A key property of Nash equilibria in normal form games is that all players are indifferent among all of their supported choices.

\(\newcommand{\game}{\Gamma} \newcommand{\players}{\mathcal{N}} \newcommand{\play}{i} \newcommand{\pures}{\mathcal{A}} \newcommand{\pure}{\alpha} \newcommand{\purealt}{\beta} \newcommand{\pay}{u} \newcommand{\others}{-i} \newcommand{\strat}{x} \newcommand{\strats}{\mathcal{X}} \newcommand{\from}{:} \newcommand{\R}{\mathbb{R}} \newcommand{\findex}{\play \pure_{\play}} \newcommand{\supp}{\text{supp}} \newcommand{\BR}{\text{BR}}\) Let $\game = \game(\players, \pures, \pay)$ be a finite normal form game with players $\players$, pure profiles $\pures$ and payoff function $u$, and let $\strats$ denote the mixed strategy space.

For any player $\play$, fixed a choice $\strat_{\others}$ from the opponents, the payoff function $\pay_\play(\cdot, \strat_{\others}) \from \strats_\play \to \R$ is the linear, real-valued function on $\strats_i$ given by $\pay_\play(\strat_\play, \strat_{\others}) = \sum_{\pure_\play \in \pures_\play} \strat_{\findex} \pay_\play(\pure_\play, \strat_{\others})$.

\[\BR_\play(\strat_{\others}) := \arg\max_{\strat_i \in \strats_i}\pay(\strat_\play, \strat_{\others}) \subseteq \strats_\play\]

DefinitionThe set ofbest repliesto any $\strat_{\others} \in \strats_{\others}$ is

By linearity of $\pay_{\play}(\cdot, \strat_{\others})$, the set of best replies to any $\strat_{\others}$ is a non-empty union of faces of $\strats_{\play}$, containing a vertex of $\strats_{\play}$ at least and the whole $\strats_{\play}$ at most.

\[\pay_\play(\pure_{\play}, \strat_{\others}) = \pay_\play(\purealt_{\play}, \strat_{\others}) \quad \text{for all } \play \in \players, \pure_{\play}, \purealt_{\play} \in \pures_\play\]

DefinitionA mixed strategy $\strat \in \strats$ is calledslope strategyif each player $\play$ is indifferent among all of their choices, given that the others play $\strat_{\others}$:

\[\pay_\play(\pure_{\play}, \strat_{\others}) = \pay_\play(\purealt_{\play}, \strat_{\others}) \quad \text{for all } \play \in \players, \pure_{\play}, \purealt_{\play} \in \supp_{\play}(\strat_\play) \subseteq \pures_{\play}\]

DefinitionA mixed strategy $\strat \in \strats$ is calledslope-supportif if each player $\play$ is indifferent among all of theiravailablechoices, given that the others play $\strat_{\others}$:

\[\strat_{\play} \in \BR_{\play}(\strat_{\others}) \quad \text{for all } \play \in \players\]

DefinitionA mixed strategy $\strat \in \strats$ is calledNash equilibriumif it is a best reply to itself, in the sense thatA Nash equilibrium is

\[\{\strat_{\play}\} = \BR_{\play}(\strat_{\others}) \quad \text{for all } \play \in \players\]strictif

Folk result on Nash equilibria and slope strategies

- A slope strategy is a Nash equilibrium
- A Nash equilibrium is a slope-support strategy
- A fully supported Nash equilibrium is a slope strategy.

**Proof**
For 1. Let $\strat$ be a slope strategy. Then for each player $\BR_{\play}(\strat_{\others})$ is the whole $\strats_\play$, that contains in particular $\strat_\play$.

For 2. Since $\strat$ is Nash $\strat_\play \in \BR_\play(\strat_{\others})$, so by linearity $\BR_\play(\strat_{\others})$ contains the whole face of $\strats_\play$ that contains $\strat_i$, for each player $\play$. In particular, it contains all of its vertices, that are precisely the pure strategies supported by $\strat_i$.

Point 3. is a direct consequence of 1. and 2.