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Animals are associated with a microbiome that can affect their reproductive success. It is, therefore, important to understand how a host and its microbiome coevolve. According to the hologenome concept, hosts and their microbiome form an integrated evolutionary entity, a holobiont, on which selection can potentially act directly. However, this view is controversial, and there is an active debate on whether the association between hosts and their microbiomes is strong enough to allow for selection at the holobiont level. Much of this debate is based on verbal arguments, but a quantitative framework is needed to investigate the conditions under which selection can act at the holobiont level. Here, we use multilevel selection theory to develop such a framework. We found that selection at the holobiont level can in principle favor a trait that is costly to the microbes but that provides a benefit to the host. However, such …