DPOS for sure

If GHASH split into 3 pools with the same combined hashing power but owned and operated by the same entity would that make Bitcoin more decentralized?

I am not even sure what you mean.  Attacks aren't "made" on the main chain.  A 51% attack involves the attacker building AN ALTERNATE chain.

In the main chain = attacker mines no block.  
In the attack chain = attacker mines all block = no penalty

When the attack chain is longer the attacker broadcasts it at which point the "attack chain" becomes the main chain and the "honest" chain just becomes an orphaned minority fork.

At no point is the attacker subject to any penalty.

What I mean is:

On the main chain, "attacker A" TRIES to attack
at the time block 1000 is created and fails... the network notices
and on block 1001, it is recorded in the main chain that attacker A's
stake is penalized.

He can still keep trying to re-attack at block 1000
or earlier, but not 1001 or greater... thus his
opportunities to attack become diminished.

I understand its about building an alternate chain,
but obviously it is one that diverges from the main
chain at some point... and what point that is,
is what i'm talking about here.

Why would he do that?  If the attacker has a majority of the stake he will eventually have the longest chain.  There is no reason to "give up".  All that matters is variance and time (well # of blocks) but the longer the chain the higher the probability the attacker will be head and that rapidly approaches 100%.  The exact same principle in in play when we assume the "good guys" have the majority.  In Bitcoin why do you trust a tx with 20 confirmations more than one with 1 confirmation?  It is because the probability that someone with a minority of the hashrate (even 49%) could build a longer chain to perform a reorg and "reverse" that tx decreases as the chain length increases.   In a 51% attack the attacker is the one with the majority so the roles are reverse.  The attack has a >50% chance of being ahead after 1 block but that rapidly rises to ~100% as the chain becomes longer.  The more of a majority the attacker has the quicker the confidence rises (both 51% and 70% will eventually produce the longest chain but we can say at a given chain length the later will have a higher computed probability of being ahead).

So why would the attacker just give up and try later?

It is but then the attacker would operate differently.  Since there is no cost he would simply support both chains.  Yes he is working against himself but with a minority of the hashrate the probability of producing a longer chain of x blocks was already <50%.  Still the odds are 0% and he isn't penalized because he is supporting the "main" chain (chain most likely to remain the longest).  So he loses nothing, gets no penalty and some % of the time will be able to out mine/mint the main chain and perform a double spend at no cost.  Granted the odds may be low (they are low for Bitcoin as well) if his share of the stake is low but say you got away with a double spend once a year with no cost or risk.  Other than ethics why wouldn't you try?  Attempt 10,000 double spends and if 9,999 are unsucessful and 1 is the fact that there was no cost means you are ahead.


I have no idea what you mean by "any block".  An attack can come from any block.  An attacker can continue to extend multiple chains extending from multiple blocks all in parallel and if any of them by "luck" end up longer than the main chain they will be published.  If the attacker has a minority of the effective stake (even when boosted by computing power), the chance of success if low.  That means the chance of penalty is high and it would be optimal to build on the main chain as well.

The same concept applies to selfish mining.  A minter produces a valid block and he will be penalized if he doesn't publish it on the main chain however the block doesn't allow him the chance of forging the next block so he hangs on to it for a while.   If prior to him finding a better block another miner publishes a block at the same height that is inferior then he publishes his block as well and suffers no penalty.  If prior to another miner finding a better block he finds a block which allows him to forge the next block he publishes that instead and then immediately attempts to solve the next block to double his reward.

There would be no penalty.  Lets look at the case of a majority and minority attacker separately.  

An attacker with a minority of the stake will optimally support ALL chains including the main chain.  He will suffer no penalty.  Yes this will (marginally) reduce his chance because it improves the effective stake of the main chain relative to his chain but the chance was already low.  In PoW the real security comes not from the fact that a double spend with a minority of the hashrate is possible.  It is very possible but it will happen infrequently and the COST for all the times it fails offsets any gain from the double spend.    For example an attacker with 10% of the Bitcoin hashrate would be able to successfully double spend a tx with 6 confirmations 1 out of 1694 attempts.   It is impossible to guarantee that an attacker (even one with a minority of the hashrate) can NEVER perform a double spend on a tx 6 confirmations deep however the cost means that if the amount being double spent is <168,000 BTC the attacker ends up losing money.  He will eventually be successful however the gain from the success will be less than the loss of coins not mined.  

With PoS, a minority attacker WILL support the main chain when required and thus he will never suffer a penalty.  His chance of success will be low (and will be lower the more confirmations that are required) but with no (real) cost and a chance >0% he will eventually pull off a double spend at no cost.  There is nothing at risk precisely because he can support all chains in parallel simultaneously.  Note: a PoS system w/ a penalty marginally reduces the profitability of the attacker when the attacker has a minority of the hashrate but it doesn't remove the nothing at stake risk because the attacker can still support all chains (he probably would anyways even w/ no penalty to collect the block rewards on the main chain).

An attacker with a majority of the stake will optimally only extend the attack chain(s) and not the main chain because he doesn't need to.  You can penalize him all you want on the main chain but once one of the attack chains are longer then it will be published, the network will reorg to use that chain as the "best chain" and all the "penalties" will be limited to an orphaned fork.  Is there an effective penalty if all the penalties are limited to a chain which has been orphaned?  The irony is that after the attack chain is published all the honest miners who did not complete blocks when required will be subject to a penalty and unable to mine for 2 days.

I revised the post to be more clear.  It is important to consider the optimal behavior for both an attacker with a minority and majority of the stake.   In neither case will there be an effective penalty although how the attacker proceeds will vary (it does in Bitcoin as well for different reasons).

No that isn't the penalty.  The "penalty" is if a miner doesn't mint a block when he can then he is prevented from minting future blocks for a period of 2 days ON THAT CHAIN.

"supporting the main chain" = minting blocks on that chain = no penalty = doesn't prevent the attacker from simultaneously working to extend one ore more candidate attack chains at the same time.