This algorithm will prevent cheating if one of the 3 nodes is bad, or no-2 nodes are in coordinated cheating.
If two out of 3 nodes are from the same cheater or same cheating organization, then you can't do anything with p2p decentralized trustless system. Use a centralized trust system instead.
Of course you can. Your consensus system can be designed around consensus, this is the very nature of Bitcoin - if 2 nodes cheat on the Bitcoin network, they get rejected as outliers.
You've literally described a system that
requires you to trust that there's not collusion between 2 parties, that is
not a trustless system, which is a twist of irony given that you called that other piece of junk out as not being trustless. There is no Byzantine fault tolerance with this system, which is the very problem Bitcoin solves. You're trying to solve this problem:
http://en.wikipedia.org/wiki/Two_Generals'_Problem - but the described system does not do so.