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On February 11 2011 08:54 SirazTV wrote:
The optimal number of workers per base is 2 per mineral patch. When above 2 workers per patch you get diminishing returns per worker. I mean why would a player not want to have a worker pay for itself asap. If you have more then 2 workers per patch it takes longer for the workers to pay for themselves. I don't understand why people think it is anything else. The only reason to go above 2 per patch is if you can not safely expand(granted this happens a lot).
Why do people buy stocks now that won't give them any returns for a year or more? It's called an investment. Less minerals now = more later.
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Out of all the responses, that's the one you quote? 
Reread please.
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On February 11 2011 08:59 Nagano wrote:There is a HUGE flaw in the numbers that I think most people have just simply overlooked. In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero." If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else? ![[image loading]](http://i52.tinypic.com/xpomtj.png) This would mean that the graph above would be completely false up to 16 workers. Am I missing something?
It's a graph of ESTIMATES, not true values. It's meant to show the shape of the returns function, not the specific values themselves.
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On February 11 2011 08:51 Soulish wrote:Show nested quote +On February 11 2011 08:09 natewOw wrote:
You now have 40 workers mining at the natural, but any amount of workers above 27 gets you nothing, and may even decrease your net income due to cluttering.
what do you mean by cluttering? There is no such thing as cluttering when workers mine cause they pass through eveything, including other workers. just curious
Even if there's no cluttering (which I agree, I don't think there is, but I've heard that other people have suggested that there is), it doesn't detract at all from what I said.
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On February 11 2011 09:02 natewOw wrote:Show nested quote +On February 11 2011 08:59 Nagano wrote:There is a HUGE flaw in the numbers that I think most people have just simply overlooked. In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero." If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else? This would mean that the graph above would be completely false up to 16 workers. Am I missing something? It's a graph of ESTIMATES, not true values. It's meant to show the shape of the returns function, not the specific values themselves.
You are COMPLETELY missing the point here. There should be ZERO DR on MR at any point before 16 workers. Half the data is wrong, and therefore the entire graph should look completely different.
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Well the thing is, tons of players in this thread are right.
The OP is 100% correct that in a game of infinite length, 27 workers > 24 workers.
However, as has been pointed out, it takes over 20 minutes for that last worker to pay for himself, while the opportunity cost of building those additional workers early on stays the same.
And appart from the opportunity cost, the other issue you have, is that minerals will actually mine out rather fast from a single base with optimal saturation.
So while its true that 27 SCVs would be the optimal number to stop at in a game of infinite time where mineral patches never mine out, and you stay on a single base for the whole game, unfortunately, it is quite irrelevant for the actual gameplay itself, since in an actual game, you either want to expand relatively soon, at which point building extra workers past 27 is still a good idea, or you want to 1 base all-in relatively soon, at which point the opportunity cost for those extra 3 workers is not worth it.
However though, I believe that the OP's formula probably can be used quite well, as long as it is solved for a more reasonable number than T=infinity.
I tried it out, and it seems to take 17 minutes to fully mine out a base, anjd 15 minutes to fully mine out a base with mules.
So perhaps the OP, or someone else that is good at maths could retake the base formula, and solve it for T=15 minutes, instead of for T= infinity, and we could check the results?
Perhaps we will get some new information, the number might be smaller 
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On February 11 2011 09:03 Nagano wrote:Show nested quote +On February 11 2011 09:02 natewOw wrote:On February 11 2011 08:59 Nagano wrote:There is a HUGE flaw in the numbers that I think most people have just simply overlooked. In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero." If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else? This would mean that the graph above would be completely false up to 16 workers. Am I missing something? It's a graph of ESTIMATES, not true values. It's meant to show the shape of the returns function, not the specific values themselves. You are COMPLETELY missing the point here. There should be ZERO DR on MR at any point before 16 workers. Half the data is wrong, and therefore the entire graph should look completely different.
The data isn't wrong. I could have posted the raw numbers, but that graph would have looked very spiky and strange and wouldn't have made sense to people.
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Since we do not know how long a game will last, we want to define the marginal cost function as the game time gets infinitely greater:
lim(50/T) as T → ∞ = 0
This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero.
no. the expected value of the marginal cost is the integral over all feasible values for T, the integrand is the marginal cost function as a function of T, multiplied with a probability density function which, roughly speaking, assigns each value of T a certain probability. that the marginal cost function converges to zero for T -> infinity does NOT imply that said expected value is zero. a very simple example: at any given point in time during a game, the remaining duration of the game can theoretically vary between 0 and 10000000 minutes. but in reality, it is over 9000 times more likely for the game to end in 10 minutes than to end in 200 minutes. the probability for high values of T approaches zero at a very fast rate, fast enough to make the zero marginal cost for these very large T-values irrelevant for the overall integral.
