1. Buy sth for $100 end sell for $200
2. Buy for $1mln end sell for $1 000 100
Option 1. seems much more accessible.
The opposite of this is true, as you are talking about "centralised" in the context of "a smaller group of people" and not "a smaller cluster of attacking machines". The most sophisticated adversaries are the hardest to deal with.
For example, if there was a pure public good project that could provide random input / output, then it would still work.
Primecoin is an example of that. While finding those specific primes is of extremely minimal value, it would still work even if solving those primes had considerable value - because it would be equally valuable to all.
The issue is economic, not technical. If I can sell the “useful” side effect of mining, then I can subsidize any attack I perform. The actual security of the network is only the work that has non-reimburseable costs.
a) it is economically rational to mine on the correct chain
b) there are substantial economic net costs to mining on a wrong fork of the chain (and these are high enough that it is prohibitively expensive to mount an attack which would fool people as to what chain is correct)
Right?
Finding primes aren't a great example. ... but unfortunately, they are the best example we have so far.
Also, I think those large primes do have economic value (https://en.wikipedia.org/wiki/Largest_known_prime_number#Pri...). I'm not sure if that's the purpose of PrimeCoin or who would get the prize, but they nonetheless have some economic value.
To see this, note that all POW and POS chains have "difficulty" functions that are actually just cost functions in external markets (hashpower or capital). What secures the difficulty of the attack in all casds is the supply curve of some block production factor in an external market.
For example, if it were solving an incredibly complex mathematical or scientific problem that benefits all of mankind - then the incentive is equal for everyone.
If you come up with something useful, but noone is willing to buy it at any price, what's your basis for claiming that it's useful?
It also happens that both these conditions (intellectual product + must be published freely) are the typical approach of PoW blockchains.
https://org.saito.tech/saito-economics
The approach works exactly because money is guaranteed to be pulled away from the block producer. Given that it always costs more money to produce blocks than attackers can earn from them, only honest nodes (who spend other people's money) have am incentive to do so. Cost of attack is the money attackers cannot get back if they spend their own money to censor the network. Check it out!
Others have given the example of PrimeCoin which is asymmetric, but somewhat useful as it discovers prime numbers.
A great metaphor for this is a Sudoku puzzle. It is much harder to solve than it is to check the solution.
If there is a lot of work but one worker is performing over half the work, that worker can compromise the network.
If there are a lot of workers but little work, a new worker who is more capable can compromise the network.