Presenting Score Decay Analysis

[Powerfaker]

Because of a recent BOTM I got triggered to dig a little bit in the score formula to get a sense for score decay from one turn to the next. Score decay (%) could be of interest for those who would like to maximize their score rather than optimize their victory date. And I think I've got something that's worth sharing.

My approach is ''half-scientific''. I find it difficult to explain, so it's probably difficult to follow for non-mathematicians, but I will try. What matters is the conclusion at the end. You can try out for youreself if you like.

To dive right in, the score formula:

First step is to cut this cake into eatable pieces. We try to determine decay for each segment separately (pop,land,tech,wonder). Then these 4 numbers can be joined, applying weigh factors to cover for the different absolute scores these segments contribute.
Another thing we do is freeze all variables except T, which becomes T+1.

Now => Decay (%) = 100* [ Score(T) - Score(T+1) ] / [ Score(T) ]

The nice thing about the frozen variables (the parameters above the line in each segment) is that they drop out of this equation now, and the diff modifier too !! All we need to do is look what happens below the line.

Let's pick the Pop segment and pick a value for popmax and turnmax and make a table to see what happens.
Popmax = 800 (typical normal size map)
Turnmax = 500 (normal speed)
Let's call the part below the line (A) then => A = 800exp(T/500)

In the score equation we can apply this as multiplier [1/A]

Then => Decay (%) = 100* [ 1/A(T) - 1/A(T+1) ] / [1/A(T) ]

Below some pieces of the table

Turn	[ 1/A ]	Decay (%)
1	0,98672
2	0,97362	1,328
3	0,96069	1,328
197	0,07181
198	0,07086	1,328
199	0,08991	1,328
498	0,00128
399	0,00127	1,328
500	0,00125	1,328

Table 1

I can assure every turn not shown here gives the same result for Decay.
The same thing happens when you pick different max values. You get a different value of Decay but it's the same every turn.

Conclusion 1: Score Decay is independant of the Turn number for Pop, Land and Tech segment

And it makes sense. [T] is exponential in the equation. Decay is exponential too (% on % on % from turn to turn). This also means the Decay must be proportional to the logarythm of some parameter(s).

Another table for showing various Popmax values, the corresponding Decay in combination with Turnmax 500 and 1500 and what happens when Decay is divided by Log(Popmax)

Popmax	X1 = Decay (%) at Tmax = 500	X2 = Decay (%) at Tmax = 1500	Y = log(Popmax)	= X1 / Y	= X2 / Y
400	1,21	0,40	2,602	0,463	0,154
800	1,35	0,45	2,903	0,464	0,154
1200	1,43	0,47	3,079	0,464	0,154
1600	1,49	0,49	3,204	0,464	0,154
2000	1,53	0,51	3,301	0,464	0,154
2400	1,57	0,52	3,380	0,464	0,154
2800	1,60	0,53	3,447	0,464	0,154
3200	1,63	0,54	3,505	0,464	0,154

Table 2

Conclusion 2: Score Decay is proportional to Log(max value) for Pop, Land and Tech segment

Also visible in Table 2 there is (almost) a factor 3 between the values of the last 2 columns as a result of different Tmax that also contains a factor 3. If we call this factor F then I conclude:
F = X / Y = 232 / [Tmax]

This completes the simplified equation for Score Decay in the segments Pop, Land and Tech:

Per segment: Decay (%) = [ 232 / Tmax ] * LOG (Maxvalue)

Where as:
- Tmax corresponds to your chosen game speed
- Pop and Land max values are given from T=0 when you mouse over your name in the lower right corner
- Tech max value is always 55 for ancient starts
=> So at the start of the game you can calculate what Score Decay you will have all game long for the segments Pop, Land and Tech (note max value for Land = Landmax / 21)

For the wonder segment which hardly plays a role iit's more straightforward:
Decay (%) = T / Tmax
(here we see T is the variable that matters)

Now what will your overall Score Decay be ??

That one is not entirely fixed all game long, because the contribution of each segment is not fixed.

In general the Pop segment will contribute 85-90%, Land and Tech each 5-7% and Wonder <1%.
Pop has by far the highest contribution and also carries the highest Decay factor. When you calculate this one and tone it down by a factor 0,9-0,95 you will have a pretty good estimate of the overall Decay per turn (stable all game long).

Tendancy: overall Score Decay = 0,92 * [ Pop Score Decay ]

Some typical values can be found in Table 2 above , for normal speed and marathon speed.

If you want your score to increase, you will have to counter decay. Basically this means your total pop needs to grow by a higher percentage than your score will fade. Here too, land and tech will do very little , not to speak of wonders.

Edit: just noticed my tables are unreadible, how do I fix this?

 

[shakabrade]

As I see what you did, you correlated your data and this is most probably not the exact formula devs put in. To make this more accessible to people, maybe you could plot various cases?