This is rather a theoretical question in order to save the trouble in trying to do empirical testing and is part of a bet, so I hope I am right...
Say there are M classes in the data BUT you want to classify JUST between to subsets of these classes: M1 and M2 (M1+M2 = M). For example all M1 are different types of fraud and all M2 are different types of genuine users. Notice that the type of fraud/user is nuisance information and only the label fraud/genuine is of importance.
One approach would be to use a multi-class classifier and then see if the estimated class is within M1 or M2. An alternative would be to use a binary classifier, disregarding the multiple labels and just using label = 1 for M1 and label = 2 for M2.
Which classifier will work better in the general case? If the answer depends on data distribution, please explain.
p.s. my intuition says that binary will work better: the hypothesis space is smaller, so generalization error is smaller too.