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A general theory of stochastic extensive form games reconciling two concepts of exogenous information – those of “nature” in classical game theory and filtrations in stochastic analysis – is constructed. Here, “nature” does not take decisions but rather behave as a one-shot lottery equipped with a dynamically updating oracle, and “personal” players take decisions in an extensive form adapted to that oracle. This requires to substantially modify the existing theory. I introduce, discuss and analyse the notions of stochastic game forests, exogenous information structures and adapted choices. The notion of random moves is introduced and, based on this, subgame-perfect equilibria can be formulated. The generality of the theory is illustrated via several examples, including stochastic differential games, timing games in continuous time, and games with asymmetric information.
