We discuss several results concerning on-line algorithms for ordered sets and comparability graphs. The main new result is on the problem of on-line transitive orientation. We view on-line transitive orientation of a comparability graph G as a two-person game. In the ith round of play, 1 i V(G), player A names a graph G_i such that G_i is isomorphic to a subgraph of G, V(G_i) = i, and G_i−1 is an induced subgraph of G_i if i> 1. Player B must respond with a transitive orientation of G_i which extends the transitive orientation given to G_i−1 in the previous round of play. Player A wins if and only if player B fails to give a transitive orientation to G_i for some i, 1 i V(G). Our main result shows that player A has at most three winning moves. We also discuss applications to on-line clique covering of comparability graphs, and we mention some open problems.