# Stéphane Le Roux

**Talk on 15/7/15 : From winning strategy to Nash equilibrium**

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 usually replaced faithfully with the notion of (pure) Nash

equilibrium. This talk will show that every determinacy result over an

arbitrary 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 or

order-theoretic conditions on the strategy sets and the preferences,

respectively, whereas counter-examples show that every requirement is

relevant, albeit possibly improvable. When the outcomes are finitely

many, the proof provides an algorithm computing a Nash equilibrium

without significant complexity loss compared to the two-outcome

case. As examples of application, this talk generalises Borel

determinacy, positional determinacy of parity games, and finite-memory

determinacy of Muller games. (From an article with the same title in

MLQ 2014.)