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We introduce a new mathematical representation of an extensive game situation, called an information protocol, without using the hypothetical underlying structure of nodes and branches. Its necessity has been emerging in our study of inductive game theory. It has two main differences from a standard extensive game: one is the use of information pieces (symbolic expressions) rather than information sets, and the other is the replacement of a game tree by a causal relation. We will give a set of axioms to show that our new formulation is equivalent to an extensive game. Also, by deleting some axioms, we can capture some weaker forms of extensive games, which are crucial to describing inductive game theory. Some theoretical results for inductive game theory become drastically simplified in the present formulation relative to previous formulations by the authors relying on extensive games.