Many procedures have been proposed in the literature to select the simulated alternative with the best mean performance from a finite set of alternatives. Among these procedures, frequentist procedures are typically designed under either the subset-selection (SS) formulation or the indifference-zone (IZ) formulation. Both formulations may encounter problems when the goal is to select the unique best alternative for any configuration of the means. In particular, SS procedures may return a subset that contains more than one alternative, and IZ procedures hinge on the relationship between the chosen IZ parameter and the true mean differences that is unknown to decision makers a priori. In this paper, we propose a new formulation that guarantees to select the unique best alternative with a user-specified probability of correct selection (PCS), as long as the means of alternatives are unique, and we design a class of fully sequential procedures under this formulation. These procedures are parameterized by the PCS value only, and their continuation boundaries are determined based on the Law of the Iterated Logarithm. Furthermore, we show that users can add a stopping criterion to these procedures to convert them into IZ procedures, and we argue that these procedures have several advantages over existing IZ procedures. Lastly, we conduct an extensive numerical study to show the performance of our procedures and compare their performance to existing procedures.