Friday, 17 February 2012

Make Mine Mining

I really like mining math.  The reason I like it so much is that the math is super simple and the consequences of it are incredibly complex.  Here is a quick rundown of how mining works:
  • When you mine you generate a random number of chunks
  • To that number is added a bonus for each time someone used the "Help Mine" action
  • The total number of chunks received is subtracted from the number of chunks remaining in the rock  if the rock is out of chunks it disappears
These simple rules give rise to so many unintended consequences.  For those of you who don't like reading through swaths examples and math, here is what you need to know about mining, in short:
  • If you had your choice, you would better off being by yourself in the area you are mining; but
  • Given that other glitchen are around, you are all better off helping each other mine

How Many Chunks are Produced?
There are three questions to ask about how many chunks are produced.  The first is how many chunks you get each time you use the mine action.  The second is how many chunks are produced from each rock.  Finally, we want to know how many chunks you get for your mining time.

The number of chunks you produce per mining action is determined first by the kind of rock you are mining.  Sparkly hands out less chunks per mine than dullite does.  The ranges overlap substantially, though, so on any two ticks, the sparkly might have more than the dullite.

You also get bonus chunks depending on the number of helpers you have.  Each time someone chooses the "Help Mine" action your bonus increases.  The bonus goes away if the glitch cancels the mining action, but does not go away if they finish the mining action.  When glitchen have the same rank of mining this usually doesn't matter.  When they have different ranks of mining, it is possible for the glitch with the higher rank of the skill to help mine more than once.

Rocks usually contain 50 chunks, but about 1 in 6 rocks contains 150 chunks instead.  Once the number of chunks collected equals or exceeds the number of chunks in the rock the rock disappears.  That does not mean that you can only get 50 or 150 chunks from a rock though, you can get more when mining alone and substantially more when mining in a group.

Why you get more than 50 chunks
The rock may disappear when the chunks are gone, but the number of chunks left in the rock has no effect on the number of chunks you get with your action.  The rock can even disappear before your action completes and you will still get your chunks.  If you have already successfully mined 49 chunks from a rock and then you get 10 with your next action then you received 59 chunks despite the fact that the rock only had 50.  If while you were taking that last action another person chose to help, giving you one bonus chunk and generating 10 chunks of their own, then in total your two person team managed to get 70 chunks out of the 50 chunk rock.

You'll notice that the difference in the number of chunks you got out of that rock had far less to do with the number of bonus chunks you generated and far more to to with the fact that there was an extra mining action on that rock after it was already gone.  In fact the bonus chunks make virtually no difference whatsoever to the number of chunks generated.

The average number of chunks generated by a lone glitch mining a normal sized sparkly rock is 52.6.  If you add one to the size of each mining action the average number of chunks per rock increases to only 53.1.  The reason the difference is so small is that all of those bonus chunks generated before you reached the limit of 50 had no impact at all on the number of chunks in the rock.  The total number of chunks was always going to be at least 50 and at most 49 plus the number of chunks you can generate in one mining action.

That extra tick that your mining partner gets after you finish off the rock, however, is generating extra chunks out of nothing.  More chunks are in the world because they did it than would have been if they had not done it.  The average number of chunks generated by a partnership mining sparkly is actually just over 59.1, or just under 29.6 each.

So I Should Mine with Friends?
The trouble is that while the rocks generated more chunks when you were with your friend, you actually got fewer of them.  Of course you got more chunks per second mining and per energy spent mining, so it sounds like a good deal.  And it would surely be a good deal if you had an infinite rock to stand by so that you never had to walk to the next rock.  Unfortunately, walking to the next rock is a pretty substantial amount of your mining time, and waiting for rocks to respawn is a very big factor.

Rocks respawn somewhat randomly, but a small sample suggests the average respawn time is around eight minutes.  Suppose you have found a nice little spot to mine where three streets have four rocks each.  It takes about 15 seconds to walk the length of a street, and it takes about 4 seconds to get from rock to rock - just slightly longer than running the street because of irregularities in terrain.

The average number of ticks it takes you to clear a rock is by yourself is 8.8, so it takes 34.8 seconds.  With the 4 second walk that means you have 38.8 seconds per rock.  You clear all twelve rocks rocks in seven and three quarter minutes and then take 45 seconds to walk from the end of the third street to the beginning of the first just in time for the first rock to respawn and for you to start mining again.  Every 8.5 minutes you clear 12 rocks and generate 631.6 chunks - 1.24 chunks per second.  It's solo mining heaven here, apparently.

Since you've found mining heaven you invite your friend.  It now takes only 7.6 ticks to clear a rock because of the bonus chunks, but you do 8.6 ticks per rock because of the extra tick on the empty rock.  With two of you, though, 8.6 ticks takes only 17.2 seconds.  So you spend your four seconds walking and get to the next rock in 21.2 seconds.  You clear all 12 rocks in 4.25 minutes and then you walk back to the beginning and wait 3.25 minutes for the rocks to respawn.  You clear 12 rocks in 8 minutes generating 709.2 chunks - 1.48 per second.  But you only get half of those chunks, so 0.74 per second.

