Ok.  Most of my original post was about what happens at the beginning and end of a "series of games".  The confusion has come from my saying "game" for your "series of games" and "round" for your "game".  Certainly there is no funny business going on shortly before or after snake-eyes.


Good point, I have not been reading these posts in full detail but just trying to get a feeling for the mathematical approach taken.  Your approach really is fine if you are talking about the binomial distribution but calling this a Poisson process is technically an error.  This is a discrete process, not a continuous one.  My apologies for not reading your post thoroughly.


Ah, we're just talking about different random variables.  The expected total payout for a throw is certainly p*B.  You're right, I made an error here.

Of course, E(X_0) is 1/n*(p*B) and hence

When focusing on how the binomial distribution is a sum of Bernoulli ones it's very easy to write "E(X_i) = p" without thinking.


The sum is, in my opinion, an important part of this problem.  Without it my solution would read:

Assuming there have been at least 9 previous throws and will be at least 9 future throws, the expected value of a throw is
Because it costs $1 to play the expected profit from any turn is exactly $0 and hence there is no winning strategy (possibly unless we allow ourselves throws close to the beginning or end of a series of games).

The interesting bit is in seeing why the expected value of a throw is $1.  If someone cannot see that it really is $1 then I doubt using words like "binomial" and "distribution" won't help.  Instead, I think the simplest reasoning is to consider how much you expect to get from each of the next 10 throws (including your own).  In each case you would be one of 10 people sharing an expected 1/100 * $100 = $1 reward so your expected income from the throw is $0.10.  Summing the expected incomes of all 10 of these throws yields the expected return of $1.

I think it's unfair to expect more than this when the question was


It seems that people have found a winning strategy and provided sufficient justification to show that it is winning.  They may not have found the best winning strategy or justified that their strategy is the best but this is not what is asked for.  You do say the winning strategy but only at one point and you never ask for the best strategy.  I would say the 1 BTC has been earnt.

The 4 BTC on the other hand would certainly require some elegant mathematical reasoning.

I think you missed my point.

Firstly, I don't question your logic on the use of the Binomial distribution.  I omitted talking about it because it is not needed to reason out the expected value of a throw (I'm trying to keep things simple).  I will now expand on my thoughts and use standard mathematical notation but I still prefer my original form.

Suppose a player takes a turn and throws the dice.  Suppose also that there have been at least 9 past and at least 9 future throws.  We want to calculate the expected amount that the player will receive.

Let X_i be the expected amount the player receives from the i-th throw (where i=0 is the player's throw and i=1..9 are the 9 future throws).

We want to calculate E(X_0 + X_1 + ... + X_9).  (Where E is the expectation operator).

The X_i's are independent identically distributed Bernoulli random variables with parameter p = 1/100 (probability of success).

There are two main ways to calculate this expected value.  The first is to consider X = X_0 + ... X_9 as a random variable in it's own right.  This will be a binomial random variable (just as you describe) and you can use the known formula for the expected value of such a random variable (number of trials * probability of success; n*p).

The other way is to note that the expectation operator is linear so
E(X_0 + ... + X_9) = E(X_0) + E(X_1) + ... + E(X_9) = n*E(X_0) = n*p
Indeed, this is the main way in which the formula for the expected value of a binomial distribution is derived.  The beauty of taking the second method here is that it the expected value of a Bernoulli random variable is intuitive so formulae can be largely avoided.  I find it difficulty to express myself properly on a forum but I feel this is the elegant way to reason about this game.

The majority of my post was discussing the assumptions needed to make the mathematics rigorous.  If you make an assumption that this game extends infinitely in both directions then everything is fine but most people will assume that the game starts at some point and ends at some point.

What happens, for example, if we actually start playing this game and I get snake eyes on my first throw?  Would I recieve $100 or would I recieve only $10 (the other $90 theoretically given out to the previous 9 players).  In the latter case there is no problem but in the former case the first 10 throws give the player a positive expected profit.  Instead of being 10 * 1/10 * $1 the expected value would be (1 + 1/2 + 1/3 + ... + 1/9 + 1/10) * $1.

