I think the circular logic might make them implode, rendering Patronus 2.0 unnecessary.

"I expect the Dementors to play poker with me because I expect the Dementors to play poker with me because I expect..."

But if you were weak-minded enough (or strong-minded enough?) to trick yourself into genuinely believing that would work, I suppose it would. But in that case, what exactly is the limit of their power? Can you expect a Dementor to turn into an all-powerful friendly AI?

It should be possible to test whether or not the mental gymnastics are even possible by getting rationalists to make homeopathy work by believing in it. Then again, it might be harder to expect that a cloaked monster that eats happiness will play poker than to believe that pills packaged just like normal pills will cause symptom relief when you're sick. So you might need to get someone who didn't know about dementors and tell them that they play poker. Be sure to tell it with a straight face. In a textbook might work best.

As Locke says, the question is how much they're capable of doing just because you expect it. Can they break the laws of physics? That sounds like an awesome thing to test.