BCH needs a new DA about difficulty-algorithms HOT 12 OPEN

For that last graph, what did you use for LWMA window size? I think you need more than 144 to match the same 'averaging material' as SMA-144. A factor of 2x longer would be needed to keep the largest weight (the most recent block) the same. Or, you can try to just match variance in which case sqrt(2) longer would be enough.

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@markblundeberg It was N=60. Your math is good because I had noticed SQRT(2) does actually give about the same std dev for constant hashrate. I'll make it a little higher at N=220. The spreadsheet I used for the above can't do the N=220 for LWMA, so I have to use a different method.

Expanded out some is below. The blank at the beginning is a startup, 2*N.

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Here's a newer image to show the BCH miner motivations better.

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Neat! It seems quite clear that the steady miners aren't being punished nearly as much in LWMA. So:

• LWMA-220 has about equal statistical power as SMA-144 (OK slightly better, it would be equivalent for LWMA-203 :-) ). In a fully steady hashrate condition (majority-chain) they should show equal fluctuations and essentially be indistinguishable.
• In conditions of opportunistic switch mining (minority-chain), LWMA is much much better in fluctuation amplitude. Also it doesn't create a runaway economic incentive problem for steady miners.
• Note that LWMA-220 actually responds to changing externalities (halving events, price changes) faster than SMA-144. We can see this in the weighting coefficients of recent blocks: 0.009 for LWMA-220, versus 0.007 for SMA-144.

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An interesting aspect is that BTG with LWMA N=45 is a lot more stable in difficulty than smaller coins with LWMA N>60 because it is more liquid and big miners are more keen to mine it when difficulty drops a little low, preventing it from dropping lower, and leave if it get a little high, preventing it from getting higher. Std Dev in difficulty with N=60 with constant hashrate modeling is about 0.13. In BTG it is about 0.09.

I was remembering wrong. Std Dev in difficulty is the same when N is the same.

Under attack conditions, you can see the Std Devs decrease

In the text above the images you can see my start and stop conditions were they being when the difficulty is 30% above the difficulty that "dedicated" miners would have had, and stops when it's 35% above. That could use some tweaking to be more like BCH's current conditions.

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The code that did these is "test_DAs.cpp" in my code area.

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I was remembering wrong. Std Dev in difficulty is the same when N is the same.

Interesting, I hadn't actually done the math and was making an educated guess. Still it seems odd to me, I will do the math now. :-)

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Oh OK, did the math now, it should be

LWMA-N stddev: sqrt((4/3) / N)
SMA-N stddev: sqrt(1 / N)

So if I have it right now, it's only a sqrt(4/3) factor, not sqrt(2). Note that I'm assuming big-N limit: this is based on calculating the variance in the average solvetime only, assuming the difficulty is reasonably constant (so solvetimes are IID); also I've approximated terms like N+1 as N.

I don't know why in your case the standard deviations are so much closer. 🤔 Perhaps the next-order term inside the square root, of order 1/N^2, is significant even at N ~ 144. Also though your sample size isn't huge, could just be the noise.

Anyway this makes it even better for my last point, which is that LWMA is more recent-responsive (2/N) than SMA (1/N), even for equivalent variance (e.g., LWMA-192 vs SMA-144, if I have the math right).

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Difficulty is a recursive equation, so I have not dared attempt it. 100,000 runs on 192/144 gives 0.072 vs 0.086 and 0.73 and 0.82 on second run of 100k. Equal N still gives really close results at 100k.

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Your difficulty algorithm might be to LWMA what LWMA is to BTC's crappy algo. I'll do an article on it sometime.

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Haha, I am honored you say that. I thought it was just a funny toy algorithm but yeah it has some interesting properties.

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I expected LWMA to be equally stable at around 200 because I had noticed that LWMA at 60 has same stability as SMA at N= 45 which is what your results are predicting. Next time I do a comparison
I need to look for any possible I may have made..

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