  We extend Approval voting to the settings where voters may have intransitive
preferences. The major obstacle to applying Approval voting in these settings
is that voters are not able to clearly determine who they should approve or
disapprove, due to the intransitivity of their preferences. An approach to
address this issue is to apply tournament solutions to help voters make the
decision. We study a class of voting systems where first each voter casts a
vote defined as a tournament, then a well-defined tournament solution is
applied to select the candidates who are assumed to be approved by the voter.
Winners are the ones receiving the most approvals. We study axiomatic
properties of this class of voting systems and complexity of control and
bribery problems for these voting systems.
