Borel Sets Via Games

Down the Borel hierarchy: Solving Muller games via safety games

We transform a Muller game with n vertices into a safety game with (n!)3 vertices whose solution allows to determine the winning regions of the Muller game and to compute a finite-state winning strategy for one player. This yields a novel antichain-based memory structure and a natural notion of permissive strategies for Muller games. Moreover, we generalize our construction by presenting a new ...

A Course on Borel Sets

Stochastic Games with Borel Payoffs

Thus H∞ = (A×B × S)× (A×B × S)× . . . . Let f be a bounded Borel-measurable function defined on H∞, where the sets A,B, and S are given the discrete topology and H∞ is given the product topology. The stochastic game S(f)(z) begins at state z ∈ S, player 1 chooses a strategy σ, player 2 chooses a strategy τ , and then 2 pays 1 the expected value of f , namely Ez,σ,τf . Theorem 1 (Martin [5]) The...

Borel Determinacy of Concurrent Games

Just as traditional games can be represented by trees, so concurrent games can be represented by event structures. We show the determinacy of such concurrent games with Borel sets of configurations as winning conditions, provided they are race-free and bounded-concurrent. Both properties are shown necessary. The determinacy proof proceeds via a reduction to the determinacy of tree games, and th...

Borel Sets with Large Squares

For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ: if a Borel set B ⊆ R × R contains a μ-square (i.e. a set of the form A × A, |A| = μ) then it contains a 20 -square and even a perfect square. And also for ⊗′μ: if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute...