|
On February 11 2011 07:22 kNightLite wrote:
Uh no offense, but marginal cost is the derivative of total cost with respect to quantity. Not time. There's a big difference between the two. I understand that you think MC lim 0 because you don't know how long the game will last. However that question comes more into play in the MR curve, not the MC curve. The longer the game lasts, the greater the MR, not the less the MC.
You're certainly correct that at economic optimization you want MC=MR, it's just that you're calculating MC incorrectly.
You're absolutely right in that this is a very unorthodox way of calculating MC. The problem is that you can't calculate it normally because this is a video game. So, one must make some assumptions and change things around a bit.
Perhaps a better way of defining the marginal cost here is to think of it as the marginal cost per minute. The reason is because MR is defined by the game itself as marginal revenue per minute. We need to put MC in the same unit of measurement. So if you calculate MC normally, and then set it as a function of time, and take the limit (as I did), you will see that the marginal cost per minute (which is what we are really measuring here) is in fact zero.
Good question.
|
The only thing I can say is... you put in work for a useless statistic for the game of SC2. The one thing that is ignored is expansions. Which then puts the optimal mining workers at 16. simply because:
Base 1 = 16 works
Base 2 = 11 works
Mines more minerals than:
Base 1 = 27 workers
Simply due to how the first 2 workers on a mineral patch mine FAR more minerals than any workers after that. Even taking into consideration the cost of a 2nd base, the mineral gains are still much higher my making a 2nd base.
Even if you look at your own graph... the margianl minerals per minute of the first 16 works > cost of a CC/Nexus/Hatch.
Basically, expanding at 16 on minerals > going to 27 workers... ALWAYS.
This basically just means that your work its sort of useless unless in the odd scenario where there is only 1 mining base left on map and you have X amount of workers, killing all but 27 will result in the highest mineral gains...
Can you please point out where I said that rushing to 27 workers was better than expanding? Because I don't believe I ever said that.
Also, you are quick to point out how what I did was useless, but I was wondering if you could please tell me how you have contributed in any way to the community? You seem to be on an awfully high pedestal.
|
While its an interesting read, there are several few ideas that are not really considered when calculating your model, and because i'm tired i'll just keep it brief. Simply put, you don't factor in opportunity cost very heavily here - which is the prime consideration when looking at the ideal number of workers to make - sacrificing attacking units / tech / additional expansions (to increase the MP of each additional workers, thus increasing efficiency) / what have you ends up being a large consideration - Maybe some time I'll write on this, but the eventual conclusion is that its highly situational and sometimes its simply better to expand to take advantage of the increase in MP of each worker than make 3 more workers for a small increase in Total Product, given that you have the ability to defend the expansion. Again, it gets complicated, but you need to consider the opportunity cost, perhaps measured in terms of "risk" or "perceived risk", which determines what applicable strategy is best sought after.
Oh economics, i love you.
Or, TLDR - its pretty fucking difficult coming up with an applicable economic model making conclusions without ignoring important assumptions in starcraft, so all conclusions found need some sort of asterisk.
|
|
|
Let me just take Drunken.Jedi's comment regarding tradeoff and elaborate a little on the conclusions of the op.
I have to say that even if the numbers are correct, I do not believe your analysis is enough to prove that "27 workers constitutes optimal saturation." I actually believe you just proved otherwise, that 24 workers constitutes optimal saturation.
Since we are being analytical here, let's step back for a moment and get Webster's definition for optimal: "the best or most favorable point; most favorable or desirable".
When you take the concept of mineral patch saturation, the discussion is obviously taking place in the context of Starcraft II and taking into account real and practical applications, otherwise it is just mindless theorycrafting. Although this may seem like nitpicking, I actually believe it is not, as the usefulness of Stacraft II related definitions directly helps subside every other single discussion, may it be here in the TL forums or not, regarding the game.
That being said, I would kindly ask for any example whatsoever of a real game situation in which it is optimal for any player to go beyond 24 workers to get the supposedly optimal 27. Your example was "If at some point your macro slips [...]" - well, that is just not optimal, is it? I am most certain that in the tradeoff between building the 25th worker in your base and building another unit or building, it will always be truly optimal to do the latter.
All that just to say that your analysis proves that the gain from any worker beyond the 24th is so small that it should be optimal to just do something else with those 50 minerals and 1 food. Although mathematically the conclusion may be true, it is just useless in the context of the discussion.
In my opinion, 24 workers still constitutes optimal saturation.
Edit: let me take the "useless" back, and change it to counterproductive. I do not mean to just critic the op's very cool work, just point out my opinion on the need have productive definitions to discuss the game.
|
Again, it gets complicated, but you need to consider the opportunity cost, perhaps measured in terms of "risk" or "perceived risk", which determines what applicable strategy is best sought after.
I addressed opportunity cost in a previous post. The problem with what you are proposing is how exactly do you measure risk? You, along with several other people, seem to think that I am advocating spending all resources on workers until you get 27 workers. This is simply not the case, and I never said this anywhere. All I said was that 27 workers gets you more income than 24. I didn't say HOW to get 27 workers, I didn't say WHEN to get 27 workers, I just said that 27 workers gets you a higher income per minute than does 24.
|
Thread resolved: http://www.teamliquid.net/forum/viewmessage.php?topic_id=192072¤tpage=5#96
You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue.
Why does this produce false information?
Because I know whatever formula you used does not take into account SCV travel time between mineral patches and the CC, and between each of the mineral patches when said patches are occupied.
The fundamental flaw here is that your model does not take into account the factors which require a much more complicated mathematical skill set than what I believe you have based on your post. (If you're a PhD in mathematics, please correct me)
Realize the mechanics of the game: when occupied by two workers timed correctly, a mineral patch only has a few milliseconds, if not zero ms, when they are not occupied and thus a third worker would occupy the patch. Three workers per patch is already pushing it, and any more is just overkill. Now I'm not saying you have to match the results of your theory to what would be expected, but you have to realize that your work is not complete and you cannot come stomping into the forums demanding you be heard if your calculations do not take into account ALL of the factors at play during mining time.
|
All that just to say that your analysis proves that the gain from any worker beyond the 24th is so small that it should be optimal to just do something else with those 50 minerals and 1 food. Although mathematically the conclusion may be true, it is just useless in the context of the discussion.
I don't mean to sound rude here, but I honestly don't see what is so difficult to grasp about this post. You keep trying to bring in "contexts" and "situations" when all I did was show that 27 workers gets you more income than 24 does. If you have the option of getting 27 workers instead of just 24, then you should take it. If you don't have that option (because of the situation or the context or whatever), then stay at 24. You said it yourself: mathematically speaking, 27 gets you more than 24. That's all I showed here.
|
On February 11 2011 07:29 natewOw wrote:Show nested quote +The only thing I can say is... you put in work for a useless statistic for the game of SC2. The one thing that is ignored is expansions. Which then puts the optimal mining workers at 16. simply because:
Base 1 = 16 works
Base 2 = 11 works
Mines more minerals than:
Base 1 = 27 workers
Simply due to how the first 2 workers on a mineral patch mine FAR more minerals than any workers after that. Even taking into consideration the cost of a 2nd base, the mineral gains are still much higher my making a 2nd base.
Even if you look at your own graph... the margianl minerals per minute of the first 16 works > cost of a CC/Nexus/Hatch.
Basically, expanding at 16 on minerals > going to 27 workers... ALWAYS.
This basically just means that your work its sort of useless unless in the odd scenario where there is only 1 mining base left on map and you have X amount of workers, killing all but 27 will result in the highest mineral gains... Can you please point out where I said that rushing to 27 workers was better than expanding? Because I don't believe I ever said that. Also, you are quick to point out how what I did was useless, but I was wondering if you could please tell me how you have contributed in any way to the community? You seem to be on an awfully high pedestal.
Can you point out a single time in game where this will EVER be useful? Can you point out a time in the game that you will ever concievebly want 27 works on any one base when you have another base open for expansion?
It is all fine and dandy that 27 workers = where the marginal cost = 0... but its not something you will ever use in game.
You WILL on the other hand expand and keep 16 workers per base simply because the MC is highest for those first 16 workers at any base.
In a one basing situtation you will either:
a) Cut workers for your build = does not hit 27 workers
b) Continue to make workers upto 44 workers at the high end so they can be maynarded over to an expansion.
In game, you will NEVER go "man... I really need to get some more income, time to get up to 27 workers on 1 base" you will on the other hand go "man... I really need some more income... time to expand" simply because the mineral output of an expansion is greater then gettting any where near 27 workers on a single base.
- - -
This is interesting yes, but it is useless for the game of SC2 as it will never be used.
|
Even if you're theory is correct, there are several large practical concerns which will likely prevent this from being useful. Several people have already mentioned expansion timing. However, the biggest concern that I have is whether the additional income you gain from workers 25-27 overcomes the cost before mineral patches start becoming mined out.