What happens when a game ends?  If I make the last throw of the game then surely $1 is too much to pay for only a 1/100 shot at receiving $10.

A real PPLNS pool has to start at some point and must end at some point too.  A naive PPLNS pool will screw up here, giving out high value to the initial miners and constantly borrowing from the future until, when the pool closes, the people holding the last N shares lose out.  This is a trapezoid scheme.  A carefully implemented PPLNS pool would try to avoid this, probably by creating a pool kitty which is funded by the initial miners and pays out to the final miners.  A PPLNS pool has other factors to worry about, such as changing N and changing difficulty.  Just as with starting and ending a pool, these changes need to be handled gracefully to make sure that the expected value of each share is correct.

That's very true.  I was used to this effect from solo mining and I admit it was much easier to be sure everything was working when you could see shares rolling.

Still, if a server is having resource issues then dropping to difficulty-2 shares seems like a much better idea than renting/buying a second server.

Perhaps I've misunderstood something here.

By matching Bitcoin and Namecoin block I'm thinking of an SHA-256 hash of [a list of Bitcoin transactions, some Bitcoin address information, and Namecoin data in a merkle tree] which happens to be a particularly low number (begins with quite a lot of 0s).  It was my understanding that any hash small enough to generate a Bitcoin block would also be small enough to generate a Namecoin block and consequently I expected any merged mining Bitcoin block to have a corresponding Namecoin block (at least while Namecoin difficulty is less than Bitcoin difficulty).

This belief was made stronger by posts such as this:
https://bitcointalk.org/index.php?topic=42667.msg566146#msg566146
where slush proudly announced the first Bitcoin merged mining block (which had a Namecoin counterpart) Bitcoin 148744 = Namecoin 19274.

Can you guys explain why Bitcoin and Namecoin block generation events are not linked in this way?  Are the corresponding random variables independent (in the sense of probability) and if so why?

Thanks.

I noticed that you put GamingG's missing Bitcoin block (149249) on the list.  Shouldn't there be a Namecoin block that coincides with this Bitcoin block?  There is no Namecoin block on this list within a 1 hour radius of this Bitcoin block.

Also, what is the matching Namecoin block of trasp's Bitcoin block (149155)?  Is it trasp's adjacent Namcoin block (19893) which is reported appearing more than a day earlier?  Is it your Namecoin block (20125) reported 12 seconds after?  Is it perhaps some other missing Namecoin block?

For anyone else out there trying to piece this together there was (and still may be) a +-1 offset bug to the reported block numbers here.

I certainly understand if it's not an easy parameter to change and the servers can take it anyway.

I don't see how share granularity is much of a plus though.  There is a bonus in that the pool can quickly detect when a user has stopped mining and send out a cautionary e-mail.  Low difficulty shares can be helpful for people trying to measure stales too.  Other than that I don't see the problem with submitting no shares between longpolls or why the pool needs to know user's hashrates.

Given the low fee that most pools ask for I might expect that a pool server would have to watch it's BTC/Watt in a similar way to a miner so surely there is incentive for making the server's more efficient.

Ah well, this is just curiosity.  I don't run a pool server nor do I intend to start.

Hmm...  I'm using the blocks page to analyse this pool's luck so if there are missing blocks that could explain the disturbing figures I'm coming up with.  While you're looking into that could you check to see if the PPLNS pool found any Namecoin blocks between 20180 and 20463, assuming a constant 30 GH/s this is an incredible gap.

The PPLNS pool stats page is now reporting another Namecoin block discovery (comradehls, 34 confirms, 20661) but none of my personal stats have changed.  I remember there being some kind of significant lag between these two but I thought that had been fixed.

Also, my Namecoin estimate is currently 0.

Interesting.  I wonder why pools use such low difficulty for their shares then.  With most pools people are submitting many more shares than they are receiving payments so the reason is certainly not related to variance.  Higher difficulty shares would help with server resources which would allow pools to operate more cheaply too.

Is there some technical reason why difficulty-1 shares are preferred?

It's a shame solidcoin failed to deliver again.  After this I can't say I'd be eager to give them a third try either and I'm glad to see that you handled the instability well this time.

