Positional voting methods satisfying the weak mutual majority and Condorcet loser principles

We consider a voting problem, where the personal preferences of electors are defined by the ranked lists of the candidates. For the social choice functions we propose positional domination principles (WPD, PD), which is closely related to the scoring rules. Also we formulate a weak mutual majority principle (WMM), which is stronger than the majority principle, but weaker than the mutual majority principle (MM). We construct two modifications for the positional median rule, which satisfy the Condorcet loser principle. The WPD and WMM principles are shown to be fulfilled for the first modification, and the PD and MM principles for the second modification. We prove that there is no rule to satisfy both WPD and MM principles.