The reason you didn't come across this in your analysis is that you asked how long the worker will be in existence for, not how long the worker will be USEFUL for. I don't know the timing exactly, but it seems quite likely that if the marginal revenue from the 27th worker is LESS THAN 10 MINS PER MINUTE (as indicated in your graph) that the mineral patches would be long mined out before that worker mined for the 5-10 minutes necessary to regain its cost.
EDIT: the table gives the 27th worker as providing an additional income of 2.46 minerals per minute, so it would need to mine for a whopping 20 minutes to regain the 50 minerals spent on it
|
On February 11 2011 07:41 natewOw wrote:Show nested quote +All that just to say that your analysis proves that the gain from any worker beyond the 24th is so small that it should be optimal to just do something else with those 50 minerals and 1 food. Although mathematically the conclusion may be true, it is just useless in the context of the discussion. I don't mean to sound rude here, but I honestly don't see what is so difficult to grasp about this post. You keep trying to bring in "contexts" and "situations" when all I did was show that 27 workers gets you more income than 24 does. If you have the option of getting 27 workers instead of just 24, then you should take it. If you don't have that option (because of the situation or the context or whatever), then stay at 24. You said it yourself: mathematically speaking, 27 gets you more than 24. That's all I showed here.
Fair enough, I'll simply ask this - while MR is positive, shouldn't it be considered that the cost of adding such small minutely revenues is not particularly logical barring some sort of inability to spend money, or in other words, the time it takes for the Total revenue for that individual worker may not be worthwhile for the total marginal cost of that worker, especially when considering the time value of money (aka money is worth more than later)? Or otherwise, perhaps it's never logical to get 27 workers in a mineral patch.+ Show Spoiler + (which I don't believe, you just have to create reasoning for it, such as the potential for the loss of mining, the usage of defence, expanding to other expansions).
However, this is kind of meaningless, I'm just talking for the sake of talking. I don't really disagree with your conclusions.
|
You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue.
I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend.
Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27?
|
On February 11 2011 07:45 natewOw wrote:Show nested quote +You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue. I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend. Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27?
No one is attacking the fact that 27 workers mines more minerals than 24 workers... we are attacking the usefullness of this fact.
If you cannot explicitly tell us when you will ever purposefully get 27 workers on a sinlge base when there are availiable expansions still unmined, please by all means let me know and I will resend any doubt I had about your fact and will even add it to my signature so that all can see that we should get 27 workers on a single mining base in scenario X.
If you cannot give an scenario for its usefullness, then it is in fact a useless fact and adds nothing to the game and its just a fun little fact to know but impacts nothing gameplay wise.
|
On February 11 2011 06:44 Drunken.Jedi wrote:
Well, the problem with this analysis is that it assumes that the value of money or minerals is static, which isn't the case. In real world economics, you have to consider opportunity cost, i.e. when you invest X amount of money into Y, you lose out on the profit that could be made by investing X into something else.
In "craftonomics" this also holds true, as the money spent on a probe could also be spent on something else and maybe that something else would increase your chances of winning more than getting the extra revenue of one drone.
Oh God finally someone else who sees SC2 choices in terms of opportunity cost. Once I tried discussing it with Cloud and he thought I was trolling him -_-"
|
On February 11 2011 07:01 natewOw wrote:Show nested quote +Btw, the cost is actually wrong it's 50 minerals for the worker +100/8 = 12.5 for the supply. The real cost is 62.5 pr worker. Please explain, I don't understand how you come to this conclusion. I don't even know what your unit of measurement is.
You have to build a supply depot/overlord/pylon to be able to build a worker, that is where the extra cost of 12.5 minerals comes from.
I do wonder how you ended up with going towards infinity in an equation with a set amount of minerals on a particular map. The biggest maps have about 14 bases with around 8 patches each, at 1500 minerals for a total of 168.000 minerals total.
|
The point everyone else is making is that the situation in which you would actually choose to get more than 24 workers per base to exactly 27 is just plain nonexistent, and thus rendering your little post here utterly null
|
On February 11 2011 07:45 natewOw wrote:Show nested quote +You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue. I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend. Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27?
What about the rest of my post? You clearly didn't want to acknowledge the rest of it because you wanted to take my first paragraph out of context.
Can you please provide me evidence that there isn't a flying spaghetti monster between Earth and Mars? You see what you did?
