increase in science costs. In other words, it takes more science to achieve the same effect (getting a tech). Another way to put this is that the value of a single unit of science is going down (aka a discount). Since we know exactly how much science it takes to get a tech by looking at the tech tree, this gives us a feeling for how science discounts over time.
So calculating the total science for a given era is easy. But now the real question, how long do we spend in each era? This is a tricky question, often we don't complete an era fully before going to the next one, and its not exact at all parts of the game. But as a ballpark we can make some assumptions, and I could really use some feedback on this part to get the right feel.
Turn Times per Era (FEEDBACK REQUESTED)
So this is the Era, and the number of turns a player spends researching that's Eras tech (whether I do it initially, or backpedal and fill in the tree later, doesn't matter). The total is 375.
Ancient 60
Class 60
Medieval 55
Ren 55
Ind 50
Modern 40
Atomic 35
Info 20
My general feeling is that eras go a little quicker each time as the game goes on (and Information blitzes by when you go SV), but again I could really use some thoughts on this table, its the crux of the whole model.
Average SPT at Different Points in the Game
So with these numbers and the totals from the tech tree we can now calculate roughly how much science per turn (SPT) we need to have at various turn numbers in order to stay competitive.
0 15.75
60 35.33333
120 112.6
175 410.4545
230 871.5
280 1981.875
320 2955
355 9161.25
Our Discount Formula
With these numbers in hand, we can now do an Excel regression to give us a formula, telling us the expected SPT at any given turn.
= 14.595e0.0176T
T = Turn Number
To keep the numbers easy, I am going to normalize the numbers by diving by 14.595. This means our Turn 0 number is now 1 SPT, and therefore its easy to compare at different turns.
= 14.595e0.0176T / 14.595
Some Example Numbers
So with our formula, the question is....how much SPT is "equal" to 1 SPT at the very beginning of the game. Aka how much more science does it take to move the needle?
0 1.0
50 2.4
100 5.8
150 14.0
200 33.8
250 81.5
300 196.4
350 473.4
374 722.3
So to show how you can use these numbers. If we look at Turn 100, the value is 2.46 (so roughly 2.5) times higher than the number at 50. This means if you could find a way to gain an extra 10 science on Turn 50, it would be equivalent to getting an extra 25 science on Turn 100.
Another example, if you built a council (1 SPT) on Turn 50, you would have to build roughly 197 councils on Turn 350 to have the same overall impact (not including the growth science, just the base +1).
One More Chart
So this chart shows the speed at which yields lose value.
50 0.83%
100 0.41%
150 0.28%
200 0.21%
250 0.17%
300 0.14%
So what this means is that on Turn 50, every turn that passes your yield loses .83% of its current value. On Turn 200, its .21%. This is the snowball, the earlier you can get yields the more they are worth. This means that even a small different in the first 100 turns makes a lot more difference than the same difference in the next 100 turns.
The Failures of the Model
To be clear, this is a rough model, as doing a true in depth model would take much more work than I am willing to do. This model looks at each era as a block, a superior model would look at what turn times we get each and every tech.
The model also doesn't consider discounts for getting the same tech as some one else, spy techs, or TR rubber bands. All of that washes out across the whole model, but it also means that each data point is going to be a little off. Those numbers are pretty small compared to the main numbers so the effect should be small but it is present.
And lastly, the model greatly depends on the amount of turns spent in each era, so once again I implore anyone looking at these numbers to give me feedback there if you think I'm off, that is the most critical data for the model to work well.
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