In the lecture given by Jon Gruber at MIT, he talks about the governmental regulations on monopolies, and he states that
for government to regulate a monopoly, it needs to know the price of that good in the case of a perfectly competitive market.However, in reality, the government does not know the demand and supply curve. Nevertheless, to get the demand curve the government can do experiments on the consumer and get the demand curve.
If so, can't the government do experiment on the monopolist ? I mean, it can set a price first, and then change the price. Doing so, observing how much the firm produce, we can learn the supply curve of the monopolist. Of course, this would in some sense be not logical, but in the long run, you the government could set all the monopolies in the optimal price.
Moreover, if we look from a broader picture, this can be accomplished more easily by cooperation of different governments on collecting the data.