OK, I'm not above making a fool of myself, since if this is a stupid question then I really ought to know better (I studied stats and probability at a university everyone has heard of).
Thanks for your interest, catfish. I am absolutely certain that pool hopping works. I was even before I understood the probability behind it, by using mining simulations.
How to hop 5 gives some examples of this sort, and
How to hop 6 derives the expected value for a share. Having studied probability yourself I encourage you to read through these posts which I hope will help explain why the expected value of a share varies.
When I discuss pools in this reply I am only referring to pools with a standard proportional payout although some information is standard for all pools.
However, if the model is effectively a discrete-time set of independent events, then unless the sample size (number of players) is statistically small, what makes hopping between pools any different than hopping back into the same pool? You're joining a series of independent events, so joining at any time should make zero difference.
Intuitively, this is wrong, since humans have an innate psychological 'law of averages' which is a fallacy. Innately, your average human will see a pool that has a 4 day block running and think 'must be done soon', just like after seeing 10 heads on a coin-toss experiment will make most humans expect a tail to have a higher probability - it hasn't though.
I understand that the reason why my logic is faulty is that *the game stops* when a certain condition is found, making it possible to analyse the probability distribution of the lengths of each round (i.e. the number of events before the game-stopping event is found). And given that is a binomial distribution, we have the analyses here.
But bitcoin mining is NOT like coin tossing. Not even inside one pool. The statistical models here present each independent event as having a specific probability which is invariant (correct me if I'm wrong, because my entire post depends on this contention). However the probability depends on the *total* number of participants in the game *across* all pools - this is the bitcoin-specific 'tweak' to the probability function (otherwise doubling the number of miners would double the money supply, etc.)
But bitcoin mining *is* like coin tossing, even in one pool. You make the statement that bitcoin mining probabilities do not follow a geometric distribution but I'd like to know why you think that. Maybe you're confusing shares with hashes? Each share is a hash of difficulty 1, and so has a probability of 1/D of solving a block. A series of events each which have the same probability of ending a trial is a standard bernoulli trial.
So there's inter-pool variance in these distributions. A steady medium-sized pool's 'width' of the probability distribution (i.e. standard deviation of time taken to find a block) will rise if other pools gain large amounts of hashing power, making the pool a smaller proportion of the entire network, and vice versa.
A pool with a hashrate of ten times another pool will solve blocks ten times more quickly, on average, because it is accumulating shares ten times more quickly. The variance you mention is related to the time taken to solve a block, not to the number of shares taken to solve it. Over a large enough series of rounds, the share variance at a small pool is the same as at a large pool, only it takes more time for the mean and variance to settle.
If you can hop so frequently that you can rely on the probability staying invariant, then these strategies make sense.
Why is frequent hopping helpful?
But as soon as you introduce variables to the probability of each independent event, and the pool's variance depends on its total hashpower
No, just the time based variance[/quote], which is also affected by pool-hoppers moving, then simplified models become less accurate.
I've seen far too much of this in quantitative finance - a model that works very well with the nice little line 'assume that P (or whatever) is invariant' fails big-style when P turns out *not* to be invariant, but volatile as hell. Isn't that JUST as relevant here?
It seems to me that the entire pool-hopping 'strategy' relies on exploiting variance in small-to-medium sized pools. The largest pools must be getting towards the law of large numbers by now, surely?[/quote] To quote Starlightbreaker - "hop all the pools!". In fact because the time based variance of small pools is so large, they are not preferable. The larger the pool, the more constant the hopping profit.
Wouldn't one potential strategy simply be to monitor the hash-power of a select range of pool hashpowers (not so small that payouts are very infrequent or the pool is at risk of closure, and not so big that there is little variance in their block-finding times) and jump to a new pool when the variance in the new pool hits a local maximum? The risk here is that if the small pools have enough 'hoppers' following this strategy on board, then a different pool's local maximum will cause all the hoppers to leave, resulting in the 'old' pool maximising its variance through loss of hashpower. And then the hoppers return... ad infinitum. Which would result in hoppers *losing* due to network downtime and too much switching between pools.
Even if this sort of variance was useful to a pool hopper, one round's variance is not reliant on previous rounds so that a maximum in average variance doesn't predict variance in the next round
Apologies, I haven't read the literature already out there on mining analysis, and my knowledge of probability *ought* to be a damn sight better, but it was decades ago...
Great questions, catfish! It took me a lot of reading to get up to speed. The best place to start is
Analysis of bitcoin pooled mining reward systems. Another thing you could do is collect a hundred or so block lengths (deepbit publishes theirs) and do some calculations for yourself. At the minimum you could generate a histogram to show you that the block lengths are distributed according to a geometric probability distribution. If you can prove that to yourself, everything else should follow.
