If you have more hashrate then all other miners combined, you can start from any block and grow a chain that will arbitrarily be longer than the "regular" chain.

This in theory would work (and I remember it was discussed a few times) but it requires insanely improbable assumptions: a very large entity that can drive difficulty up and down that has a very, very cheap hardware (so the depreciation is small and it can sit idly without incurring large losses) and very expensive electricity or very inefficient mining. Moreover, a price/difficulty ratio for bitcoin must be close to marginal electricity price. All together I think probability of such a scenario is astronomically small.

Moreover, anybody with so huge capacity, well above >50% network can do a much simpler and more rewarding thing:
1. Grow a parallel chain, independently to Bitcoin
2. Stop when they find 2016 blocks.
3. Inject this chain shortly before difficulty change.
4. Collect all the 2016 blocks.
5. See that the difficulty did not change much.

It's a myth. Every getwork is unique. There is no duplication ever. You can run 10 miners with 10 separate bitcoind or 10 with a single bitcoind and the probability of finding a block is the same.

Nobody knows except them.

Good for you but for each miners with 108% efficiency, there is one with 92%. But solo mining obviously brings more income on average. You have no pool fees, you collect extra blocks fees and there is usually less downtime. And you have extra fun with unpredictable character of Bitcoin lottery (and some frustration with the same).

The problem with pool concentration is a bit overblown. There is no motivation for pool operators to play dirty. Any potential gain will be overshadowed by loss of reputation. The only problem I can think of is that some rogue player hacks into one of the pool and uses its power to perform >50% attack in order to damage the Bitcoin reputation. This is rather problematic and can be theoretically detected by pool users (and obviously pool operators). >50% attack requires building upon parallel blockchain and not distributing it to the Bitcoin network until after it has grown by more than a few blocks. Therefore, pool users can detect it if they find that there are winning shares (in their logs) that were not subsequently contributed to the Bitcoin network by the pool. It takes some time to build the parallel chain, so even for a 60% total hash entity, and 6 blocks per hour for the whole network, the rogue player advantage is 1.2 blocks per hour (3.6 vs 2.4 blocks). It will take more than 4 hours on average to build a 5 block longer chain (with some luck it can be shorter, though). Anybody that detects such a problem can raise the alarm. Even nonpool users should detect such a bizarre slowdown in the block generation of the main Bitcoin network. There will need to be some automation in this detection but it is possible to do. If such building of parallel blockchain is detected, it is enough for several percent of the "rogue pool" miners to switch to another to stop it.

If one looks at the bitcoinwatch pie chart, the "other" part of the pie shrank significantly. Since this calculation from reported hashes and global hashrate is rather inaccurate (e.g., due to changing speed between pools and a "luck factor"), I counted blocks solved by pools during 24 h starting 30 May 20:00 UT.
The whole Bitcoin network found 142 blocks. Individual pools:
deepbit: 55 blocks
slush: 22 blocks
btcmine: 18 blocks
btcguild: 12 blocks
eligius: 5 blocks
swepool: 2 blocks
bitcoins.lc: 0 blocks (unlucky day apparently)
total: 114
other miners and smaller pools: 28 blocks = 20%

I do not have some hard data but I think this is an all time low.

What we can guess from this statistics? Well, apparently with current difficulty the variance with mining solo is rather brutal for small miners. Even a few high-end rigs may be not enough to find a block in a week. And it seems people are rather risk averse. I find it interesting that a lot of people spend quite a lot of money on gambling and unfair lotteries (unfair=with payoff to winners less than ticket revenue). Yet Bitcoin, which is a fair lottery (and better than fair, with extra fees in the blocks), has so few players that want to take  the risk.  

Also, it seems that a notion that mining is rather concentrated is not true. Large miners prefer mining solo (we do not have stats for all pools but for those that publish statistics, largest miners are in teens GH/s max). Those big solo miners are apparently at best 20% of the global hashrate and likely less since there are definitely some hard-core solo miners with small hash rates. Therefore, mining income is rather widely distributed. I think it is rather good for wide adaption of Bitcoin. It is also good for safety and robustness of the network (even though it is concentrated in pools). Recent dynamics is for concentration of mining in pools. If it reverses, this may be a signal of a start of mining concentration.

