Mathematics – Logic
Scientific paper
2012-03-08
Mathematics
Logic
15 pages
Scientific paper
Game theory is usually considered applied mathematics, but a few game-theoretic results, such as Borel determinacy, were developed by mathematicians for mathematics in a broad sense. These results usually state determinacy, i.e. the existence of a winning strategy in games that involve two players and two outcomes saying who wins. In a multi-outcome setting, the notion of winning strategy is irrelevant yet faithfully replaced with the notion of (pure) Nash equilibrium. This article shows that every determinacy result over a game structure, e.g. a tree, is transferable into existence of multi-outcome (pure) Nash equilibrium over the same game structure. The equilibrium-transfer theorem requires cardinal and order-theoretic conditions on the strategies and the preferences respectively, whereas counter-examples show that every requirement is important, including the players being two only. As examples of application, this article generalises (by invoking!) Borel determinacy, positional determinacy of parity games, and finite-memory determinacy of M\"uller games.
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