What's the psychology behind parents who brag about their children's achievements to everyone

but to their kids' faces they act indifferent or even criticize them (for example "you could have done better")?

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Survival mechanisms. They praise them to others because they are proud, but don't want to inflate the egos of the child and also want to harden them to the real world of adulthood.

Not that I agree with it, but I understand the logic.