Smart sanctions are an increasingly popular tool in foreign policy. Countries and international institutions worldwide issue such lists to sanction targeted entities through financial asset freezing, embargoes, and travel restrictions. The relationships between the issuer and the targeted entities in such lists reflect what kind of entities the issuer intends to be against. Thus, analyzing the similarities of sets of targeted entities created by several issuers might pave the way toward understanding the foreign political power structure that influences institutions to take similar actions. In the current paper, by analyzing the smart sanctions lists issued by major countries and international institutions worldwide (a total of 73 countries, 12 international organizations, and 1,700 lists), we identify the hierarchical structure of influence among these institutions that encourages them to take such actions. The Helmholtz--Hodge decomposition is a method that decomposes network flow into a hierarchical gradient component and a loop component and is especially suited for this task. Hence, by performing a Helmholtz--Hodge decomposition of the influence network of these institutions, as constructed from the smart sanctions lists they have issued, we show that meaningful insights about the hierarchical influence structure behind smart sanctions can be obtained.