In this paper, we examine relations among various refinements of a perfect equilibrium. First, we compare two refinements of a perfect equilibrium, a strictly perfect equilibrium (SPE) and a truly perfect equilibrium (TPE). We show that true perfectness implies strict perfectness and even strict properness. We also introduce the concept of a restrictive perfect equilibrium allowing only some or all sequences of totally mixed strategies satisfying a certain property. A proper equilibrium is an example of a restrictive perfect equilibrium. Then, true perfectness imposing the most severe robustness against perturbations implies any restrictive perfectness. This proves that every TPE is proper.
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