We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u+V_n when it visits a particular state 0 at time n, with V_n representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. Typically the probability of an excursion of length n for the underlying Markov chain is taken to decay as a power of n (called the loop exponent), perhaps with a slowly varying correction. A particular case not covered in a number of previous studies is that of loop exponent one, which includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ at least at low temperatures. The work is joint with N. Zygouras.