user51857: How about a scenario where a few miners change their protocols by setting difficulty to something easy. Let's take it to the extreme and say they are able to accept any block. 
They would keep publishing blocks which they think are valid and those would get rejected by the network. What if more miners start joining that scheme though (by changing their protocols)? 
Because they are able to publish blocks extremely quickly (because they set their difficulty low) they might actually be able to compete with the whole network and their branch might overtake the network branch even if they are not that many of them.
What am I missing here?

user48884: Contrary to popular belief, the longer blockchain is not the one with the most blocks. Rather it is the blockchain which has the most proof of work. Each block has a field for indicating the difficulty (i.e. work) for that block. The longest blockchain is the one that has the largest sum of these difficulty fields. The only way for a fork to overtake the current blockchain and replace it with the fork is if the miners on the fork have equal to or greater hashrate than the miners on the current blockchain. Otherwise the miners on that fork won't be doing as much work as the miners on the current blockchain so they can never overtake the blockchain.

user11253: The rest of the network, which uses the existing difficulty parameters, will reject the new blocks generated by your modified miners. There will be a longer blockchain, but there will not be a longer valid blockchain according to the established rules.

51857: <p>How about a scenario where a few miners change their protocols by setting difficulty to something easy. Let's take it to the extreme and say they are able to accept any block. </p>

<p>They would keep publishing blocks which they think are valid and those would get rejected by the network. What if more miners start joining that scheme though (by changing their protocols)? </p>

<p>Because they are able to publish blocks extremely quickly (because they set their difficulty low) they might actually be able to compete with the whole network and their branch might overtake the network branch even if they are not that many of them.</p>

<p>What am I missing here?</p>
 48884: <p>Contrary to popular belief, the longer blockchain is not the one with the most blocks. Rather it is the blockchain which has the most proof of work. Each block has a field for indicating the difficulty (i.e. work) for that block. The longest blockchain is the one that has the largest sum of these difficulty fields. The only way for a fork to overtake the current blockchain and replace it with the fork is if the miners on the fork have equal to or greater hashrate than the miners on the current blockchain. Otherwise the miners on that fork won't be doing as much work as the miners on the current blockchain so they can never overtake the blockchain.</p>
 11253: <p>The rest of the network, which uses the existing difficulty parameters, will reject the new blocks generated by your modified miners. There will be a longer blockchain, but there will not be a longer <em>valid</em> blockchain according to the established rules.</p>
