I have just seen the following sentence in a prestigious grammar book.
We use might ( not ‘may’) + infinitive to talk about what was typically the case in the past:
-During the war, the police might arrest you for criticising the king
I am not too sure about the 'typically'. Shouldn't we say:
During the war, the police could arrest you ...
(or, even better:)
During the war, you could be arrested for criticising the king
You're almost there. Might has a lot of optionality in it - could doesn't cover might, and is very similar to may(both are about possibilities). May is permissive, might is too, but is also passively sufferable.
Might in the present means it is an open possibility. Your first example is set in the past meaning wise with "During the war", since there's no war publicly known or present in current days - 2016 to be precise.
So with during the war, the sentence's meaning is thrown back to a war of past times.
The other way to express it is: During the war, you could have been arrested by the police for criticising the king.
It's a strange way of expressing intent, but is valid.
During the war, the police might arrest you for criticising the king.
I can do best by quoting from The English verb, by F. R. Palmer (2nd edition):
Might is also used, though mainly in literary contexts, to refer to habitual activity:
In those days we might go for a walk in the woods.
Again, could is a possibility here.
So. yes, might has the sense of what was typically the case in the past and it can be replaced by could. As Palmer says, the usage is by and large restricted to written English. In fact, I didn't recognize it as a genuine use of might till I read it in Palmer. I would expect some other construction to show that past time was being referred to, such as might have arrested/might have gone, and that's how I would use might here in spoken English.
Palmer has a hard time categorizing this usage of might and suggests it may be either "a special dynamic use of might, unless it can in some way be interpreted in terms of existential 'sometimes'".
An exampe of existential 'sometimes' is
Lions can be dangerous
which can mean
Sometimes Lions are dangerous.
In general, the use of might to refer to past typical or habitual activity is just one way that English speakers use modals in ways that do not fit neatly into a category: might as present possibility (He might be in his office), might as a tentative form of epistemic may (Might I borrow a pen?), etc.
The words Might and Could carry two different connotations. Might has a connotation of possibility, usually of likelihood. Could has a connotation of permissibility and available options. Might would be better used to describe what is typically possible because could only describes an action as an available option, not carrying any connotation of likelihood or chance. Just because someone can DO something doesn't mean they will. Might implies that the action in question has possibility, and that it's not just an option, but one that might happen.
Thus, when describing an action that is possible and likely, might is a better word to use than could.
