  The Generalized Second Price (GSP) auction used typically to model sponsored
search auctions does not include the notion of budget constraints, which is
present in practice. Motivated by this, we introduce the different variants of
GSP auctions that take budgets into account in natural ways. We examine their
stability by focusing on the existence of Nash equilibria and envy-free
assignments. We highlight the differences between these mechanisms and find
that only some of them exhibit both notions of stability. This shows the
importance of carefully picking the right mechanism to ensure stable outcomes
in the presence of budgets
