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1.4.3 Nash Equilibrium, Dominant Strateg... > 1.4.3 Nash Equilibrium, Dominant Str... - Pg. 23

1.4 Some Fundamental Notions of Game Theory 23 This characterization, formalized by Nash (1950), indicates that Nash equilibria are solutions to a fixed point problem. This explains why standard existence theo- rems are based on topological and geometrical assumptions (see Chapter 2 for more details). In the two definitions above, s represents a strategy profile in the broad sense. For instance, it may be a vector of actions, a vector of probability distributions, or a vector of functions. When profiles of pure actions (resp. mixed strategies) are involved, the Nash equilibrium is said to be a pure (resp. mixed) Nash equilibrium. A special type of pure/mixed Nash equilibrium is the dominant strategy equilibrium, which is defined as follows: ·························································································· · · · · · · · If each player has a dominant strategy, and each player plays the dominant strategy, · · then that combination of (dominant) strategies and the corresponding utilities are said · · · to constitute the dominant strategy equilibrium for that game. · Definition 29: Dominant Strategy Equilibrium As a comment, it can be seen that a mixed Nash equilibrium of a strategic form game corresponds to a pure Nash equilibrium of its mixed extension. By definition (see Def. 19), mixed strategies correspond to independent randomizations; that is