The analytic hierarchy process (AHP) initialed by Saaty (1980) has recently been attracting attention as a useful support method for multi-objective decision making. This paper presents some useful theorems for the sensitivity analysis of priority that is playing an important role in the AHP. These theorems are derived from the principle of hierarchical composition which is expressed in the form of a reachability matrix. Using the theorems, we can easily calculate the degrees of effects caused by local or global changes in the priorities of some criteria, and also we can examine the possibility of rank reversal among alternatives. Furthermore, by applying these theorems to a dwelling selection problem, this paper verifies their effectiveness.