TLDR: that the marginal cost function converges to zero for T -> infinity does NOT imply that the expected marginal cost of producing one more worker is zero. therefore, your analysis completely fails on a very basic mathematical level. sorry if i sound harsh now, but i´d say back to calc and stats 101.
if an additional worker does not increase the mining rate at all (ie we already have 24+ workers on minerals per base), it is simply useless to build more workers if one does not plan to expand in the near or intermediate future or is at immediate risk of losing workers to harass. in reality though, these conditions will almost always be fullfilled so that it is almost always advisable to continue worker production unless one has very specific and good reasons not to.
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On February 11 2011 09:08 natewOw wrote:Show nested quote +On February 11 2011 09:03 Nagano wrote:On February 11 2011 09:02 natewOw wrote:On February 11 2011 08:59 Nagano wrote:There is a HUGE flaw in the numbers that I think most people have just simply overlooked. In the OP: "This is saying that the limit of the marginal cost function, as time gets infinitely greater, is zero. Thus, the expected marginal cost of producing an additional probe is zero." If MC is zero, so then why is there a diminishing return on MR when there are 16 or less workers (assuming 8 mineral patches)? Does this make no sense to anyone else? This would mean that the graph above would be completely false up to 16 workers. Am I missing something? It's a graph of ESTIMATES, not true values. It's meant to show the shape of the returns function, not the specific values themselves. You are COMPLETELY missing the point here. There should be ZERO DR on MR at any point before 16 workers. Half the data is wrong, and therefore the entire graph should look completely different. The data isn't wrong. I could have posted the raw numbers, but that graph would have looked very spiky and strange and wouldn't have made sense to people.
Generally if a graph is very spiky and strange you have inconclusive data...
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You aren't listening. There are reductions in MR at every point before 16 workers. There should be NO reduction in MR before 16 workers assuming 8 mineral patches if you assume zero MC.. At best the first half of that graph (up to 16) should be flat. Even if you were smoothing it for aesthetic purposes, there should be no negative slope up to 16 workers! Therefore how can you trust ANY of the numbers, including your conclusion of 27 workers, which is derived from your graph?
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Workers cost 62.5 minerals because each supply building (or overlord) costs 100 minerals and provides 8 supply.
In other words, your first four workers cost 50 minerals each as Protoss and Zerg because you start with 6/10 supply. For Terran the cost is pretty much the same because, to get more supply, you have one worker that does nothing for a little bit more than the time it takes to create an additional worker (so the +1 supply in the early game matters for little)
So, after the 10th worker, each worker costs 62.5 minerals until an expansion goes up and provides you with more supply. This is clearest with Zerg, since hatcheries only provide 2 supply each.
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an additional aspect which the analysis in the OP does not account for is the opportunity cost of supply: additional workers take up one supply which cant be used for army. the closer a player gets to the supply cap of 200, the higher the opportunity cost of taking up a supply spot by a nonfighting unit. factoring in the increasing opportunity cost of supply, the marginal cost of additional workers can never be zero and thus the increase in mining rate needs to be ABOVE zero to justify the production of this worker, which obviously means that from an economic point of view it is no good idea to supersaturate.
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On February 11 2011 09:11 Nagano wrote:
You aren't listening. There are reductions in MR at every point before 16 workers. There should be NO reduction in MR before 16 workers assuming 8 mineral patches if you assume zero MC.. At best the first half of that graph (up to 16) should be flat. Even if you were smoothing it for aesthetic purposes, there should be no negative slope up to 16 workers! Therefore how can you trust ANY of the numbers, including your conclusion of 27 workers, which is derived from your graph?
You don't have to trust me. Run it yourself, you will see that 27 workers gets you higher income-per-minute than 24.
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care to show more info on the cubic in the op? Also after 27 the income/minute wont go down, the negative part of your cubic does not kick in even if it tends to 0
Income tends to a maximum rate after which its a waste of the 50 minerals to make a worker once 100% saturation is hit
Theres plent of graphs where it shows worker income increasing untill 28-32 before flatlining, but in a real game you wont go past 20-24, so 3/ patch is a good rule of thumb limit
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[B]that the marginal cost function converges to zero for T -> infinity does NOT imply that said expected value is zero. a very simple example: at any given point in time during a game, the remaining duration of the game can theoretically vary between 0 and 10000000 minutes. but in reality, it is over 9000 times more likely for the game to end in 10 minutes than to end in 200 minutes. the probability for high values of T approaches zero at a very fast rate, fast enough to make the zero marginal cost for these very large T-values irrelevant for the overall integral.
This is my concern as well. I don't believe you can draw the conclusion that a 27th probe will always be worthwhile since it's based on the assumption of a game of infinite length (as others have pointed out). I'm not sure you can actually draw any conclusions in this area since there's no exact method of estimating how long a game will last at any point before the final moment of the game.
However, that's not to say the numbers have no value. I think a more meaningful statistic would be to determine that amount of time it takes for each probe to pay itself off. That value has a number of practical applications, especially for all-in builds that win or die at a specific moment in time.
For example, if you have an all-in build that attacks at 6 minutes, is it worth build a probe at 5 minutes that takes 2.5 minutes to pay itself off? Probably not...you'd be better off spending the 50 minerals and 1 supply on something else.