Yes, it was. The graph was made by analyzing activity for this wallet
http://blockexplorer.com/a/ALEwEXxVGo

It can't be right. First of all, hopping reward is completely independent of difficulty.  1GH/s hopper @ difficulty 100000 gets the same reward as 2GH/s hopper @ difficulty 200000. Secondly, even with infinite number of pools, there is a finite number of blocks. With 6 blocks/hours you can only make 6 hops an hour and some of the hops are impossible to do (you cannot hop to solo miners). My guess based on the intuition I gained working on this problem is that in the limit of infinite number of proportional pools, all global hashrate contributed by the pools you can hop into, and optimal hopping, you can probably get about 100-150% more. I'm yet to derive it or program a simulator. This number is reduced if some hashrate is from solo mining and if the pool operators introduce countermeasures. I doubt that anon4758 can get further beyond his 70% by more than a few percent because he probably hops among almost all the hashrate available for hopping.

OK. Now I get it what you mean by 10-20% more. This is 10-20% more out of slush's pool. If you mined solo for the rest of your hashrate, it would be 6% of your total hashrate. When slush implemented the pool, there were no other pools and the hopping reward was somewhat smaller. With multiple pools, it might be worth the fuss. 

I've just did a Monte-Carlo simulation for decay constant 0.0871 and hopping cut-off 0.14 and you get
reward equal to 19.4% of your total hashrate out of the pool, while one should have gotten only 13.2%. So it is 6% of your hashrate extra (assuming you mined solo or pay par share pool the rest of the time) but more than 40% more out of the shares submitted to the pool. Pool owner can reduce it by decreasing decay constant while increasing variance but it's hard to reduce it completely without increasing the variance excessively.



This geometric score requires either huge fees or introduce a payoff variance for the pool operators which may be unacceptable for them.

Pay per share (PPS) is very dangerous for pool operators, especially the small ones. In PPS, the pool operators have to pay for bad lack streaks with their own pockets. It can make them go bankrupt because 10-20% bad luck streaks are not uncommon. Moreover, there can be malicious users who do not submit winning shares. With difficulty over 400,000, the loss for the user is negligible, for pool operator it is huge.

I haven't checked all your math but although slush's method is hopping prone and it can be proved, by no means you can get 10-20% more out of it. You must have forgotten how rare are the short rounds and for long rounds, you'll get virtually zero by staying during 14% of the round. We have done simulations with Slush and it resulted in about 2-3% gain at that time.



It may be easier to understand but was much easier to implement by slush because he does not have to keep all the shares in memory, only the score. Moreover,
your proposal is nothing more than an elaborate pay par share method which may be good for pool users but is very risky for the pool operators.

You are free to do what you want in your client. The 0.3.21/22 version is much more newbie friendly because the transaction is going to be delayed unless they pay a fee. As an advanced user, you are free to change your client. If you want to revert to 0.3.20 behavior, replace
by
in AllowFree of main.h and recompile.

And if you are lucky, you will be ahead. It all averages out.

By the way, difficulty changes are completely irrelevant if you mine regularly (it does not have to be constantly). I really don't know why people care about difficulty changes at all. The only thing that matters is what is your hashrate ratio relative to the current global hashrate. This is all continuous. If the current difficulty correspond to, say 3TH/s hashrate but the network is at 4TH/s, you will get the same number of coins per month as if the difficulty was corresponding to 4TH/s and hashrate equal to 4TH/s because in the former case the round will be shorter. 

This is not true. If you mine intermittently on score based pool you only increase your variance but the average income will be proportional to your contributed shares. You will quickly lose score if you stop mining but you quickly gain if you start. If you mine from the start of the round to the half time of the round, you'll get close to zero. But if you join at half time of the round, you'll get almost full payout while contributing only half of the shares. In long term, it'll all average out.

Then the following things will happen (or rather their combination):
1. Difficulty increase will slow or stop
2. Mining will go to professional centers
3. Mining will be spread among more people

By the way, there is no need to "keep up with difficulty increase".

It's a non-reference Sapphire card.

Does not work:


My card is

I would like to see numbers that prove it.  The current network consumes 2MW of power constantly assuming (on average) 2MH/s/W. It's about 2 million USD per year. Of course that's not much compared to the money used for fiat money flow but for a 45 million USD money supply it's a lot.  And then you have a few million of USD in equipment that based on Moore law will be worth a fraction of the current value in 12-18 months. When you add human work (building and maintaining miners), you have a lot of cost for the amount of money supply that is handled by a small bank.

At the very best, Bitcoin is as efficient as the fiat currency system dollar per dollar and likely less efficient.

P.S I know about the cost of bailouts but Bitcoin is yet to start handling loans. And it were loans not money transfers and money flow that needed a bailout.

The web hosting I use, started to become pretty unreliable recently. I need to find something different. But if you PM me your email, I can email the paper to you.

One can argue (and it probably works perfectly on the stock market) that a large value of bitcoin days destroyed may mean that the Bitcoin "founding fathers" (insiders) start to run for the exits.