We analyze the on-line chain partitioning problem as a two-person game
. One person builds an order one point at a time. The other person res
ponds by making an irrevocable assignment of the new point to a chain
of a chain partition. Kierstead gave a strategy showing that width k o
rders can be on-line chain partitioned into (5(k) - 1)/4 chains. We fi
rst prove that width two orders can be partitioned on-line into 5 chai
ns. Secondly, we introduce a variant of the game. We impose the restri
ction that the new point presented by the first player has to be a max
imum element in the present order. For this up-growing variant we prov
e matching upper and lower bounds of ((k+1)(2)) on orders of width k.