>>12636458>Identifying a game with its initial position, it is completely described by the sets of left and right options, each of which is another game. This leads to the recursive Definition 2.1 (1). Note that the sets L and R of options may well be infinite or empty. The Descending Game Condition (2) simply says that every game must eventually come to an end no matter how it is played; the number of moves until the end can usually not be bounded uniformly in terms of the game only.Definition 2.1 (Game). (1) Let L and R be two sets of games. Then the ordered pair G := (L, R) is a game.
(2) (Descending Game Condition (DGC)). There is no infinite sequence of games
G^i = (L^i, R^i) with G^i+1 ? L^i ? R^i for all i ? N.
Logically speaking, this recursive definition does not tell you what games are, and it does not need to: it only needs to specify the axiomatic properties of games. A major purpose of this paper is of course to explain the meaning of the theory; see for example the creation of games below.
Definition 2.2 (Options and Positions).
(1) (Options). The elements of L and R are called left resp. right options of G.
(2) (Positions). The positions of G are G and all the positions of any option of G.
