Gold, Beakers, and Deficit Research

In this article, I hope to clarify the research value of gold. The discussion won’t involve any hidden game mechanics, and it may already be well understood by experienced players. Still, we will give an unambiguous answer to certain key questions. Can you cut research times by clever manipulation of the slider? Can running merchant specialists be more beneficial to your tech rate than running scientists? How much do buildings like courthouses or markets affect your research? What is a fair trading price for techs?
But before we begin, what is gold even good for in the first place? We can certainly use gold in trades, but why does gold have value for our trading partners? One important answer is that through Universal Suffrage or unit upgrades, gold can be an empire-wide substitute for production. These abilities can be critical, but they are not used on most turns, and we will not consider them much here. But every turn from the time the second city is founded, gold is necessary to pay the costs of our empire.

The Ideal Slider Rate

If you don’t already have enough gold in your treasury, you must use a fraction of your incoming commerce as gold to pay maintenance each turn. We will consider an ‘ideal’ percentage of commerce at which there is no net change in the treasury. Since the slider is limited to increments of 10% this is rarely exactly possible, but in practice it is no different from adjusting the slider above and below the ideal point.

Now, if you do already have enough gold, you can just leave the slider at 100% and let the maintenance eat away at your treasury. If we consider our ‘standard’ science rate to be that at the ideal slider point, after a number of turns we find we have lost a certain amount of gold from the treasury and we have produced more beakers than we would have at the standard rate. It is as if the gold has been converted into beakers – this is what we mean by deficit research.

So this situation defines a beaker value for any given amount of gold – the amount of extra beakers we produce if we set research to 100% until the given amount is fully spent on maintenance. But gold also has a beaker value in terms of the beakers we would lose if we set research to 0% until we have produced the given amount of gold. If you do the math (it’s easy to check any statement I make here), you will find that the two beaker-per-gold ratios are exactly the same.

Instead of researching at the ideal slider rate, we might turn off research for several turns to build a treasury, and then use that treasury for deficit research. But the fact that there is one beaker-per-gold ratio means that we produce exactly the same amount of beakers using either method (ignoring rounding errors). This is true for any two paths of research that begin and end at the same treasury value. And if they end at different treasury values we can easily find the difference in beakers produced by converting the treasury difference using the ratio. This is why it makes sense to talk about an ideal slider rate. Constantly adjusting the slider above and below the ideal point so that there is not much change in the treasury will produce almost exactly the same tech rate as the ideal.

These concepts allow us to blur the distinction between gold and beakers. In an abstract sense, we always research at the ideal slider rate, regardless of the position of the slider. If the slider is set below the ideal slider rate, some of our beakers are ‘stored’ as gold. If the slider is above the ideal rate we unlock some extra beakers through deficit research. If we are in the happy situation of running a surplus at 100% science, it is as if the ideal slider rate is higher than 100%. Since we can’t ever set the slider higher than this ideal point, some of our ideal beakers must always be stored as surplus gold. We can never unlock extra beakers through deficit research in this situation, but the equivalence implied by the beaker-per-gold ratio still has abstract meaning.

Finding the Ratio

This magic ratio is much easier to find than you might think. It is simply the total amount of beakers that can be produced due to commerce, divided by the total amount of gold due to commerce. To find this, turn the slider to 100%, and record the net change in gold (typically negative), which we’ll call M, and also find the total amount of beakers produced, B, using the Financial Advisor. Now switch the slider to 0%, and find the total amount of gold produced, G, and also record any beakers, b, which are produced at 0% research. The ratio we are looking for is just (B – b)/(G – M)

So in this case our beaker-gold ratio is (736-38)/(359+93) = 1.54

My choice of parameters is conventional. Instead of using the numbers G and M, we may use the gold values listed in the Financial Advisor. And as will soon become clear, we may compare the four parameters at any two distinct slider values.

Gold and Beaker Multipliers

If you have no beaker or gold multiplying buildings (libraries, markets, etc), this ratio will be exactly 1, since commerce is converted into beakers and gold at the same rate. This suggests the beaker value of gold depends in some sense on the presence of multiplier buildings in your empire. In fact the ratio is just equal to the average beaker multiplier divided by the average gold multiplier, where the average is weighted by the fraction of commerce a city produces. So the beaker value of gold is fairly stable over time, it only changes if you build new multiplier buildings or significantly change the commerce distribution of your cities. It does not depend on many economically significant actions like hiring or firing specialists, building courthouses, producing wealth or research from production, and so forth.

But what happens when the beaker-per-gold ratio does change? If you already have gold in your treasury, its value for deficit research also changes. You can take advantage of this by manipulating the amount of gold in your treasury. If your ratio increases, you will get more extra beakers from deficit research than before, but you will also lose more potential beakers in accumulating a given amount of gold. So if you accumulate the gold before the ratio change, you will gain some extra beakers.

This is a little abstract, so let’s consider an example. Suppose you’re building Oxford in a commerce city (beakers from specialists complicate this example, but not the idea in general). You can set the slider to 0% a few turns before Oxford is complete, and then spend the gold on deficit research afterwards. Since you are producing no beakers to multiply, the situation is no different than if you had built Oxford a few turns earlier, when you had started 0% research. Spending the gold produced during these turns will give you the same amount of beakers as researching at the ideal rate, but here it is as if you already get the better science multiplier a few turns before Oxford is built!

It is important to realize that it doesn’t matter how or when you accumulate the gold before the ratio increase. The advantage comes from the change in the value of deficit research. Setting the slider to 0% just before a new beaker multiplier comes into effect simply allows you to plan fewer turns in advance (and makes the example above easier to understand).

