Infinite Games

Innnite games are widely used in mathematical logic 2, 8, 12]. In particular, innnite games proved to be a useful tool in dealing with the monadic second-order theories of innnite strings and innnite trees 3, 4, 7]. Recently 1, 15, 13], innnite games were used in connection to concurrent computational processes that do not necessarily terminate. For example, an operating system may be seen as playing a game \against" the disruptive forces of users. The classical question of the existence of winning strategies turns out to be of importance to practice. Here we attempt to explain basics of innnite game theory. Quisani: What is an innnite game? Author: In the simplest form, there are two players, Player 1 and Player 2. Player 1 starts by choosing a binary bit a 1 , then Player 2 chooses a binary bit a 2 , then Player 1 chooses a binary bit a 3 , then Player 2 chooses a binary bit a 4 , and so on ad innnitum. If the resulting innnite string X (i.e. the function X(i) = a i on positive integers, the play) belongs to an a priori xed set W of innnite strings then Player 2 wins the game; otherwise Player 1 does. Q: Do you mean that diierent sets W give diierent games? A: Yes. The set W deenes the goals of the players. Let W 1 be the complement of W and W 2 = W. Then the goal of Player " is to ensure that the play belongs to his winning set W ". Q: I guess, your simplest form is a little too simple. As one of those disruptive users, I want more than 2 keys on my keyboard. A: You are right. Let me describe a more general setting. We are given an innnite countable tree A called the arena and a set W of branches of A. Nodes of A are possible positions of the game ?(A; W); the root is the initial position. Player 1 begins by choosing a child x 1 of the root, then Player 2 chooses a child x 2 of x 1 , and so on. If