In this paper, we show how Ehrenfeucht-Fraïssé games can be successfully exploited to compare (finite) strings. More precisely, we give necessary and sufficient conditions for Spoiler/Duplicator to win games played on finite structures with a limited order relation, that lies in between the successor relation and the usual (linear) order relation, and a finite number of unary predicates. On the basis of such conditions, we outline a polynomial (in the size of the input strings) algorithm to compute the "remoteness" of a game and to determine the optimal strategies/moves for both players.