Lab Efficiency Research

[alexzzzz]

I have a suggestion about labs and their efficiency.

Problem: Unbalanced mid-game research difficulty

In all my games the only time I actually carefully think about what to research next is the early game. This is because there's no enough labs yet or no enough science packs or resources to make them. In the middle game and later there's no need to think much about the research. You just click on whatever technology is available for research and it's done in a couple of minutes. Click - done, click - done, click - done.

Solution: Make the dependency between the number of labs and the research speed nonlinear

Something like:
1 lab gives x1 speed,
2 labs - x2 speed,
10 labs - x5 speed,
50 labs - x20,
100 labs - x30,
etc

This won't affect the early game, but will bring mid-game research speed to more sensible level. The late game technologies however require higher speed, but here come Lab Efficiency Techs - they can gradually bring the research speed dependency back to linear level.

[BlakeMW]

In all my games the only time I actually carefully think about what to research next is the early game. This is because there's no enough labs yet or no enough science packs or resources to make them. In the middle game and later there's no need to think much about the research. You just click on whatever technology is available for research and it's done in a couple of minutes. Click - done, click - done, click - done.

To be fair the Marathon/Deathworld presets do not suffer from this problem. The techs are expensive enough relative to the available ore (basically, 8-10x as expensive as normal) that you really do need to think about what you research. If there is a solution which doesn't involve research at all it'll usually be the better choice (i.e. building more laser turrets, rather than upgrading the laser turret damage/speed).

Once you've unlocked prod3 modules and beacons you can start being a bit indiscriminate but that's basically late game.

[Nasabot]

It is fine as it is. You can view it this way: If you research lab speed, you save Mk 3 Modules.

[Frightning]

It is fine as it is. You can view it this way: If you research lab speed, you save Mk 3 Modules.

It's cheaper than both to just build more labs if all you want is speed. If you intend to use productivity modules in the Labs to stretch resources, then things get more interesting. The speed bonus from the research is multiplicative with other speed adjustments (e.g. from modules). So that actually makes the math simpler to do, still rather involved, mainly because of comparison with speed beaconed setups.

Speed bonuses:
Lab speed 1: +20%
Lab speed 2: +30%
Lab speed 3: +40%
Lab speed 4: +50%

Cumulative speed bonuses:
Lab speed 1: +20%=*1.2
Lab speed 2: +50%=*1.5
Lab speed 3: +90%=*1.9
Lab speed 4: +140%=*2.4

Costs:
Lab speed 1: 100 Science pack 1s, 100 Science pack 2s (650 Iron ore, 250 Copper ore=900 ore)
Lab speed 2: 150 Science pack 1s, 150 Science pack 2s (975 Iron ore, 375 Copper ore=1350 ore)
Lab speed 3: 250 Science pack 1s, 250 Science pack 2s, 250 Science pack 3s (10450 Iron ore, 3000 Copper ore, 500 Plastic bars=13450 ore+500 Coal+10000 Petroleum gas)
Lab speed 4: 500 Science pack 1s, 500 Science pack 2s, 500 Science pack 3s (20900 Iron ore, 6000 Copper ore, 1000 Plastic bars=26900 ore+1000 Coal+20000 Petroleum gas)

Cumulative costs:
Lab speed 1: 100 Science pack 1s, 100 Science pack 2s (650 Iron ore, 250 Copper ore=900 ore)
Lab speed 2: 250 Science pack 1s, 250 Science pack 2s (1625 Iron ore, 625 Copper ore=2250 ore)
Lab speed 3: 500 Science pack 1s, 500 Science pack 2s, 250 Science pack 3s (12075 Iron ore, 4350 Copper ore, 500 Plastic bars=16425 ore+500 Coal+10000 Petroleum gas)
Lab speed 4: 1000 Science pack 1s, 1000 Science pack 2s, 750 Science pack 3s (22975 Iron ore, 10350 Copper ore, 1500 Plastic bars=33325 ore+1500 Coal+30000 Petroleum gas)

That covers research, what about prod moduled Labs?

