This note studies a Poisson arrival selection problem for the full-information case with an unknown intensity lambda which has a Gamma prior density G(r, 1/a), where a > 0 and r is a natural number. For the no-information case with the same setting, the problem is monotone and the one-step look-ahead rule is an optimal stopping rule; in contrast, this note proves that the full-information case is not a monotone stopping problem.