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Hi! This is PART of the research I have done on ideal mining in Starcraft (sorry for not being able to post all my findings but yeah man... that means I'll be wiser ;D), and also this can be treated as a follow up on IMPERVIOUS' thread on Ideal Mining, so do check that out too.
Although this is only a part of my research, I bet this is all you need to know to play. 24 workers every mineral line with 8/9 patches is a pretty well-known number, no?
Now, I'll just show you some of the data that i have collected. Note that there are erroneous data under 'average minerals gathered/worker/minute', which could be due to human reaction time as my research team was using a stopwatch. Although it does not affect the final argument, DO NOT take these readings as standard.
Taking the ceteris paribus assumption...
Number of Workers | Average Mineral Gathered/Worker/Minute | Average Mineral Spent/Worker
4............................................78.5.................................................175
8 ...........................................60.0.................................................112
12..........................................55.0.................................................100
16..........................................49.5..................................................87.5
20..........................................44.8..................................................80.0
24..........................................44.7..................................................79.2
32 .........................................38.0..................................................75.0
36..........................................33.6 .................................................72.2
40..........................................38.0..................................................70.0
44..........................................33.6..................................................70.5
48..........................................30.1..................................................68.8
Note: Average Mineral Spent / Worker = (400 + No. of Supply depots*100 + No. of Workers*50) / No. of Workers, where 400 = cost of a command centre / nexus and not including hatchery.
We will then derive the average cost of a worker, which equals (average minerals spent/worker) - (Average Minerals Gathered/Worker/Minute)
No. of Workers | Average Cost of a Worker
4....................................96.5
8....................................52..0
12...................................45.0
16...................................38.0
20...................................35.2
24...................................34.5
28...................................36.4
32...................................37.0
36...................................38.6
40...................................32.0
44...................................36.9
48...................................38.7
Plotting the points on the graph, you will roughly get a U shape curve. Because some data are erroneous as mentioned earlier, you will not get a smooth curve, which is not suppose to be so. BUT the trend of the readings can be seen going in a U shape curve, with those points that are plotted way off being the erroneous points. But anyway, that curve is the short-run average cost curve (SRAC). It shows the cost efficiency of each worker against the number of workers in a SINGLE base (considering early game where you only have one command centre). The minimum point of the U shape curve is where the cost of the worker is at the lowest, and it is at its lowest when the number of workers in a SINGLE base is 24.
[Picture in courtesy of The Open University, GREAT HELP there...]
Another reason why it's 24 without using the Theory of Economics is.... the mineral line will simply reach saturation.
Ok... so when it reaches mid-game, a player will most likely take multiple expansions (duh). We will now examine the long run average cost curve (LRAC), taking the ceteris paribus assumption again, and the assumption that all mining bases are at maximum efficiency (24 workers each).
No. of Workers | Average Cost of a Worker
48.................................34.5...................
48.................................32.3...................
72.................................33.1...................
96.................................32.0...................
160...............................31.7...................
(You cannot go beyond 160 workers, man.....)
[Note: Average COst of a Worker = (No. of Command Centre * 400 + No. of Supply Depots * 100 + No. of Workers * 50)/No. of Workers]
Picture in courtesy of The Open University... BIG HELP again... :D
[Note that the LRAC is made up of infinitely many SRAC at its most efficient/minimum point)
Ignoring the erroneous data, when these points are plotted you will get a 1/x curve. This curve will have a very large minimum efficient scale (minimum point, which is the lowest point of the curve) of infinity (because, as no. of workers reaches infinity, the cost of each worker approaches 0). This shows that the more bases you get, the more cost efficient your workers will be as there are no dis-economies of scale, while the reason on why the cost of each worker will come infinitely close to 0 as the no of workers increases lies in the idea of the economies of scale.
As each base should contain 1 command centre, it can only be employed efficiently if it is producing workers at the lowest cost, which is when the number of workers = 24. That means, to be at its most efficient, the command centre has to produce at least a certain amount of output. A command centre producing less than 24 workers will not be as efficient as a command centre producing 24 workers. This is the idea of factor indivisibility in the idea of 'economies of scale'.
*EDIT: Sorry to the zergs but these readings do not include hatcheries, drones and overlords. But using the same methods, in theory the zergs will need less workers to be efficient as the cost of the hatchery is 100 minerals less than the cc/nexus. I do not know the actual number as it requires another set of calculations (but of the same methods).
Conclusion:
24 workers is the best number because it saturates the mineral line, which will result in a higher cost efficiency, leaving you aside with more minerals to produce units.
And also, the more bases the better.
(Thanks Whiplash for reminding)
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Having more than 80, maybe 100 workers is really prohibitive. Very interesting stuff.
A suggestion, though. For research in this area I think using probes is better as they are a bit less buggy in their mining patterns and have the fastest mining rate. (IE, a really fast terran can forcibly change the mining patterns of their like first 10 or so scvs to achieve quicker mining. It only gives you like an extra 30-50 minerals though). You would probably get less erroneous data with probes. Drones have the intermediate mining rate if you are interested in that, but zerg economy has an entire extra variable beyond the 3 that terran/protoss have.
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You would probably get less erroneous data with probes.
I'll put that in mind. Thanks for the input.
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Can you have a conclusion or summary of what your trying to say here? I kinda read over most of it and I still don't understand. Is there an optimal number of workers we're trying to hit here? (keep in mind im posting this at 6:30 am and I haven't gone to sleep yet).