Lab with 2x Productivity module 3: 2202 Iron ore, 4015 Copper ore, 260 Coal, 5650 Petroleum gas

How much are Beacons with 2x Speed module 3s?

Beacon with 2x Speed module 3: 2376 Iron ore, 4135 Copper Ore, 280 Coal, 6050 Petroleum gas

All math assumes standard costs in 0.15 (according to Factorio wiki numbers, anyways; and barring any errors on my part). I think I will work up the theoretical answers to the relevant questions in another post here, but I wanted to put these raw numbers up first for others to see.

[Frightning]

It's apparent from the numbers I posted a bit ago that at some point, the multiplicative speed increase from Lab speed research becomes cost efficient relative to adding the number of Labs required to achieve the same total increase in research speed, so the question to ask is, how many Labs does it take before it's worth it? The answer depends on whether or not you use prod modules in the Labs (I will only math out tier 3 case), and whether or not speed beacons are used on the Labs as well.

The simplest case is that of unmoduled Labs:
For simplicity, and versatility we will derive a standard model for computing the theoretical 'break-even' point where researching the Lab speed research is cheaper than making the required number of Labs to achieve the same increase in total research rate. We do this for Lab speed 1 first.

Lab speed 1 provides a 20% increase in research rate, which means that your new research rate is equal to your current number of labs times 1.2, the net gain in speed is thus 0.2*Labs, or 1/5 the number of Labs worth in additional research rate. Thus for every 5 Labs you have, the Lab speed 1 upgrade provides 1 'free' Lab worth of extra research rate (I should point out that it's 1 unupgraded Lab worth of additional research rate). Thus we seek the number of Labs, L, such that the cost of Lab speed 1 is the same as the cost of making L/5 new Labs.

Let R denote the cost of the research, Lab speed 1. Let C, denote the cost of making a Lab, and let L be the number of Labs you already have. Then we have the equation:
R=C*(L/5)
Solving for L, we obtain
L=5R/C

Using the number from my earlier post, we can look at Iron and Copper break even points, as well as the ore break-even (where iron and copper are valued equally). These numbers are:

Iron ore:
L=5(650)/(36)=1625/18=90.2(7)...
Copper ore:
L=5(250)/(15)=250/3=83.(3)...
Ores:
L=5(900)/(51)=1500/17=88.(2352941176470588)...

So it takes nearly 90 Labs before it's cheaper to research Lab speed 1 then just adding more Labs...yikes. What about Lab speed 2? Rather than try and compute marginal benefit under the assumption that Lab speed 1 is done, it is easier to use cumulative costs and benefits and compare directly with no research at all (this also gives us an idea if the level 2 research is more or less efficient than level 1). Here the cumulative speed increase is +50%, so we get a free Lab worth in extra research rate for every 2 unupgraded Labs we had. Thus our comparison factor is L/2 instead of L/5. Of course we must use the much larger cumulative cost to make an accurate comparison.

R=C*(L/2)
L=2R/C

Using my earlier numbers, we obtain:

Iron ore:
L=2(1625)/(36)=1625/18=90.2(7)...

Copper ore:
L=2(625)/(15)=250/3=83.(3)...

Ores:
L=2(2250)/(51)=1500/17=88.(2352941176470588)...

