The contest success function (CSF) is a mapping from contestants' efforts to the probabilistic allocation. An axiomatization of the CSFs with luck is provided. It extends the classic axiomatization of Tullock CSF and connects to works of CSF with draws. The central axiom is homogeneous relative externality, which restricts the relative externality from one contestant's effort to others' probabilistic allocation to be scale-invariant. More specifications on parameters are also discussed.
Haoran Yu (2025) studied this question.