Games of Timing and Random Walks on Interval

We consider a timing game, whose payoff function depends on the difference of arguments-strategies and, possibly, has break lines outside the diagonal of the square. It is shown that the solution of such game is closely related to a random walk on the unit segment with two absorbing screens. It is proved that the value of the game is inversely proportional to the time the point is on the segment, and the distribution function of the player’s optimal equalizing strategy is proportional to the time the point is on the corresponding segment. As an example, a game with a piecewise constant payoff function is considered: advertising campaign game.