  Acyclicity of individual preferences is a minimal assumption in social choice
theory. We replace that assumption by the direct assumption that preferences
have maximal elements on a fixed agenda. We show that the core of a simple game
is nonempty for all profiles of such preferences if and only if the number of
alternatives in the agenda is less than the Nakamura number of the game. The
same is true if we replace the core by the core without majority
dissatisfaction, obtained by deleting from the agenda all the alternatives that
are non-maximal for all players in a winning coalition. Unlike the core, the
core without majority dissatisfaction depends only on the players' sets of
maximal elements and is included in the union of such sets. A result for an
extended framework gives another sense in which the core without majority
dissatisfaction behaves better than the core.
