Questions tagged [nash-equilibrium]

Let $(S,f)$ be a game with $n$ players, where $S_{i}$ is the strategy set for player $i$, $S=S_{1}\times S_{2}\times \ldots \times S_{n}$ is the set of strategy profiles and $f(x)=(f_{1}(x),\dotsc ,f_{n}(x))$ is its payoff function evaluated at $x\in S$.

Let $x_{i}$ be a strategy profile of player $i$,$x_{-i}$ be a strategy profile of all players except for player $i$. When each player $i\in \{1,\dotsc ,n\}$ chooses strategy $x_{i}$ resulting in strategy profile $x=(x_{1},\dotsc ,x_{n})$ then player $i$ obtains payoff $f_{i}(x)$. Note that the payoff depends on the strategy profile chosen, i.e., on the strategy chosen by player $i$ as well as the strategies chosen by all the other players. A strategy profile $x^{*}\in S$ is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is

$$\forall i, x_i \in S_i: f_i(x_i^{*}, x_{-i}^{*}) \ge f_i(x_i, x_{-i}^{*}).$$