[SPOILERS] There Can Only Be One Suttree - French Fredrick - LOL, Sut?

[Commodore]

Your signs are a bit arcane, where is that again?

Looks nice over-all. Phi abuse incoming?

[suttree]

Something like this:

Forgot the backline commerce site - that's t94. Everything other than t86/t94 is provisional. Hoping there is more food near marble and I might have to fight for the wines.

[suttree]

Much has happened. It shouldn't be a surprise that Oxy (Ghandi) beelined Code of Laws and took Confucianism. I thought about bulbing philosophy, but I think I will go without religion for now. The first great general was born, though I'm sure you can turn to other threads for that story. The Byzantines are in Representation - they're lagging behind in power, but I'm not sure how to convince Sisu to do something about it. I think I will give a shot at renewing our peace t90 - I don't think it will constrain me and it might ease some of the tension from my Hereditary Rule power surge. Sisu is matching me - nice because he has to pay, but not so nice if he ignores BXB and stays focused on me.

SISU
He keeps offering me open borders and built a road though (for now) neutral territory connecting Charlies Bunion. Perhaps he means well, but the culture from Superscript makes things a little unstable. I'm certainly not going to open borders, let his religion spread, and give him visibility on my cities when he pops a great prophet from the oracle. I also don't see the point of having a road there, so I razed it.

As t100 approaches I need to micro out the growth on my cities, perhaps I'll make time to do that for my t90 report.

DEMOS

OVERVIEW

[SevenSpirits]

Why bulb math? That seems like overkill to me, if you just want to chop units. Why not just skip it instead, since your next goal is monarchy, and math isn't a prerequisite for that. And just save the forests until you do have math.

Yay, someone took the bait!

1. I want math now because it powers the chops that provide the units that provide the hereditary rule happiness. After the happy cap is relieved with currency/calendar, those units support the catapults that allow for aggression in the early 100's. Also chops are very important now to set up new cities.

2. I don't understand why bulbing mathematics is bad.

Here are my handwavy thoughts:

HANDWAVE MK I - Math costs 401 beakers in this game. Discounting the prereq bonus, that's ~334 raw beakers. If we approximate the growth rate in the game at 1/t = 30, that's equivalent to ~11b/t. The return on a settled great scientist with library is, say, ~9b/t, and at 50% science I would need a city producing an average of 44 commerce/t before t108 to match with an academy - t108 is my estimated due date for another great scientist.

HANDWAVE MK II - I'm not actually bulbing mathematics. That is, the bulb gets me math no earlier than if I research it myself. Rather, it gets me mathematics and then Monarchy 401b earlier. Is bulbing Monarchy bad?

I'm sure this is bullshit, but it describes how I think about the game. Why am I wrong?

For me, it comes down to if I really believe my investment in the tech tree doubles every 30 turns. If that's true, then bulbing a 400b tech t78 is the same as bulbing a 800b tech t108 is the same as bulbing a 1600b tech t138. And that seems like a pretty good deal.

I've written about this before, and your thread is great, so I want to chime in and correct a logical inconsistency which you might have picked up from me.

Suppose we are taking investments as doubling every 30 turns (normal speed), like in your second post. Then the interest rate on our stuff is 3.53%. With that interest rate I double my money in 30t turns if I keep reinvesting gains.

.... sooooo, yeah, 1/t does not equal 30. If it did, I could take my 30, sell it for 1/t, loan out the trickle of income at 3.53%, and 30t later sell my 1/t machine for 30. Now I have 30 from selling the machine, 30 from collecting 1/t, that's 60 like everyone expected. But I also have the interest I collected on the 1/t while I was accumulating, muhahahaha everyone else in this economy is a fool!

At first I thought, well it's approximate and maybe it cancels out with the fact that the 1/t often sits in a building under construction not doing anything. I don't believe that anymore. I think you get a better result by doing the math right: how much do you gain if you continually invest 1/t (and the proceeds) for 30t? The answer is about 43 and personally I just call it 45 for the sake of simplicity.

So, numbers I use for normal speed:
1 now is worth as much as 2 30t later
1/t = 45

(And for quick speed, 1 = 2 in 20t, 1/t = 30)


What this means for a great scientist in most cases is, the first one should probably build an academy unless it's bulbing philosophy for immediate pacifism or something like that. (Part of that is that the academy is going to keep getting better.) Though, looking at your capital, it's not that strong for commerce, so I don't know.

