Refinement (sigma Algebra)
Refinement may refer to: Mathematics * Equilibrium refinement, the identification of actualized equilibria in game theory * Refinement of an equivalence relation, in mathematics ** Refinement (topology), the refinement of an open cover in mathematical topology * Refinement (category theory) Other uses * Refinement (computing), computer science approaches for designing correct computer programs and enabling their formal verification * Refining, a process of purification ** Refining (metallurgy) * Refinement (culture), a quality of cultural sophistication * Refinement (horse), a racehorse ridden by jockey Tony McCoy Sir Anthony Peter McCoy (born 4 May 1974), commonly known as AP McCoy or Tony McCoy, is a Northern Irish former National Hunt horse racing jockey. Based in Ireland and Britain, McCoy rode a record 4,358 winners and was Champion Jockey a recor ...

Equilibrium Refinement
In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are Economic equilibrium, equilibrium concepts, most famously Nash equilibrium. Many solution concepts, for many games, will result in more than one solution. This puts any one of the solutions in doubt, so a game theorist may apply a refinement to narrow down the solutions. Each successive solution concept presented in the following improves on its predecessor by eliminating implausible equilibria in richer games. Formal definition Let \Gamma be the class of all games and, for each game G \in \Gamma, let S_G be the set of strategy profiles of G. A ''solution concept'' is an element of the direct product \Pi_2^; ''i.e''., a function F: \Gamma \rightarrow \bigcup\nolimits_ 2^ such that F(G) \subseteq S_G for al ...
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