  We propose a dynamic information manipulation game (DIMG) to investigate the
incentives of an information manipulator (IM) to influence the observation
rules of a partially observable Markov decision process (POMDP). DIMG is a
hierarchical game where the upper-level IM stealthily designs the POMDP's joint
state distributions to influence the lower-level controller's actions. DIMP's
fundamental feature is characterized by a stage-wise constraint that ensures
the consistency between the unobservable marginals of the designed and the
original kernels. In an equilibrium of information distortion, the IM minimizes
cumulative cost that depends on the controller's informationally manipulated
actions generated by the optimal policy to the POMDP. We characterize ex ante
and interim equilibria of information distortion and show their connections.
The IM's impact is characterized by a tight upper-bound on the performance
degradation of the POMDP, which consists of the stage-wise instantaneous
informational differences scaled by the amplifications at subsequent stages.