Btw, I am pretty sure I have written this anywhere yet, but there are a couple subtle side effects of bulbing vs building an academy. When you bulb, you end up increasing the average cost of techs you haven't yet researched, and as a result your natural beakers per turn become less effective: it ends up taking you longer to complete each tech, so the beakers spend more time sitting around doing nothing. Building the academy on the other hand gives a short-term boost in the other direction - for a while after building the academy you will be completing techs faster than average and therefore your beakers will not be sitting around useless so long.

In the long term, these effects start to matter less and less, because the increasing cost of techs means that you're eventually going to find yourself once again researching techs that take you a normal amount of turns, but I think the short-term effect is impactful enough to be worth mentioning.

[suttree]

@SevenSpirits

Thanks for the reply! At the moment, I'm just using my gut to guide decisions and checking myself using your numbers. Your post made me want to think things through for myself.

QUESTION:
What is the value of 1/T?

MODEL:
*Since 1 or 1/T can contribute to only one investment at a time, we can model economic growth in Civ4 as a single sequence of investments I1->I2->I3->....
*Each investment will have a COMPOUNDING TIME, a number of turns before that investment actually starts to earn interest.
*Each investment will have an INTEREST RATE, the actual return once that investment starts compounding.

SIMPLIFYING ASSUMPTIONS
*Assume that all investments have the same interest rate, the MEAN INTEREST RATE.
*Assume that all investments have the same compounding time, the MEAN COMPOUNDING TIME

DISCUSSION
The mean interest rate, I think, is determined by the player. That is, the player sets the goal of maintaining a certain growth rate and tailors his decisions to achieve that growth rate. In many cases, the player aims to maximize the growth rate, so game data is useful to determine an upper bound on the mean interest rate.

The mean compounding time, I think, is either a constant or a function of the number of game turns elapsed. Much of the micro-strategy in Civ4 comes from working to minimize the compounding time on investments. Compounding time, however, is limited by the game design - the game is designed with a certain rhythm in mind.

I don't have enough experience with Civ4 to make claims about the game rhythm, but I do think that it varies by era. A player who pays attention to average time to tech in a given era or building cost as a function of average city production should be able to make a decent estimate of the mean compounding time for that era.

The important thing to note, however, is that the mean compounding time is never one. This brings us to your numbers.

Let's work normal speed, and take it as given that 1=2 in 30T. This is an estimate of the mean interest rate. Please note that if investments compound per turn, then the normal speed interest rate is not 3.53% as in quick speed. Instead it is 2^(1/30)=2.38% which I think foreshadows the importance of a mean compounding time >1. I suspect that the mean interest rate is the same or similar when comparing quick and normal speed. It is the mean compounding time that scales between the two settings and gives 1=2 in 20T (quick) <=> 1=2 in 30T (normal).

As you explained in your post, taking into account the compounding time explains why 1/t > 30. 1/t=30 iff we estimate the mean compounding time >= 30T, which seems to me to be false even in the very early stages of the game. Unless you're playing FFH2, maybe And if the mean compounding time is >= 30 then how does 1=2 in 30T make sense?

But I'm not sure I agree that using the per turn compound interest rate is any better. In what real game situation will your 1/t contribute to a sequence of investments that compound each turn on average?

It seems better to make a rough estimate of the mean compounding time (if beakers, say, look at average tech rate for the era) and derive accordingly. I'll jump right to the conclusion, but if you're interested the math is below.

To calculate the value of 1/T

Step 1: Estimate N, the mean compounding time
Step 2: Cacluate I, the mean interest rate: I=1/N * [ 2^[1/(31/N -1)] - 1 ]
Step 3: Calculate r, the compounding coefficient : r=(1+NI)^-1/N
Step 4: Calculate S, the present value of 1/T: S=1 + [r^2 * (1+NI)] / (1-r)

Here's a spreadsheet for you to play with
Value of 1/T

Note that values for N approaching 30 lead to very high interest rates. We really are assuming that investments double every 30T, so if the mean compounding time approaches 30, those are some darn good investments!

What do we get for all this trouble? You should probably round down to 40 rather than up on normal speed.

MATH

Let's say in a certain era we get a tech every N turns on average. If the size of our investment relative to the size of our economy is small, the investment will stagnate for N turns, then earn interest I thereafter.

In brief, the value of 1 compounding every N turns at interest rate I on turn tN-1 is V_tN-1=(1+NI)^(t-1).

Proof by induction or words to see the pattern:
1 on turn N-1 is still 1. It then starts earning interest at I per turn, so after 2N-1 turns we have 1+NI. This investment then earns interest at (1+NI)I per turn, so on turn 3N-1 turns we have 1+NI+(1+NI)NI or (1+NI)(1+NI)=(1+NI)^2. To drive the pattern home, on turn 4N-1 we have (1+NI)^2+(1+NI)^2*NI=(1+NI)^2*(1+I)=(1+NI)^3.

