SUMMARY
Influenza virus can completely escape most antibodies with single mutations. However, rare antibodies broadly neutralize many viral strains. It is unclear how easily influenza virus might escape such antibodies if they became widespread due to therapeutic use or vaccination. Here we map all single amino-acid mutations that increase resistance to broad antibodies targeting an H1 hemagglutinin. Crucially, our approach not only identifies antigenic mutations but also quantifies their effect sizes. All antibodies select mutations, but the effect sizes vary widely. The virus can escape a broad antibody targeting hemagglutinin’s receptor-binding site the same way it escapes narrow strain-specific antibodies: via single mutations with huge effects. In contrast, broad antibodies targeting hemagglutinin’s stalk only select mutations with small effects. Therefore, antibody breadth is not necessarily an indicator of the difficulty of viral escape. Broadly neutralizing antibodies targeting hemagglutinin’s stalk are quantifiably harder to escape than the other antibodies tested here.