  By the Gibbard--Satterthwaite theorem, every reasonable voting rule for three
or more alternatives is susceptible to manipulation: there exist elections
where one or more voters can change the election outcome in their favour by
unilaterally modifying their vote. When a given election admits several such
voters, strategic voting becomes a game among potential manipulators: a
manipulative vote that leads to a better outcome when other voters are truthful
may lead to disastrous results when other voters choose to manipulate as well.
We consider this situation from the perspective of a boundedly rational voter,
and use the cognitive hierarchy framework to identify good strategies. We then
investigate the associated algorithmic questions under the k-approval voting
rule. We obtain positive algorithmic results for k=1 and 2, and NP- and
coNP-hardness results for k>3.
