Skaperdas (1996) characterized the Tullock (1980)-Hirschleifer (1989) contest success functions (CSFs), which stipulate the winning probabilities of the contestants, using respectively the scale invariance and translation invariance axioms. This paper first characterizes the entire family of CSFs that fulfils an μ-independence axiom, a convex mixture of the two invariance axioms, where 0 ≤μ≤ 1 is a value judgment parameter. This family contains the Tullock and Hirschleifer CSFs as special cases. Next, we consider two axioms related to ranking, scale consistency and translation consistency, and characterize the respective classes of CSFs. It has been demonstrated that scale consistency and translation consistency, in the presence of other axioms, characterize the same functional forms identified by scale and translation invariances respectively. Finally, we define an intermediate μ-consistency condition and classify all CSFs satisfying the same. We also explore the possibility of existence of Nash equilibrium of the contest game satisfying μ-independence and the corresponding equilibrium efforts.