Thanx, Gandalf, for your help. I still do not get it and plead guilty of having caused the misunderstanding.
But I still do not get it and would like to work it out.
Let´s make it a bit easier: assume, as in Menis example, that 4000 blocks are found per month by individual participants, under the assumption that your non-parallelizable PoW is in operation. Assume further that all of these people just meet up and decide to share the revenue equally to smoothe out their income stream.
Fine. Still with you. Assuming this.
According to what you have been arguing, the total number of blocks they find would go down to 1 purely due to the fact, that they are colluding in terms of revenue-sharing.
No. Why should the number of blocks go down? I do not claim that they go down.
Maybe the misunderstanding is earlier. A non-parallelizable PoW means that the participants CANNOT collude on the PoW. Of course, they still can share their revenues, that is a completely different issue.
In current parallelizable PoW, all 4000 participants work by looking for a nonce with a specific property. For this goal, they test large numbers of candidate nonces. Every test has a certain success probability (determined by difficulty). Individual tests are, of course, independent of each other. Hence the PoW can be brute forced. This can be done in parallel. A block is found sooner or later, according to the Poisson process in place. So, the time to find a block follows and exponential distribution.
Now let us consider a strictly sequential, deterministic PoW (not as a suggestion for Bitcoin, but to see the difference). Here, a specific computational result must be obtained. To obtain this result, a large number of arithmetic operations must be performed in strict sequence. The number is adapted to the average speed of a single core CPU. The participant who reaches the result first wins the block. This cannot be done in parallel. However, it is always the participant with the fastest single core CPU, who wins the block. This is boring and not what we need. This is time lock cryptography and not exactly useful for Bitcoin.
Now let us consider a non-parallelizable PoW. Here, every participant must make a large number of sequential steps to reach a goal. However, contrary to the sequential, deterministic PoW, there are still random aspects, branching points in the computation. So it still depends on random choices of the participants, who will win the block (which is what we need). Of course, participants can still pool. Two participants will still get twice as many blocks in the long run. The block rate does not go down magically.
However now comes the crucial difference. Assume I have 2^256 participants, numbered 0, 1, 2, 3, ... How long will they need for the first block? In the current (parallelizable) PoW used in Bitcoin they need a few microseconds. Every participant uses his own number as nonce in the first round...and most likely one of them will produce a hash which is smaller than the current target value. In the non-parallelizable PoW I am thinking of, they will still need more or less 10 minutes as they should, since this corresponds more or less to the number of operations they have to do before they get a realistic chance for reaching the goal. However, since there is some variability, also a slower CPU with better random choices gets a chance.