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 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.

Let ${\displaystyle \Gamma }$ be the class of all games and, for each game ${\displaystyle G\in \Gamma }$, let ${\displaystyle S_{G}}$ be the set of strategy profiles of ${\displaystyle G}$. A solution concept is an element of the direct product ${\displaystyle \Pi _{G\in \Gamma }2^{S_{G}};}$ i.e., a function ${\displaystyle F:\Gamma \rightarrow \bigcup \nolimits _{G\in \Gamma }2^{S_{G}}}$ such that ${\displaystyle F(G)\subseteq S_{G}}$ for all ${\displaystyle G\in \Gamma .}$

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