I'm saying that your work is incomplete because the formula you used to derive 27 workers is fundamentally flawed. Does it model worker movement, travel time between its destinations, the calculation the client uses to determine whether or not a worker moves or stays on a mineral patch if it is currently occupied? I didn't think so. I am not going to sit for a few days trying to determine it myself because I have better things to do, but I can assure you that no one has to do so in order to see that what you did is incomplete. Therefore your number of 27 is questionable AT BEST and PLAIN WRONG AT WORST.
|
On February 11 2011 07:49 Insanious wrote:Show nested quote +On February 11 2011 07:45 natewOw wrote:You need to stop being so hostile. It's obvious to me that you're trying to put yourself in a teacher role where you believe you contributed in some meaningful way and that the rest of us just have it wrong. I understand you just completed or are in the process of completing your Introduction to Economics class, but you need to understand that your approach, while notable, is fundamentally flawed. Why? Because your conclusion is based upon the formula you used to determine the marginal revenue. I'm being hostile? I was under the impression that people were attacking my work (thus far without merit, I have yet to see a single post that has undermined anything I said) and I was defending it. It's a standard academic procedure. You criticize, I defend. Also, if I am wrong, can you please provide evidence that 24 workers gets you more minerals per minute than does 27? No one is attacking the fact that 27 workers mines more minerals than 24 workers... we are attacking the usefullness of this fact. If you cannot explicitly tell us when you will ever purposefully get 27 workers on a sinlge base when there are availiable expansions still unmined, please by all means let me know and I will resend any doubt I had about your fact and will even add it to my signature so that all can see that we should get 27 workers on a single mining base in scenario X. If you cannot give an scenario for its usefullness, then it is in fact a useless fact and adds nothing to the game and its just a fun little fact to know but impacts nothing gameplay wise.
all ins?
|
On February 11 2011 07:27 natewOw wrote:Show nested quote +On February 11 2011 07:22 kNightLite wrote:
Uh no offense, but marginal cost is the derivative of total cost with respect to quantity. Not time. There's a big difference between the two. I understand that you think MC lim 0 because you don't know how long the game will last. However that question comes more into play in the MR curve, not the MC curve. The longer the game lasts, the greater the MR, not the less the MC.
You're certainly correct that at economic optimization you want MC=MR, it's just that you're calculating MC incorrectly. You're absolutely right in that this is a very unorthodox way of calculating MC. The problem is that you can't calculate it normally because this is a video game. So, one must make some assumptions and change things around a bit. Sorry, but no, it doesn't. In fact it's actually easier to build an economic model for SC2 than most real-world scenarios because of the relative absence of fixed costs, scaling, and externalities. Game length has an impact upon how many total minerals you receive over time. It doesn't change the fact that building an extra SCV is always going to cost 50 minerals. If you want to view saturation as a present value vs future value question that's fine, but that's totally separate from cost analysis.
Perhaps a better way of defining the marginal cost here is to think of it as the marginal cost per minute. The reason is because MR is defined by the game itself as marginal revenue per minute. We need to put MC in the same unit of measurement. So if you calculate MC normally, and then set it as a function of time, and take the limit (as I did), you will see that the marginal cost per minute (which is what we are really measuring here) is in fact zero.
Marginal cost per minute is a meaningless statistic. Especially when you let time approach infinity. Look at your own graph. At quantity = 1-6 your measure of MR = 0 just as much at 27.
|
Fair enough, I'll simply ask this - while MR is positive, shouldn't it be considered that the cost of adding such small minutely revenues is not particularly logical barring some sort of inability to spend money, or in other words, the time it takes for the Total revenue for that individual worker may not be worthwhile for the total marginal cost of that worker, especially when considering the time value of money (aka money is worth more than later)? Or otherwise, perhaps it's never logical to get 27 workers in a mineral patch.
Sure. I mentioned that we want to investigate what happens to MC and MR as the game time, T, becomes greater. If we look at the marginal revenue provided by the 27th worker, the total revenue it provides is its total revenue provided less its total cost. I said that the total revenue provided by the 27th worker was about 2.46. So the total revenue provided by this worker is:
(2.46*T') - 50, where T' is the amount of time remaining in the game.
In order to find when it would be productive to get this 27th worker, we need to solve for T', which is the minimum amount of remaining game time in order for the 27th worker to start paying for itself. In this case, T' is about 20 minutes. After 20 minutes (game time, not real time), the probe will start netting you positive net revenue. Obviously this would only be useful in longer macro games.
So while the amount of marginal revenue provided by the 27th probe is very small, after 20 minutes or so of game time, it will start to pay for itself.
|
|
|
|
|
|
EPT
