Nash Equilibrium

MAS Definice

A strategy profile in which no player has an incentive to deviate, assuming the strategies of all other players remain unchanged.

Definition

Pure strategy NE

A pure strategy profile $s^{\ast }\in S$ is a Nash equilibrium if for every player $i\in N$ $u_i({\mathbf{s}}^{\ast })\ge u_i(s,{\mathbf{s}}_{-i}^{\ast })\forall s\in S_i.$

Mixed strategy NE

A mixed strategy profile $p^{\ast }\in \mathbf{\Delta }={\times }_{i\in N}{\Delta }_i$ is a Nash equilibrium if for every player $i\in N$ $U_i({\mathbf{p}}^{\ast })\ge U_i(p,{\mathbf{p}}_{-i}^{\ast })\forall p\in {\Delta }_i.$