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Multiagent decision-making problems in partially observable environments are usually modeled as either extensive-form games (EFGs) within the game theory community or partially observable stochastic games (POSGs) within the reinforcement learning community. While most practical problems can be modeled in both formalisms, the communities using these models are mostly distinct with little sharing of ideas or advances. The last decade has seen dramatic progress in algorithms for EFGs, mainly driven by the challenge problem of poker. We have seen computational techniques achieving super-human performance, some variants of poker are essentially solved, and there are now sound local search algorithms which were previously thought impossible. While the advances have garnered attention, the fundamental advances are not yet understood outside the EFG community. This can be largely explained by the starkly different formalisms between the game theory and reinforcement learning communities and, further, by the unsuitability of the original EFG formalism to make the ideas simple and clear. This paper aims to address these hindrances, by advocating a new unifying formalism, a variant of POSGs, which we call Factored-Observation Games (FOGs). We prove that any timeable perfect-recall EFG can be efficiently modeled as a FOG as well as relating FOGs to other existing formalisms. Additionally, a FOG explicitly identifies the public and private components of observations, which is fundamental to the recent EFG breakthroughs. We conclude by presenting the two building-blocks of these breakthroughs -counterfactual regret minimization and public state decomposition -in the new formalism, illustrating our goal of a simpler path for sharing recent advances between game theory and reinforcement learning community.

We investigate and compare F-Borel classes and absolute F-Borel classes. We provide precise examples distinguishing these two hierarchies. We also show that for separable metrizable spaces, F-Borel classes are automatically absolute.Borel sets (in some topological space X) are the smallest family containing the open sets, which is closed under the operations of taking countable intersections, countable unions and complements (in X). The family of F -Borel (resp. G-Borel ) sets is the smallest family containing all closed (resp. open) sets, which is closed under the operations of taking countable intersections and countable unions of its elements. In metrizable spaces the families of Borel, F -Borel and G-Borel sets coincide. In non-metrizable spaces, open set might not necessarily be a countable union of closed sets, so we need to make a distinction between Borel, F -Borel and G-Borel sets.In the present paper, we investigate absolute F -Borel classes. While Borel classes are absolute by [HS] (see Proposition 1.8 below), by [Tal] it is not the case for F -Borel classes (see Theorem 1.7 below). We develop a method of estimating absolute Borel classes from above, which enables us to compute the exact complexity of Talagrand's examples and to provide further examples by modifying them. Our main results are Theorem 5.14 and Corollary 5.15.The paper is organized as follows: In the rest of the introductory section, we define the Borel, F -Borel and G-Borel hierarchies and recall some basic results. In Section 2, we recall the definitions and basic results concerning compactifications and their ordering. In Theorem 2.3, we show that the complexity of separable metrizable spaces coincides with their absolute complexity (in any of the hierarchies). Section 3 is devoted to showing that absolute complexity is inherited by closed subsets. In Section 4, we study special sets of sequences of integers -trees, sets which extend to closed discrete subsets of ω ω and the 'broom sets' introduced by Talagrand. We study in detail the hierarchy of these sets using the notion of rank. In Section 5, we introduce the class of examples of spaces used by Talagrand. We investigate in detail their absolute complexity and in Section 5.3, we prove our main results.Let us start by defining the basic notions. Throughout the paper, all the spaces will be Tychonoff. For a family of sets C, we will denote by C σ the collection of all countable unions of elements of C and by C δ the collection of all countable intersections of elements of C.Definition 1.1 (Borel classes). Let X be a topological space. We define the Borel multiplicative classes M α (X) and Borel additive classes A α (X), α < ω 1 , as follows:• M 0 (X) = A 0 (X) := the algebra generated by open subsets of X,• M α (X) := β<α (M β (X) ∪ A β (X)) δ for 1 ≤ α < ω 1 ,• A α (X) := β<α (M β (X) ∪ A β (X)) σ

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