  We propose a notion of alternating bisimulation for strategic abilities under
imperfect information. The bisimulation preserves formulas of ATL$^*$ for both
the {\em objective} and {\em subjective} variants of the state-based semantics
with imperfect information, which are commonly used in the modeling and
verification of multi-agent systems. Furthermore, we apply the theoretical
result to the verification of coercion-resistance in the ThreeBallot voting
system, a voting protocol that does not use cryptography. In particular, we
show that natural simplifications of an initial model of the protocol are in
fact bisimulations of the original model, and therefore satisfy the same
ATL$^*$ properties, including coercion-resistance. These simplifications allow
the model-checking tool MCMAS to terminate on models with a larger number of
voters and candidates, compared with the initial model.