Of course that second person dramatically oversaturated the area for mining.  Let's say you add two more streets to your mining course.  Your total time to run the circuit is now 8 minutes and 19 seconds and you cleared 20 rocks.  How does this compare to the solo mining?  It's 2.36 chunks per second - 1.18 chunks per second each.

How can efficiently mining with two people be worse than efficiently mining with one person?  The answer in this case is the run back to the starting point.  With three streets that took 45 seconds, with five it took a minute and 15 seconds.

In the most popular mining areas, however, it is possible to run in a circle to mine so you never have to backtrack over empty streets.  But if we are going to talk about reality, we have to talk about the fact that there are other people there too.  In any given street area the total number of rocks available to mine at any given time is determined by the total number of rock spawn locations, the respawn time of the rocks, the time it takes to clear a rock.  When you bring a friend you end up with the same spawn locations, the same respawn time, and a faster clear time, so fewer available rocks.  You reduce the likelihood that you can consistently move from rock to rock without interruption and increase the likelihood that you run around looking for a rock.

If you are the only glitch in the whole of Callopee then you can easily run a circuit with multiple mining partners and never run out of rocks to mine, but this is never going to happen.  If you ever go to a street and end up running through it without finding a rock to mine then your mining partner is adding to your problems, not subtracting from them.

What about Super Rocks?
I ignored, in that analysis, the fact that some rocks have 150 chunks instead of 50.  This ends up helping the case of partnership mining.  The partnership doesn't really do much better on the 150 chunk rock than they do on the 50 chunk rock.  In fact, the average number of chunks for a solo miner from a 150 chunk rock is 152.6, almost exactly 100 more than the average from a 50 chunk rock.  The average for a partnership is 159.1, again almost exactly 100 more.  So while the solo miner gets 2.9 time as many chunks from a super rock as from a normal rock, a partnership gets only 2.7 times as many.  Why this helps the partnership, however, is that with more chunks in the rock, the acceleration they get from the bonus chunks actually has time to kick in.  It's pretty clear that the partnership does better when standing and mining on an infinite rock, so the closer we get to that scenario the better they are.

So we'll apply this to the examples above, giving the three street / twelve rock example two super rocks and the five street / twenty rock example three and a third super rocks.  We get 831.6 chunks in 10.75 minutes for 1.29 chunks per second for the solo miner and 1382 chunks in 9.25 minutes over two people for 1.24 chunks per second in the partnership case.  Super rocks help the partnership, but you'd still be better off if your friend didn't come.

So I Should Mine by Myself?
In my example above, your friend was only coming to mine if you asked them to.  Again, in the real world other people mine because they want to mine.  Those other people are going to be running around mining rocks and reducing your opportunity to get rocks whether you like it or not.  Given this, the best way to get more chunks is to make sure there are more chunks to go around. 

If we go back to our five contiguous street example above, but have the glitchen mine separately instead of together, then now they spend an average of 8 seconds between rocks because the next rock in line has already been mined by someone else.  They take 42.8 seconds to clear a rock and get to the next one, and still have to run back to the beginning, so the whole circuit takes nearly 8.4 minutes.  Since each rock is only giving 52.6 chunks instead of the 59.1 it was giving when mining in tandem, there are only 1052 chunks to go around, which is 526 per glitch or only 1.05 per second.  They could be getting chunks 13% faster by pairing up.

So in an area that has any real competition, partnering up with everyone you see and all mining together is the best way to go.  In an ideal world from your own perspective, though, you'd rather they just all went away so you'd never waste your time walking past an already expired rock.

Ajaya Bliss
It would seem silly for me to talk about co-operative mining without mention Ajaya Bliss.  Suppose there are 15 or 20 people there so rocks always go within about four seconds of spawning, with the final ticks finishing nearly four seconds after the rock is gone.  Each of the 15 rocks spawns every 8 minutes so a rock spawns every 32 seconds.  You are really on the ball so you are early enough in on every rock to always get the maximum of 30 bonus chunks.  Surely this is a great deal, right?

In terms of chunks it is a good deal, but not a spectacular one.  Every 32 seconds you generate 34 to 38 chunks.  That is 1.125 chunks per second.  That is the same rate of mining sparkly as mining by yourself and walking for an average of 11.7 seconds between rocks - quite a realistic rate to find sparkly rocks in Calloppe or Pollokoo.

Of course mining partners also increase the chance of getting a gem, so that can't be ignored, but it is worth noting that if you were mining by yourself and spending 11.7 seconds between rocks you would be mining an average of  0.18 times per second, or 6 times in 32 seconds.  In Ajaya Bliss you'd only get to mine once in that time.  Thus, in order for the gems to make the difference, those mining helpers would have to multiply your chances of getting a gem by more than 6.  While I don't have data on this, I wouldn't be surprised if the gems went in favor of the solo miner in this case.

For the most part not mining,

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