A similar situation holds for decreases in the ratio. When you build gold-multiplying buildings, you lose less potential beakers in producing a given amount of gold, but you also get less extra beakers from deficit research. This may be a little confusing – you do not actually get less total beakers when you spend gold on deficit research, but since your ideal slider rate increases, less of those beakers are ‘extra.’ So it may be wise to spend your treasury on deficit research before the beaker value of gold decreases. This lets you take advantage of extra beakers you couldn’t get during those turns by researching at the lower ideal rate.

Gold vs. Beakers

So now we return to the questions we set out to answer. Running a merchant specialist reduces your net expenses, thus increasing your ideal slider rate – indirectly increasing your beakers per turn. Can a merchant produce more beakers than a scientist? The answer depends on what multiplier buildings you have in the city in question, as well as the average multipliers in you empire as a whole. If your cities have many beaker multipliers (libraries, monastaries…) but you’ve been neglecting your gold multipliers (grocers, banks…), then gold has a higher value in terms of beakers.

As an extreme example, say you have a library, university, observatory, and laboratory in each one of your cities – but you have almost no gold multipliers…just a bank in one city. In that city with a bank, a merchant will produce 4 gold (rounded down) while a scientist will produce 6 beakers. But since the beaker-per-gold ratio in your empire will be very close to 2, the merchant will help your research rate more than a scientist. This will be true in any city that has better gold multipliers than your empire-wide average (unless the beaker multiplier there is also much higher than average).

How much do buildings that reduce your maintenance affect your research rate? This question is easy to answer. If a developing city costs 8 gold in maintenance, a courthouse is effectively giving you 4 gold. With the magic ratio we can find the equivalent amount of beakers. And unless the city is bringing in significant commerce, or gold has low value in your empire, the courthouse will probably help your research more than a library.

What about buildings like markets and grocers? The extra gold from the building depends on the ideal slider rate, and this building will also change the current beaker-per-gold ratio. Unfortunately there is no simple formula that exactly determines the extra beakers produced.

But we can still make some assertions. For example, say you have libraries throughout your empire but not many markets. If your maintenance is bad enough that your ideal slider rate is near 50%, then building a grocer in a city has more immediate benefit to research than a university. If your ideal slider rate is higher, less commerce is devoted to gold and so gold-multipliers have less research value. This may just be considered common sense, but what is not obvious is the role of the beaker-to-gold ratio. If this ratio is high a grocer could still be better for research even for ideal rates approaching 70%. But for even higher ideal slider rates, it is doubtful that the value of gold could ever be high enough to compensate.

Tech Buying

So far we have been speaking about the research value of small amounts of gold per turn. What about decisions like whether to use a Great Merchant for a lightbulb or a trade mission? Large amounts of gold and beakers involve more strategic elements. Gold is not ‘converted’ to beakers instantly through deficit research. Your net expenses per turn (the number M in the above formula) determines the time for the ‘conversion’ to take place. Unless you plan to use the production aspects of gold it is usually better to get the same amount of beakers now in the form of a tech, rather than over several turns by using gold for deficit research. A Great Merchant still may present a tough decision since the beaker value of a trade mission is typically much higher than the beakers gotten through lightbulbing – and the beakers eventually produced through settling will typically be much higher than both.

So the ‘value’ of large sums of gold has no simple mathematical definition. Even so, the beaker-gold ratio gives a decent rule-of-thumb for pricing techs in trades. If you are buying a tech from an AI for more gold than the value of the beaker cost, you are probably getting a bad deal. You could research the tech faster than you could produce the gold you are trading. On the other hand – if it prevents a fair tech trade – it may be disadvantageous to sell a tech for its ‘fair’ gold value. The seller potentially gives up a tech now for the same amount of beakers over a delayed time.

It is still possible to have a mutually beneficial tech buying trade, though. The player buying the tech may have a lower beaker value of gold than the seller, and the production aspects of gold may have higher strategic value for the seller. A less common situation would be where the buyer is running a surplus at 100%. In this case the buyer can’t actually research at the ideal slider rate, so it would take longer to research the tech than it would to produce the equivalent gold.

More importantly, in a tech buying deal the seller does not lose the traded tech, while the buyer loses the value of the gold. This means it often makes sense to price a tech for less than the beaker value in gold. Even so, the AI tends to give you close to the beaker value in gold when you sell to them. So tech selling can sometimes get you more delayed beakers than a tech trade, while the AI also loses a similar amount.

…………………..

I hope this post wasn’t entirely obvious to everyone. I know that I certainly didn’t have a clear appreciation for how much things like merchants and courthouses improve your research rate until I did the math. I’ve included the derivation of the beaker-gold ratio in the spoiler below, but it’s not necessary to understand the concepts in this article. And yes I spend too much time on this game – but if you got this far in the post, so do you :goodjob:

Feel free to argue or point out errors, thanks for reading!

Spoiler:

The numbers G, M, B, and b are the same as in the text.

We define the ‘ideal’ research rate, x, as that at which there is no change in the treasury.

So (1 – x)(G – M) + M = 0, which means x = G/(G – M)

At the ideal rate, we make x(B – b) + b beakers. This is x(B – b) more beakers than at 0%

And at 100%, we make B – (x(B – b) + b) = (1 – x)(B – b) more beakers than the ideal rate.

What happens if we use a given amount of gold G0 for deficit research?

We can sustain 100% science for G0/(-M) turns, so we make (1 – x)(B – b)*G0/(-M) extra beakers.
Using our expression for x, this simplifies to (B – b)/(G – M) * G0.

So (B – b)/(G – M) is our beaker to gold ratio.

What happens if we turn off research until we build up a given amount G0?

It takes us G0/G turns to build the given wealth, so we lose x(B – b)*G0/G,
simplifying to (B – b)/(G – M) * G0.

And again we see the ratio (B – b)/(G – M), and we note we can’t get extra beakers by changing the slider.

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