Breaking out a tangent discussion from OCC warmonger guide.
Let's try to express the value of the profit in gold / turn. First cut: lets assume a raw beaker is 5x more valuable than gold. So the rep specialist is providing 15g + 200% = 45g/turn. The gold itself converts 1:1. So those two hammers, with the multipliers, would need to be worth 25g each. So you'd be looking at 8:5:1. I might buy those ratios - gold is nearly worthless, two hammers is about the same as three commerce (that wouldn't normally be true, but production demands are much lower in an occ strategies that require a tech advantage).
To get to 1000x, we would need 80:50:1. That strikes me as ridiculous - you can buy beakers from the AI more efficiently than that.
If the religion happens to spread to one additional city (not unreasonable, given that the guide was advocating early Hindu), then you need 16:10:1. In addition, our simplified model puts all of the multipliers in play at once, even though Education/Steel are a fair ways up the tech tree from the point when the prophet appears.
It does add up to a lot of stuff. Which is the problem - the effect is largely additive, rather than multiplicative.
We've discussed a number of times that early advantages accumulate, producing a snowball effect. You can see this when you look at the various graphs on the info screen - healthy civs see their metrics growing exponentially. That's true of an OCC civ just as it is for one that it expanding -- the scales of the curves are different, but the shapes are fundamentally the same.
Once you recognize that you are dealing with an exponential curve, you can begin to guess how things are going to go by considering other fields with similar curves. Probably the most familiar of these is compound interest. Put money in the bank earning interest, and after some number of years, it doubles. Wait the same number of years, it doubles again.
BUT - assuming that the interest rate is constant - it doesn't matter how much money you start with. $1 to $2 takes the same amount of time as $1000 to $2000. There's a significant additive difference between the two accounts, but the ratios don't change at all.
Another way of saying the same thing is that the difference in time between the two accounts doesn't change. To keep things simple, let's pretend the amount of money doubles every day. A $1 account becomes a $4 account in two days. That two day difference between the accounts is constant: a $4 account becomes a $32 in three days, a $1 accounts becomes a $32 account in five days.
In an exponential race, your lead (distance) will get bigger and bigger over time, but it's only going to be 100X bigger if it started 100X bigger.
Yet another way of saying the same thing is that we can convert money later to money now. I can buy $4 two days from now if I stick $1 someplace where it doubles every day, and then wait. So we can (in this idealized problem where we know all the numbers) convert different amounts of money at different times to their equivalents now, and then compare them. I'll be coming back to this idea.
One of the really useful features of exponentials is that they are self replicating. If your total wealth is growing exponentially over time, then your income (ie, the rate at which your wealth is changing) is also growing exponentially, and the rate at which your income is growing is also growing exponentially, and turtles all the way down. For those of you who remember calculus, this is just saying that the derivative of an exponential function is also an exponential function.
So how do we evaluate attaching the great prophet to the city? One way to think about it is to compare our "baseline" exponentially growing city with another where we use world builder to give ourselves a free prophet on turn X. This isn't quite right, since the baseline already assumes an exponential return on the GPP we've been generating, but we can assume that the error is small, and come back later if we need to check that.
Analogy: the baseline case is our $1 bank account. We just let the money sit there an accumulate ($1,$2,$4,$8...). With the great prophet attached, we're starting with a $1 account, and then depositing a nickle every day. So we get something that looks like ($1, $2.05, $4.15, $8.35....).
Here comes the magic - compare that second account to a third account that starts with $1.05 in it. We double every day, so the progression looks like ($1.05, $2.10, $4.20, $8.40...). There's only a nickel difference between accounts two and three, forever. The effect of the fixed income is that of a slightly larger principal, with fixed error that over time becomes insignificant relative to the scale of the account.
That's a little startling, but the math checks out. Today's balance is yesterday's balance, plus today's deposit, plus yesterday's interest. The tricky bit is that yesterday's interest is made from two parts - the interest on the balance from the day before, and the interest on the balance from the day before. In other words, there's a recursion relation to be solved. With a bit of stubborn algebra (which I will spare you all), you end up with three terms - the baseline term (compounding the interest of $1), a second compounding term on a pretend nickel, and a small, non-compounding error correction because we didn't start with the extra nickel, we're only pretending.
Why is it a nickel? Happy accident in this case, because I've been choosing numbers that make the examples easy to follow. But what it means is the value, right now, of the nickel we get on day 2, and the value of the nickel we get on day 3, and the nickel we get on day 4.... This is the money later to money now conversion I alluded to earlier.
We can also convert that imaginary nickel itself to time - if compound interest takes $1 to $2 in a day, then it takes $1 to $1.05 in about 1.7 hours.
In principle, the same approach should work for evaluating the use of the great prophet. His actual yield per turn can be understood as a virtual value now, and compared with alternatives on that basis, or it can be converted into units of time. You might indeed be able to show that settling the prophet brings the win forward by 10 turns, whereas constructing the shrine brings the win forward by 1/10th of a turn, and voila... 100x
Converting money later to money now is linear -- if the settled prophet is 100x better than the shrine per turn, then the present value of a settled prophet is 100x better than the present value of the shrine. That's NOT true when you convert the values to time; 100x in value is about 6.7x in time.
Final thought: it's not at all clear to me what "100x better" ought to mean. What are the units? Sooner? Bigger? More likely to win?
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Originally Posted by lymond
(Post 12422328)
A Holy Shrine for one city? :lol: A settled rep prophet is a thousand times more valuable than wasting on a shrine in OCC.