Perhaps a bit surprisingly, the numbers are exactly the same, this because the cost and benefits of the first two levels are proportional at a ratio of 2-to-3. 90ish Labs is much larger than most people will ever build, especially outside of an endgame megabase research area. However, given how expensive endgame techs are, it's very desireable to use Productivity modules to stretch the resources used and thus reduce the amount of raw materials needed. This includes using Productivity modules in the Labs themselves, which makes adding more Labs a much more expensive proposition, as you also need to make 2x Productivity modules for the Labs. This is especially impactful with how expensive the higher tier modules are, with tier 3 being by far the most expensive, but also conferring the biggest productivity bonus, hence it is desirable to use the highest tier modules you can get to maximize resource stretch. This makes the cost of Lab speed research, as a means to increase speed look far more attractive. Without considering beacons, it's already worthwhile from a copper cost perspective to research Lab speed 1 instead of making a 2nd Lab and 2 more Productivity module 3s for it. A speed beacon is slightly more expensive than a Lab, but it can affect more than one Lab if you have them, which can mean it provides much better rate return on the investment than another Lab and two more Productivity module 3s. It is cheaper to make a 2nd Lab and 2 more prod modules than it is to make your first Beacon and the rate return is a doubling (+1 prod module 3'd Lab), instead of the increase of 5/7 (of a prod module 3'd Lab) provided by the speed beacon. However, after the 2nd Lab, the Beacon is a much better investment, because the 3rd Lab would only give a 50% increase in research rate, whereas the Beacon would give just over 57% (precisely 57.(142857)...%), which more than justifies its slightly higher cost. Once you have both Labs and Beacons however, things get more complicated, because a new Lab may be able to be placed in a manner where it is immediately affected by one or more Beacons (theoretically as many as 12, though in practice, 8 tends to be about the best that can be achieved in a scalable layout, and is far more economical because the Beacons can be positioned to affect as many as 8 Labs each).

Assuming an idealized scenario, where we have alternating rows and Beacons and Labs, so that every Lab is affected by 8 Beacons, and most of the Beacons are affecting 8 Labs (the side-most rows and ends won't reach those numbers, but the larger the setup, the lower the overhead, though it is generally quite non-trivial in practice). This means that we have roughly an equal number of Labs and Beacons and that each Lab is getting +40/7 unbeaconed prod 3'd Labs worth in extra throughput (8*(5/7) mentioned above). The cost of adding one beaconed Lab to this setup, tends towards the total cost of 1 Lab+1 Beacon+modules for both (2 Productivity module 3s and 2 Speed module 3s) from above as the size of the setup increases. Because the speed increase from the research is multiplicative relative to the speed adjustments from the modules and beacons, we can use the same methods as before to compute the number of Labs for the research to be worthwhile, realistically, these numbers will be a bit inflated, and the smaller they are, the more inflated they will be compared to reality because we are ignoring the overhead costs of the Beaconed setup, which are highly significant for low Lab counts, and push things further in favor of doing the research sooner rather than later. using my earlier numbers, we have:

Theoretical minimum total cost of adding 1 beaconed Lab:
C=4578 Iron ore, 8150 Copper ore, 540 Coal, 11700 Petroleum gas (12728 Ore)

Lab speed formulas:
Lab speed 1: L=5R/C
Lab speed 2: L=2R/C
Lab speed 3: L=10R/9C
Lab speed 4: L=5R/7C

Using Ore costs only (Iron+Copper), we obtain:
Lab speed 1: L=5(900)/(12728)=1125/3182 (which is about 1/3 so it's very worth even if you're just starting out with modules)
Lab speed 2: L=2(2250)/(12728)=1125/3182 (which is just as worth; again, the proportionality kicks in here)
Lab speed 3: L=10(16425)/9(12728)=9125/6364 (which is about 3/2, so it's very worth as soon as you have several Labs and/or Beacons all moduled up)
Lab speed 4: L=5(33325)/7(12728)=166625/89096=3875/2072 (which is a bit below 2, so yet again, it's worth as soon as you have multiple Labs and Beacons moduled up)

I suspect Lab speed 5 and especially 6 actually require a half decent number of Labs (especially the latter), but I don't want to fuss with calculating it out, moreover, because of significant overhead, those numbers are most likely small enough that it's still very worth even if you are a bit under the theoretical Lab count. The conclusion here seems pretty clear: Lab speed isn't useful until you look to start using modules in Labs, and if you're going for tier 3 modules, then they are quite worthwhile to research, for lower tier modules, the picture is probably a bit more interesting, and with the models provided above it shouldn't be too hard to calculate, the only thing that changes is the C value, because the modules are much less expensive, so the C value is much smaller, hence number of labs goes up, by quite a lot (I suspect Lab speed 3 and especially 4 aren't worthwhile with tier 1 modules unless the setup is very large, and for tier 2, the setup probably needs to be mid-sized or larger, but this is an educated guess on my part; feel free to find the appropriate C values and run the numbers if you're interested).