Bad info vs no info - can Information/Game theory provide an answer? In light of the COVID19 pandemic, myself and some microbiology/bioinformatics colleagues have been discussing the merits (or otherwise) of testing recently.
Specifically, in a recent press conference here in the UK, Dominic Raab stated that

"no test is better than a bad test [for COVID]".

There are, of course, an enormous number of caveats and subtleties to this, as it depends on the way in which the test is bad etc. For instance, a high false positive rate is not all that problematic, but a high false negative rate could be, but I'm not interested in getting in to the weeds on that specifically at this point.
To me, it seems like it might generally be true that any new information, even if the false positive/negative rate is high, is likely to be better than not knowing anything by not testing. Though I'm not a mathematician, this feels a bit like an information/game theory puzzle, in a kind of Monty Hall-esque way.
So, my question is: can information/game theory answer this in general terms?
Is no new information at all preferable to bad information, even if that bad information has some useful info potentially mixed in too?

For the avoidance of doubt, I'm not interested in getting into the political consequences etc. of the decision making, I'm interested purely in whether mathematics could inform that specific aspect of testing strategy (or lack thereof).
I also realise this might be more of a probability question, so if this question fits better on e.g. CrossValidated or some other Stack, feel free to migrate.
 A: The expected utility of your optimal strategy given the bad information is always at least as high as the expected utility of your optimal strategy given no information. The proof of this is simple: in the case where you get the bad information you can always follow the strategy that you would have followed if you had no information.
But this logic doesn't quite apply to the example you gave, because the government's choice isn't strictly between 'bad information' and 'no information', it's between 'bad information given to you and everybody else' and 'no information'. If the government administered an unreliable test and then didn't quarantine or treat the people which the test showed had the virus, then they would be harshly criticized by the media even if their choice was the best one to combat the virus.
EDIT: The idea that you can always behave as though you don't know the information relies on some assumptions, in particular it requires that none of the scenario's rules explicitly refer to your knowledge. For example consider the game where if you don't know the atomic weight of boron then you get a dollar and otherwise you get nothing. In this game there is information that it's better to not have. But this kind of thing only happens if the rules refer to your knowledge explicitly. If the game was instead that you got a dollar if you answered a question about the atomic weight of boron incorrectly, then the knowledge would actually be useful since you could deliberately answer wrongly. So whether or not knowledge can be harmful depends on the class of games we consider.
A: It’s not the case that information is always beneficial. In game theory, it’s sometimes good not to have information, e.g. because this can prevent other players from effectively threatening you.
Here’s my favourite example of information being detrimental:
A group of people are wearing amber or scarlet hats. They can see everyone else’s hat but not their own, and they’re not allowed to communicate (no tricks). They’re tasked with lining up in a row such that all amber hats are together on one side and all scarlet hats on the other.
The solution is to form the line one by one, with each new person stepping into the line at the point between the amber and scarlet hats that are already there.
Now the task is made slightly harder. Two separate groups are to be formed, with all the amber hats in one corner of the room and all the scarlet hats in another corner. They can line up as before, but if they then try to go to their respective corners, the person who got into line last won’t know where to go.
Now here’s the clue: If only they could forget how they got there, they could solve the task. There are actually two people in the middle of the line who can’t deduce from looking at the line which colour their hat is. The reason one of them nevertheless goes to one of the corners is that she remembers on which side of her the last person got in line. If she didn’t have this information, both people in the middle would stay standing while everyone else goes to a corner. Then they could deduce the colour of their hat from the fact that it was the two of them who didn’t know where to go.
