This paper revisits the n -player rent-seeking contest with homogeneous valuations and increasing returns. Our main result says that, for any $$m\in \{2,\ldots ,n-1\}$$ m ∈ { 2 , … , n - 1 } , there are threshold values $$1 1 R ∗ ( m ) R ∗ ( m ) ≤ 2 for the Tullock parameter R such that a pure strategy equilibrium with m active players exists if and only if $$R\in [R_*(m),\,R^*(m)]$$ R ∈ [ R ∗ ( m ) , R ∗ ( m ) ] . Among other things, this observation leads to a simple characterization of the values of R for which the n -player contest has a unique pure strategy equilibrium.