Let's interpolate (substitute T=tN-1) between compounding periods and say the value of 1 on turn T is V_T=(1+NI)^[(T+1)/N - 1]

Now we can calculate I in terms of N.
If 1=2 after 30turns then
2=V_30=(1+NI)^[(30+1)/N - 1]
So
I=1/N * [ 2^[1/(31/N -1)] - 1 ]

See how our formula reduces as expected when N=1
I=2^[1/30] -1=2.34%

And so, FINALLY, we can get to something useful - what is the value of 1/t?

If the value of x is on turn T is V_T(x)=x*(1+NI)^[(T+1)/N - 1]
then we find the present value of a single resource T turns from now by solving 1=VT*(1+NI)^[(T+1)/N-1]
or VT= (1+NI)^-1[(T+1)/N-1]=(1+NI)^[1-(T+1)/N].

The value of 1/t is then the infinite series S=1 + V1 + V2 ...

I'll do this fast, and check it when I'm done,

Form a geometric sequence with r=(1+NI)^-1/N:
S=1+(1+NI)^[1-(2/N)] + (1+NI)^[1-(3/N)] + (1+NI)^[1-4/N)]
S/(1+NI)=1/(1+NI) + r^2 + r^3 + r^3 ...
S/[r^2(1+NI)]=1/[r^2(1+NI) + 1/(1-r)

So S=1 + [r^2 * (1+NI)] / (1-r)

When N=1, I=2.34%,then r=(1+.0234)^-1=0.9771 so S= 1 + [(0.9771)^2*(1+0.0234)]/(1-0.9771) = 42.7
with lots of rounding so I think it checks out.

[suttree]

Taking in to account mean compounding time also explains your final comment, though I'd add the proviso that bulbing only increases the compounding time if we are researching a beeline where costs inflate. In some situations (early bubling in the classical era for example) the player is faced with many techs with similar costs, so bulbing doesn't depreciate existing beakers per turn.

Before I sign off, it occured to me that it might be worth modifying the formula for a more general estimate of the interest rate, say 1=4 every 60 turns on normal, to allow for larger mean compounding times.

[SevenSpirits]

Nice, that is a lot of math.

Let's work normal speed, and take it as given that 1=2 in 30T. This is an estimate of the mean interest rate. Please note that if investments compound per turn, then the normal speed interest rate is not 3.53% as in quick speed. Instead it is 2^(1/30)=2.38% which I think foreshadows the importance of a mean compounding time >1. I suspect that the mean interest rate is the same or similar when comparing quick and normal speed. It is the mean compounding time that scales between the two settings and gives 1=2 in 20T (quick) <=> 1=2 in 30T (normal).

Oh, you're right, I was quoting quick speed interest rate because I'm so used to quick speed. I'm not sure if I understand what you're saying directly after that, though. IMO the normal speed interest rate is going to be correspondingly lower (so that the doubling time is about +50% compared to quick), but the compounding time will not be increased so much due to the higher time resolution (therefore leading to a slightly more efficient economy on normal speed). (For example, on both quick and normal speed, it takes a minimum of 2 turns to start and then whip a settler.)

Regarding the mean compounding time, slavery has a big impact on it (and it's a major reason that slavery is good). My unresearched estimate would be that on quick it's 5 turns, and on normal it's 7.

[Krill]

Could someone please make those posts understandable in terms of workers and settlers?

[SevenSpirits]

Could someone please make those posts understandable in terms of workers and settlers?

Normal speed:

Suppose that getting a thing now tends to be worth about twice as much as getting a thing in thirty turns. Math says that in this case, getting a thing every turn forever is worth about as much as getting 42 equally valuable things now. You can use this, along with other estimates, as a baseline when comparing options. For example, if you want to estimate the value of a settled scientist, without calculating in turn the value of the techs you get slightly faster or the buildings you complete slightly faster, you could just say, 7.5b and 1h every turn, that's about as good as getting 315b and 42h right now. So then you can compare the benefit of the settled scientist more easily to other things you could do with the scientist instead, to a reasonable approximation.

Workers and settlers are significantly more complicated than this because they interact so strongly with the map, so they aren't very helpful in explaining it.

[Krill]

I'm sensing a philosophical argument coming on about how everything in civ ultimately interacts in with the map...unless you can force a concession via economics. This is probably not the thread for that debate though.

[Leading Question] How does this apply to military units? Or is this only useful for valuing items that are not represented on the map?