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The non-emptiness of cores of voting games not only depends on the dominance relation, but also on the properties of individual preferences. In this paper, we enrich the panorama of existing non-emptiness results (for core concepts and the stability set) by providing their counter-parts when one moves from monotonic voting games to non monotonic ones; or from the assumption of individual preferences as linear orders to that of those preferences as weak orders. We mainly show how these two types of movements modify the three characteristic numbers associated with theorems on refinements of the core and the 1-stability set of voting games, namely the Nakaruma number (Nakamura [1979]), the Andjiga-Mbih number (Andjiga & Mbih [2000]) and the Andjiga-Moyouwou number (Andjiga & Moyouwou [2006]).
