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Some applications of game determinacy. (English) Zbl 0889.90177

Summary: Through a sample of three examples we show how game determinacy can be used in problems where games are not involved a priori. We hope that the variety of these examples will convince the reader in the interest of such a procedure. The recipe is the following: If you are interested in proving some statement of the form “\((A)\Rightarrow (B)\)”, introduce some game \(G\) with the following properties: (1) If Player I has a winning strategy in \(G\) then (non \(A\)) holds. (2) If Player II has a winning strategy in \(G\) then \((B)\) holds.
Then “\((A)\Rightarrow (B)\)” is equivalent to the determinacy of the game \(G\). Of course the recipe does not give you any indication how to invent the game \(G\)…On the other hand not all games are determined. But since as we shall see “Borel games” are determined, this procedure will be more successful if you deal with “nice” properties of Borel sets. However as we shall see in one of the examples “natural” properties of Borel sets might create non Borel games. We shall also describe the main classical trick to produce Borel games, and even closed games, when dealing with analytic sets. But more than a nice approach for solving concrete mathematical problems, we shall show how the proof of some result using a closed game has interesting descriptive consequences.

MSC:

91A44 Games involving topology, set theory, or logic
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