Enhanced indexation is a structured investment approach that combines passive and active financial management techniques. We propose an enhanced indexation model whose goal is to maximize the excess return that can be attained with high reliability, while ensuring that the relative market risk does not exceed a specified limit. We measure the relative risk with the coherent semideviation risk functional and model the asset returns as random variables. We consider that the probability distributions of the index fund and excess returns are imperfectly known and belong to a class of distributions characterized by an ellipsoidal distributional set. We provide a game theoretical formulation for the enhanced indexation problem in which we maximize the minimum excess return over all allowable probability distributions. The variance of the excess return is calculated with a computationally efficient method that avoids model specification issues. Finally, we show that the game theoretical model can be recast as a convex programming problem and discuss the results of numerical experiments.