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Quote:
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Originally Posted by VoiceOfUnreason
(Post 12422715)
I disagree with lymond's hyperbole - a settled rep specialist is better than a shrine, but not nearly 100x better.
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Quote:
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Originally Posted by lymond
(Post 12423089)
Ha.. I always love to exaggerate, but really I'd be interested in the math, because I do think it is a very significant number. Without getting strictly into math, for an OCC city, a settled GP gives 3B2H5G. Take into account that hammer and beaker multipliers are far more important to an OCC city than gold multipliers. You have a Library, Academy, Uni and Ox giving 200% on those beakers and Forge, Factory, Plant, Ironworks on the hammers. Count that over the full game, X turns, and that is a lot of stuff.
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To get to 1000x, we would need 80:50:1. That strikes me as ridiculous - you can buy beakers from the AI more efficiently than that.
If the religion happens to spread to one additional city (not unreasonable, given that the guide was advocating early Hindu), then you need 16:10:1. In addition, our simplified model puts all of the multipliers in play at once, even though Education/Steel are a fair ways up the tech tree from the point when the prophet appears.
It does add up to a lot of stuff. Which is the problem - the effect is largely additive, rather than multiplicative.
We've discussed a number of times that early advantages accumulate, producing a snowball effect. You can see this when you look at the various graphs on the info screen - healthy civs see their metrics growing exponentially. That's true of an OCC civ just as it is for one that it expanding -- the scales of the curves are different, but the shapes are fundamentally the same.
Once you recognize that you are dealing with an exponential curve, you can begin to guess how things are going to go by considering other fields with similar curves. Probably the most familiar of these is compound interest. Put money in the bank earning interest, and after some number of years, it doubles. Wait the same number of years, it doubles again.
BUT - assuming that the interest rate is constant - it doesn't matter how much money you start with. $1 to $2 takes the same amount of time as $1000 to $2000. There's a significant additive difference between the two accounts, but the ratios don't change at all.
Another way of saying the same thing is that the difference in time between the two accounts doesn't change. To keep things simple, let's pretend the amount of money doubles every day. A $1 account becomes a $4 account in two days. That two day difference between the accounts is constant: a $4 account becomes a $32 in three days, a $1 accounts becomes a $32 account in five days.
In an exponential race, your lead (distance) will get bigger and bigger over time, but it's only going to be 100X bigger if it started 100X bigger.
Yet another way of saying the same thing is that we can convert money later to money now. I can buy $4 two days from now if I stick $1 someplace where it doubles every day, and then wait. So we can (in this idealized problem where we know all the numbers) convert different amounts of money at different times to their equivalents now, and then compare them. I'll be coming back to this idea.
One of the really useful features of exponentials is that they are self replicating. If your total wealth is growing exponentially over time, then your income (ie, the rate at which your wealth is changing) is also growing exponentially, and the rate at which your income is growing is also growing exponentially, and turtles all the way down. For those of you who remember calculus, this is just saying that the derivative of an exponential function is also an exponential function.
So how do we evaluate attaching the great prophet to the city? One way to think about it is to compare our "baseline" exponentially growing city with another where we use world builder to give ourselves a free prophet on turn X. This isn't quite right, since the baseline already assumes an exponential return on the GPP we've been generating, but we can assume that the error is small, and come back later if we need to check that.
Analogy: the baseline case is our $1 bank account. We just let the money sit there an accumulate ($1,$2,$4,$8...). With the great prophet attached, we're starting with a $1 account, and then depositing a nickle every day. So we get something that looks like ($1, $2.05, $4.15, $8.35....).
Here comes the magic - compare that second account to a third account that starts with $1.05 in it. We double every day, so the progression looks like ($1.05, $2.10, $4.20, $8.40...). There's only a nickel difference between accounts two and three, forever. The effect of the fixed income is that of a slightly larger principal, with fixed error that over time becomes insignificant relative to the scale of the account.
That's a little startling, but the math checks out. Today's balance is yesterday's balance, plus today's deposit, plus yesterday's interest. The tricky bit is that yesterday's interest is made from two parts - the interest on the balance from the day before, and the interest on the balance from the day before. In other words, there's a recursion relation to be solved. With a bit of stubborn algebra (which I will spare you all), you end up with three terms - the baseline term (compounding the interest of $1), a second compounding term on a pretend nickel, and a small, non-compounding error correction because we didn't start with the extra nickel, we're only pretending.
Why is it a nickel? Happy accident in this case, because I've been choosing numbers that make the examples easy to follow. But what it means is the value, right now, of the nickel we get on day 2, and the value of the nickel we get on day 3, and the nickel we get on day 4.... This is the money later to money now conversion I alluded to earlier.
We can also convert that imaginary nickel itself to time - if compound interest takes $1 to $2 in a day, then it takes $1 to $1.05 in about 1.7 hours.
In principle, the same approach should work for evaluating the use of the great prophet. His actual yield per turn can be understood as a virtual value now, and compared with alternatives on that basis, or it can be converted into units of time. You might indeed be able to show that settling the prophet brings the win forward by 10 turns, whereas constructing the shrine brings the win forward by 1/10th of a turn, and voila... 100x
Converting money later to money now is linear -- if the settled prophet is 100x better than the shrine per turn, then the present value of a settled prophet is 100x better than the present value of the shrine. That's NOT true when you convert the values to time; 100x in value is about 6.7x in time.
Final thought: it's not at all clear to me what "100x better" ought to mean. What are the units? Sooner? Bigger? More likely to win?