We provide the first theoretical analysis of altruism in networks. Agents are embedded in a fixed, weighted network and care about their direct friends. Given some initial distribution of incomes, they may decide to support their poorer friends. We study the resulting non-cooperative transfer game. Our analysis highlights the importance of indirect gifts, where an agent gives to a friend because his friend himself has a friend in need. We uncover four main features of this interdependence. First, we show that there is a unique profile of incomes after transfers, for any network and any utility functions. Uniqueness in transfers holds on trees, but not on arbitrary networks. Second, there is no waste in transfers in equilibrium. In particular, transfers flow through indirect paths of highest altruistic strength. Third, a negative shock on one agent cannot benefit others and tends to affect socially closer agents first. In addition, an income redistribution that decreases inequality ex-ante can increase inequality ex-post. Fourth, altruistic networks decrease income inequality. In contrast, more altruistic or more homophilous networks can increase inequality.
