The following class of games is considered: a sequence chooser produces an infinite sequence of 0's and 1's, and a predictor observes the sequence for a finite time, stopping when he pleases and choosing an action from a finite set. The predictor wins an amount depending only on the action chosen and on the first two unobserved terms of the sequence. The value of such games is determined and the formula obtained is used to give a derivation of the well-known values of the 1-0 game and the two-move lag bomber-battleship game. Values for some generalizations of the latter game are given. Optimal strategies for the sequence chooser are discussed.