This paper focuses on the effect of additive contest decomposition on performance: winning probabilities and efforts of the contestants. Our main result provides a sufficient condition for invariance of contest performance to the decomposition of a contest, when the sum of the possibly differently valued prizes in the segmented contests is equal to the value of the prize in the original grand contest and the relative prizes in the sub-contests are equal for every contestant. It is shown that this condition is satisfied by the commonly used exponential logistic contest success functions. With these functions the contest designer does not have an incentive to split the prize and create additive, segmented sub-contests. We then prove that when the additive contest decomposition is asymmetric, contest decomposition may adversely affect the designer; that it, reduce the total efforts of the contestants.