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So you position seems to be that they exist so they must be useful? That seems like such a weak argument that I must have mis-understood you. Anyway, I don't deny that they are useful. Game theory has its place, but only a fool would base decisions on its results without reference to the real world. Statistics are clearly essential to any meaningful analysis and it's unfortunate that you think that I wish to dismiss it.
I would make the inverse argument. If you can't provide a formal argument for why your theory works, there's something wrong with your theory. Game theory doesn't force you to make strict rationality assumptions. It does force you to demonstrate how each part of your story relate. I'd be weary of any statistical or experimental result that can't be modeled properly. After all, even experiments have margins of error, which are made worse by likely measurement or other errors.
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I sympathise - science journalism does a terrible job of explaining the stories of the day to the layman, and I expect that the same is true for economics. I don't doubt that my understanding of the subtleties of your field is abominably poor. From what you've said, it seems that the ideological disputes are what I have a problem with. The various policies are laid out like treatments for a sick man, with each faction having their own preferred remedy. Each is confident that their cure is the best, but no real evidence is ever presented, only rhetoric. The factions seem to trot out the same 'treatments' no matter the problem that they claim to be solving. They resemble bloodletters and homeopaths more than any rational discipline, and should be treated accordingly.
For a clinical equivalent, imagine every person takes 20 different drugs. You tell that person to take another drug for problem x, and you are confident that the drug helps with problem x. But you have only a vague idea about how that drug interacts with the other 20 drugs, or whether it might even do more harm than good given the risk the other 20 drugs pose in terms of uncertainty. That's basically economics. Not my field by the way; I do poli sci.