One way for agents to reach a joint decision is to vote over the alternatives. In open, anonymous settings such as the Internet, an agent can vote more than once without being detected. A voting rule is false-name-proof if no agent ever benefits from casting additional votes. Previous work has shown that all false-name-proof voting rules are unresponsive to agents' preferences. However, that work implicitly assumes that casting additional votes is costless. In this paper, we consider what happens if there is a cost to casting additional votes. We characterize the optimal (most responsive) false-name-proofwith-costs voting rule for 2 alternatives. In sharp contrast to the costless setting, we prove that as the voting population grows larger, the probability that this rule selects the major