  We generalize the 1/3-2/3 conjecture from partially ordered sets to
antimatroids: we conjecture that any antimatroid has a pair of elements x,y
such that x has probability between 1/3 and 2/3 of appearing earlier than y in
a uniformly random basic word of the antimatroid. We prove the conjecture for
antimatroids of convex dimension two (the antimatroid-theoretic analogue of
partial orders of width two), for antimatroids of height two, for antimatroids
with an independent element, and for the perfect elimination antimatroids and
node search antimatroids of several classes of graphs. A computer search shows
that the conjecture is true for all antimatroids with at most six elements.