Right now my account is showing an unconfirmed balance of 1.52413899 BTC but the stats report no unconfirmed BTC blocks.  I have a separate confirmed balance of 1.61780172 BTC which I believe is down to trasp's block (149155).

Is there a PPLNS bitcoin block which is not yet on the stats chart Mike?

Might this "1.52413899" actually be unconfirmed namecoins from ericwmson's Namecoin block (20599)? (this block is currently unconfirmed and I have 0 unconfirmed namecoins).

I'd love to start encouraging people to come to this pool again but I'm personally finding the stats very concerning.

How much of that is down to the shares?  If you created a pool accepting difficulty 100 shares then would your bandwidth requirements drop significantly?

Assuming that there were at least 9 past throws and will be at least 9 future throws, a throw gives the player an expected return of exactly $1.  This is because a throw has a total payout expected value of $100/100 = $1 and a player will receive 1/10 of the payout for their throw and 1/10 of the payout for each of the 9 future throws.  If there are fewer than 9 past throws then you receive a large proportion of the payout for your throw so on average you will profit.  If there will be fewer than 9 future turns then you clearly lose on average.

Consequently, in most real world cases, the winning strategy is to make as many of the first 10 throws as possible and then leave (unless there is a real chance that there will be fewer than 20 throws in total).

For miners this means joining new PPLNS pools that you think have implemented the reward system in this naive way and leaving after the first N shares have been found (again, banking that the pool will receive at least 2N shares).  Hopefully new PPLNS pools will not be so basic.

Another few hours without any Namecoin blocks on any of the pools (and at this very low difficulty).  I have no doubt that something is wrong.  Please put our fears at ease Mike by fixing any stat update lag and putting the recent blocks up.

I continue to mine here but I absolutely cannot advise it to others.  If you're not happy to take the risk here I would mine at another pool and wait for block reports here to return to normal.  Merged mining itself seems to be perfectly fine and the income bonus for using it is very high (150%) so I would suggest another merged mining pool.

Hmm...

Each of these can be rationally argued as a valid choice in certain circumstances although I'm surprised by the undue popularity of many of these options.  It seems popularity itself is an attractive quality for an choice or activity for most people.  Perhaps people like the safety in numbers which reduces their need to think critically about such choices and focus on more important matters in their lives.

Agreed.  Simplecoin is reporting very few blocks for the hash rate it receives but this is most likely due to problems with the SQL stats replication server.  It would be wise to wait for simplecoin.us to start reporting normal luck before joining that pool, putting pressure on the operator to get the stats sorted as soon as possible.

I didn't know people hadn't got namecoins from slush yet but I would be startled to learn that he had anything other than honourable intentions.  Consider masterpool.eu, nmcbit.com, eligius.st and keep you eyes open for news from other merged mining pools.  Avoid raw proportional (unless your hopping), spread your resources (reducing variance) and configure backup pools; the next two days are important (low Namecoin difficulty).

Now I've seen it once before, are you sure the recent blocks are all displayed Mike?  The PPLNS pool has Namecoin round shares at more than 6 times the difficulty.

Also, Meni, we are far off topic here.  This thread is supposed to be a flame war about the ethics of pool hopping, not a place for intelligent discussion on the mathematics of alternative reward systems.  We are very much in danger of being branded trolls!

Yeah, what was up with that?  Why would weighing shares to given them all the same expected value hurt intermittent miners?  Where do people get these ideas from?

I don't doubt that, it just felt at first glance that it was an attempt to reduce server workload.  The high complexity was very off putting and it felt like it would require some careful patching and scaling to be unhoppable.

I'll look into these.  Right now I would not switch from even variable N PPLNS to double-geom if it meant giving up merged mining (obviously as it's about 2.4 times a profitable right now) but if there is a merged mining double-geom pool out there (sensible parameters) with a solid server and a reputable and attentive operator then I'd send it half of my hashing power and have it as a backup for the other half.

Yes, it seems like this round is going to last longer than I expected.  Maybe we'll still have this bonus 2 days from now!  I assumed more people would move to merged mining with such serious profit involved but it seems this assumption was faulty and that there are lots of miners out there who don't keep up with mining news.

Happy mining everyone!