  The game of best choice (also known as the secretary problem) is a model for
sequential decision making with a long history and many variations. The
classical setup assumes that the sequence of candidate rankings are uniformly
distributed. Given a statistic on the symmetric group, one can instead weight
each permutation according to an exponential function in the statistic. We play
the game of best choice on the Ewens and Mallows distributions that are
obtained in this way from the number of left-to-right maxima and number of
inversions in the permutation, respectively. For each of these, we give the
optimal strategy and probability of winning. Moreover, we introduce a general
class of permutation statistics that always produces games of best choice whose
optimal strategies are positional, which simplifies their analysis
considerably.
