Levmore recommends eliminating the liability restrictions on ISPs and forcing them to divulge the identity of IP addresses if subpoenaed. The logic underlying his recommendation is a hypothetical bargain among all users of the internet. The benefit to each person is tiny from being able to post degrading insults about others, but the cost of being a target of these insults is very high. Even if the chance of being targeted is small, the cost is large enough that the expected value outweighs the miniscule benefit. Thus, the bargainers would not immunize such conduct. Levmore focused on differences in expected costs and benefits to each person while implicitly assuming homogeneous preferences, but an alternative formulation could depend on differences in preferences. No one wants to be targeted, but only a minority wants to target others, so the majority demands the minority give up its antisocial behavior.
One obvious problem with this argument is that it makes the wrong comparison. We shouldn't be comparing the benefit of "being able to post degrading insults" with the cost of being a target; we should be comparing the benefit of the anonymity as a rational protection (e.g., on controversial topics) with the cost of being a target of abuse. There are no cases exactly analogous to internet privacy, but consider a loose analogy. Suppose someone were to come to us and say, "We are going to be massively restricting academic freedom because the benefit of abusing those protections is very small, while the cost of being harmed by such abuses are very high." Would anyone be taken in by this argument in the slightest degree, which only considers the abuses of privileges and not the beneficial uses? And there certainly are beneficial uses of internet anonymity. I'm not anonymous, and never have been (never really ever thought to be, actually), but I am benefitted by the fact that many of my online colleagues can use the protection of anonymity, and obviously they often benefit from it.
This is not to say that there should be no restrictions on internet anonymity; there genuinely are serious and worrisome issues involved with abuses. But this argument, as stated, is an obviously bad argument for that claim.
