  The Age of Incorrect Information (AoII) is a recently proposed metric for
real-time remote monitoring systems. In particular, AoII measures the time the
information at the monitor is incorrect, weighted by the magnitude of this
incorrectness, thereby combining the notions of freshness and distortion. This
paper addresses the definition of an AoII-optimal transmission policy in a
discrete-time communication scheme with a resource constraint and a hybrid
automatic repeat request (HARQ) protocol. Considering an $N$-ary symmetric
Markov source, the problem is formulated as an infinite-horizon average-cost
constrained Markov decision process (CMDP). The source model is characterized
by the cardinality of the state space and the probability of staying at the
same state. Interestingly, it is proved that under some conditions, the optimal
transmission policy is to never transmit. This reveals that there exists a
region of the source dynamics where communication is inadequate in reducing the
AoII. Elsewhere, there exists an optimal transmission policy, which is a
randomized mixture of two discrete threshold-based policies that randomize at
one state. The optimal threshold and the randomization component are derived
analytically. Numerical results illustrate the impact of source dynamics,
channel conditions, and resource constraints on the average AoII.
