  We study a social choice setting of manipulation in elections and extend the
usual model in two major ways: first, instead of considering a single
manipulating agent, in our setting there are several, possibly competing ones;
second, instead of evaluating an election after the first manipulative action,
we allow several back-and-forth rounds to take place. We show that in certain
situations, such as in elections with only a few candidates, optimal strategies
for each of the manipulating agents can be computed efficiently.
  Our algorithmic results rely on formulating the problem of finding an optimal
strategy as sentences of Presburger arithmetic that are short and only involve
small coefficients, which we show is fixed-parameter tractable -- indeed, one
of our contributions is a general result regarding fixed-parameter tractability
of Presburger arithmetic that might be useful in other settings. Following our
general theorem, we design quite general algorithms; in particular, we describe
how to design efficient algorithms for various settings, including settings in
which we model diffusion of opinions in a social network, complex budgeting
schemes available to the manipulating agents, and various realistic
restrictions on adversary actions.
