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In this paper, we present a contest success function (CSF), which is homogeneous of degree zero and in which the probability of winning the prize depends on the relative difference of efforts. In a simultaneous game with two players, we present a necessary and sufficient condition for the existence of a pure strategy Nash equilibrium. This equilibrium is unique and interior. This condition does not depend on the size of the valuations as in an absolute difference CSF. We prove that several properties of Nash equilibrium with the Tullock CSF still hold in our framework. Finally, we consider the case of players, generalize the previous condition and show that this condition is sufficient for the existence of a unique interior Nash equilibrium in pure strategies. For some parameter values of our CSF and when all players are identical, equilibrium entails full rent dissipation for any number of players.