 There exists an alternative, game-theoretic way of interpreting these connective results. Suppose the game under study is fixed, and associated with some sort of ‘game board’ representing major features of the game’s general state (think of Chess, though more abstract game boards may occur). Then the above equivalences suggest that winning strategies, i.e., a typical game-theoretic notion defined in terms of the complete extensive game tree, is equivalent to a simpler ‘invariant’ that can be defined entirely in terms of some game board associated with the tree’s nodes. Identifying useful such invariants is a well-known art in the analysis of concrete games. In terms of a main theme of this entry, invariants can live at different levels of representation associated with a given class of games.