Families of abstract decision problems whose admissible sets intersect in a singleton

An abstract decision problem is an ordered pair where the first component is a nonempty and finite set of alternatives and the second component is an irreflexive relation on that set, called dominance relation. The admissible set of an abstract decision problem is the set of the maximal elements of the reflexive and transitive closure of the dominance relation. Given a finite sequence of abstract decision problems on the same set of alternatives, we give conditions on the dominance relations that guarantee that the intersection of all the admissible sets of the considered problems is a singleton as well as conditions that guarantee that the intersection is nonempty. We show then that such results allow to deduce some interesting facts about the resoluteness of the Schulze network solution and the Schulze social choice correspondence as well as some information about the existence of a (unique) common recurrent state for finite families of discrete-time homogeneous Markov chains.