  Prior work on the complexity of bribery assumes that the bribery happens
simultaneously, and that the briber has full knowledge of all votes. However,
in many real-world settings votes come in sequentially, and the briber may have
a use-it-or-lose-it moment to decide whether to alter a given vote, and when
making that decision the briber may not know what votes remaining voters will
cast.
  We introduce a model for, and initiate the study of, bribery in such an
online, sequential setting. We show that even for election systems whose
winner-determination problem is polynomial-time computable, an online,
sequential setting may vastly increase the complexity of bribery, jumping the
problem up to completeness for high levels of the polynomial hierarchy or even
PSPACE. But we also show that for some natural, important election systems,
such a dramatic complexity increase does not occur, and we pinpoint the
complexity of their bribery problems.