To the 27 vs 24 probes argument, I can see some times when this would be useful (assuming its accurate). There are maps where it is challenging to take and hold a 3rd base. It is useful information to know that the 25, 26, and 27th probes on each base generate additional income, especially if I have an excess of minerals that I can't spend for some reason. Although again, there is a cost associated with this. Is it better to build the additional probe or to build a gateway that will be used <100% of the time? I'm not sure how to answer that question.
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On February 11 2011 09:19 natewOw wrote:Show nested quote +On February 11 2011 09:11 Nagano wrote:
You aren't listening. There are reductions in MR at every point before 16 workers. There should be NO reduction in MR before 16 workers assuming 8 mineral patches if you assume zero MC.. At best the first half of that graph (up to 16) should be flat. Even if you were smoothing it for aesthetic purposes, there should be no negative slope up to 16 workers! Therefore how can you trust ANY of the numbers, including your conclusion of 27 workers, which is derived from your graph? You don't have to trust me. Run it yourself, you will see that 27 workers gets you higher income-per-minute than 24.
Your graph is 100% wrong, it should have no negative slope before point 16. Black Gun just explained to you the enormous pitfalls in your mathematical logic.
Your method involved running a game and observing the income tab. You fabricated the rest of the information used to generate the downward sloping graph, including faking your "cubic" function and results, because the real graph of MR would look nothing remotely close to the one you created. You assumed infinite mineral values per patch and an MC of zero. The entire OP was a fabrication attempting to hide your basic assumptions and mathematical ability behind a guise of a one month depth in econ 1 knowledge.
Someone close this thread, please.
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On February 11 2011 07:20 natewOw wrote:
It's just fact? Where are you getting this "fact" from? Because I ran a simulation comparing 24 workers to 27, and 27 was netting me more minerals. Can you offer any evidence to back up your fact?
Simulation?
What happens in the game? I don't know either way, but surely it's better to look?
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@natewow: The question you set out to ask doesn't require any of the economics theory. If you choose discard all the context of an actual game of starcraft, then time doesn't factor in. If you are trying to maximize the minerals you have to spend, you should never build any workers beyond the six you start with, because the minerals on the map are finite. If you are trying to maximize the rate of mineral income at a single base, that is an academic exercise best solved by direct measurement (if you doubt the logic of 24 workers), specific to each expansion on each map, because the mineral patch placements differ. (I guess there will be some exact duplicates across the maps.) You are optimizing something arbitrary, which is fine. You set out to find the least amount of workers that gives you maximum mining rate at a single base. You stated something about the MR after 20 minutes... an expansion's minerals don't last that long. The idea of considering what happens as T increases is simply an inaccurate portrayal of the situation being modeled. This is not to say it isn't an interesting abstract perspective.
The most accurate way to measure the income rate at a given number of workers would be to see how much you mine in a set time period, say 5 minutes. Minerals mined / 5 minutes. I assume this would incline until plateauing at 24, unless my understanding of the worker AI isn't correct. From what I see, the workers will bounce quite a bit until eventually settling, where they pair or triple up. If you have every patch tripled up, every patch is being mined at every possible moment. There is no way an extra worker could access further mining time. This might be wrong, in which case the optimum worker count is near to 22-26.
@everyone else: Yes, we know. If you don't agree with OP, don't post. As many have pointed out, it makes no difference anyway. I hate watching intractable arguments over inconsequential viewpoints.
...Sorry for the outburst.
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I'm sorry I'm not a math major but I did my best to double check my numbers, feel free to check them yourself. Assuming you don't chrono boost the extra 3 (why would you) my calculations using your income numbers from the OP, put you mining out at ~16:15 with 0 chrono boost used on probes and no gas being mined. If I had built 24 probes in the same way, I would be mined out in ~16:28 with a disadvantage of 30 minerals (counting the 150 spent on the probes) at the 16:15 mark.
Now I hate numbers like these because there would never be a situation where your probes NEVER leave the mineral line, don't chrono probes, don't scout and you don't mine any gas in the first 16 min of a game. however doing any of these things will make the margin even smaller, as you will be closer to mined out when you get to the magic 27 probes on minerals.
When being held in my base I do advocate making extra havesters for the event in which the pressure is repelled, you have them ready to transfer as many people have already stated, however I can see no merit in having 27 probes per mineral line on 2 bases on say close position metal or somewhere where a 3rd is difficult.
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Just read the OP and most of the responses. First of all, i have to say i feel bad for the OP because so many people are being simply rude/passive aggressive and poiting out that 3x8=24 which somehow seems to be supposed to prove something.
Regarding the OP i am actually very bad at understanding economics and most math beyond what statistics for psychology education covers (which is basically using a computer) so my request and feedback would be to ask you to explain some of the terms a bit more specifically for those of us who are clueless when it comes to economics.
Otherwise great thread and dont get discouraged or mad at other people even if they are being condescending. It's easy to fall into that trap.
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EPT
