  We give a bijection between permutations of length 2n and certain pairs of
Dyck paths with labels on the down steps. The bijection arises from a game in
which two players alternate selecting from a set of 2n items: the permutation
encodes the players' preference ordering of the items, and the Dyck paths
encode the order in which items are selected under optimal play. We enumerate
permutations by certain statistics, AA inversions and BB inversions, which have
natural interpretations in terms of the game. We give new proofs of classical
identities such as \sum_p \prod_{i=1}^n q^{h_i -1} [h_i]_q = [1]_q [3]_q ...
[2n-1]_q where the sum is over all Dyck paths p of length 2n, and the h_i are
the heights of the down steps of p.
