  In this paper we investigate stationary probability vectors of higher-order
two-dimensional symmetric transition probability tensors. We show that there
are two special symmetric transition probability tensors of order $m$ dimension
2, which have and only have two stationary probability vectors; and any other
symmetric transition probability tensor of order $m$ dimension 2 has a unique
stationary probability vector. As a byproduct, we obtain that any symmetric
transition probability tensor of order $m$ dimension 2 has a unique positive
stationary probability vector; and that any symmetric irreducible transition
probability tensor of order $m$ dimension 2 has a unique stationary probability
vector.