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Ho ho!? Sorry 'bout dat. The conclusion is that 24 workers on each base with 8/9 patches of minerals is the best number, both because it saturates the mineral line and also it is the most cost efficient, leaving aside more minerals for producing units. And also the more expansions you have, the better 
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As a physicist I'd like to see some error analysis/uncertainties from multiple runs of the data. The LRAC numbers in particular look too close to be called definitive in comparison to the data of the SRAC. Also I'm sort of uncomfortable with how you define Average Mineral Spent/Worker, so if you could explain your methodology in a later post that would be awesome.
Also, if you want to improve your runs (or perhaps even automate them) you could potentially try via UMS maps.
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Taking the ceteris paribus assumption...
Number of Workers | Average Mineral Gathered/Worker/Minute
4............................................78.5.
8 ...........................................60.0..
How can 4 workers be that much more efficient than 8 when the only difference between the first 4 and the next 4 is the tiny increase in distance they have to travel to get to their patch?
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As a physicist I'd like to see some error analysis/uncertainties from multiple runs of the data. The LRAC numbers in particular look too close to be called definitive in comparison to the data of the SRAC. Also I'm sort of uncomfortable with how you define Average Mineral Spent/Worker, so if you could explain your methodology in a later post that would be awesome.
Also, if you want to improve your runs (or perhaps even automate them) you could potentially try via UMS maps.
Oops... I didn't do multiple runs of the data and I didn't include uncertainties because this is really for an ungraded project in college.
Thanks for the suggestions, but I think I will prefer spending time playing the game rather than doing this whole experiments again DX (dead)
Also, regarding the LRAC, I had the intention to run the experiment again because...... I just do not have the apm to put all workers from all bases to mine at the same time.... but deadline was pressing and what I posted above is basically what I submitted (sigh...). However I think the argument is there (without the erroneous data) through logical thinking. And I believe you can still see the trend of the curve if the points are plotted, though it could be just luck that they are in the shape they should be.
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How can 4 workers be that much more efficient than 8 when the only difference between them is the tiny increase in distance they have to travel to get to their patch?
That's why I said 'DO NOT take these readings as standards'....
I am really sorry but I just need to show some readings to support my arguments. The minimum I'll go is to let the data shows some kind of trend that supports my graphs.
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Very interesting and cool stuff. It'd be interesting to see the dynamics of time and how saturation is affected by depletion. I understand these graphs are based off of X amount of workers in the initial phase, but a big question would also be how the graphs change as the scvs start to scale to the number. I suppose it will be left to a project in the future sometime though :-)
Great stuff regardless.
Just a note by the way. I use to do these tests for maps to test mineral saturations and mining analysis as well. From your text, I see you used a stopwatch. What I use to do is go to single player, turn on all my cheats and kill the computer immediately. Then I'd make sure all the minerals were the same number, and I'd start mining the respective bases. For example, destination, i'd kill the computer, make all the mineral numbers the same in the computer base since they mined, then i'd get 24 scvs to go mine. I'd leave it on(using windows mode) and surf the web or do whatever, maybe even go to sleep! Then I'd look at the replay and note how many minerals were mined just before the first patch was empty, and count total amount of minerals mined, and i'd track how long the replay was on minus the time when the initial scvs started working. My assumptions while doing this would be things unique to destination pathing, 8 mineral patches, SCVs mining(others path the same).
d3 is correct however and you need duplication and statistical accuracy when you're finalizing your projects, just for future reference. 30 times generally implies safe statistical accuracy.
I love your analysis though:-) great work
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So what, there should be a noticeable improvement in prodction efficiency if you have 72 SCV's when you're at 3 bases as opposed to 60-65, which is the current standard number?
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Belgium9950 Posts
haha, this has been researched many times, and 23/24 workers is always the result they get as optimal for 8 patch expansions
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Looking at the graph of the SRAC, there is very little change in the effectiveness of the mining when you are near that point (ie, the tangent there is nearly zero).
So, in game terms, even if you aren't at the exact 24 workers/expansion, but you are close (under if you have a lot of expansions, over if you are planning on making a new expansion), you will be in a nearly ideal situation. I don't think it's even possible to model the ideal game situation.....
Also, because these results used SCVs, this is the data for Terran. Using Probes, you would find a different number (although probably very close to 24, maybe 23 or 25). Using Zerg, the number would be lower. Trials with all 3 races would be necessary to determine what number is most effective for each race.
Awesome work though. Too bad you aren't going to share everything.....
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if you want to further your research, in addition to an economic curve, have attack (damage/second) and defense curve with basic units.
the point is to find a balance on when to produce units and enough damage pts to defend against rushes.
Best done with Excel.
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A fun little read for me. (man oh man, am I not ready to go back to college).
May I ask for what class this is for? As an Econ major, I'm just curious  .
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Belgium6775 Posts
The question now is: can BW API be used to lower the optimal number ?
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On January 08 2010 05:09 Xeofreestyler wrote:
The question now is: can BW API be used to lower the optimal number ?
I expect so. By taking into account worker positioning and how long a mineral patch has until it is free for mining by another worker, there should be minimal wandering time. It's something I have been planning to code once I have some free time.
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This was very interesting. I always keep making workers, but it seems that I'm overproducing.
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On January 08 2010 22:22 Zona wrote:Show nested quote +On January 08 2010 05:09 Xeofreestyler wrote:
The question now is: can BW API be used to lower the optimal number ? I expect so. By taking into account worker positioning and how long a mineral patch has until it is free for mining by another worker, there should be minimal wandering time. It's something I have been planning to code once I have some free time.
That would be horrible to try to code - the pathfinding issues would be difficult to compensate for.....
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How do I know if I have 24 workers on my mineral line?
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EPT![[image loading]](http://openlearn.open.ac.uk/file.php/3459/DD202_3_005i.jpg)
![[image loading]](http://openlearn.open.ac.uk/file.php/3459/DD202_3_010i.jpg)
