Brilliant mathematician John Nash worked on many theory such as nonlinear parabolicpartial differential equations theory

and singularity theory. The most important work of John Nash is Nash equilibrium which is related to the non-cooperative

games. Nash won the Nobel Prize with the following study: .

# Nash Equilibrum

A Nash equilibrium of a game G in strategic form is defined as any outcome $\left({a}_{1}^{*},..,{a}_{n}^{*}\right)$ such that ${u}_{i}\left({a}_{i}^{*},{a}_{-i}^{*}\right)\ge {u}_{i}\left({a}_{i},{a}_{-i}^{1}\right)$ holds for each player i: The set of all Nash equilibria of G is denoted N(G): In a two player game, for example, an action profile $\left({a}_{1}^{*},{a}_{2}^{*}\right)$ is a Nash equilibrium if the following two conditions hold

$\begin{array}{l}{a}_{1}^{*}\in \text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{arg}\text{\hspace{0.17em}}\underset{{a}_{1}\in {A}_{1}}{\underbrace{\mathrm{max}}}{u}_{1}\left({a}_{1},{a}_{2}^{*}\right)\\ {a}_{2}^{*}\in \text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{arg}\text{\hspace{0.17em}}\underset{{a}_{2}\in {A}_{2}}{\underbrace{\mathrm{max}}}{u}_{2}\left({a}_{1}^{*},{a}_{2}\right)\end{array}$

Therefore, we may say that, in a Nash equilibrium, each players choice of action is a best response to the actions actually taken by his opponents. This suggests, and sometimes more useful, definition of Nash equilibrium, based on the notion of the best response correspondance. We define the best response correspondance of player i in a a strategic form game as the correspondence

$\begin{array}{l}{B}_{i}:\text{\hspace{0.17em}}{A}_{-i}\to {A}_{i\text{\hspace{0.17em}}}\text{\hspace{0.17em}}given\text{\hspace{0.17em}}by\\ {B}_{i}\left({a}_{-i}\right)=\left\{{a}_{i}\in {A}_{i}:{u}_{i}\left({a}_{i},{a}_{-i}\right)\ge {u}_{i}\left({b}_{i},{a}_{-i}\right)\text{\hspace{0.17em}}for\text{\hspace{0.17em}}all\text{\hspace{0.17em}}{b}_{i}\in {A}_{i}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{arg}\underset{{a}_{i}\in {A}_{i}}{\underbrace{\mathrm{max}}}{u}_{i}\left({a}_{i},{a}_{-i}\right)\\ \end{array}$

(Notice that, for each ${a}_{-i}\in {A}_{-i},{B}_{i}\left({a}_{-i}\right)\text{\hspace{0.17em}}is\text{\hspace{0.17em}}a\text{\hspace{0.17em}}set\text{\hspace{0.17em}}which\text{\hspace{0.17em}}may\text{\hspace{0.17em}}or\text{\hspace{0.17em}}may\text{\hspace{0.17em}}not\text{\hspace{0.17em}}be\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\mathrm{sin}\text{​}gleton\right).$ So, for example, in a 2-person game, if player 2 plays ${a}_{2}$ player 1s best choice is to play some action in ${B}_{1}\left({a}_{2}\right)$ $\begin{array}{l}{B}_{1}\left({a}_{2}\right)=\left\{{a}_{1}\in {A}_{1}:{u}_{1}\left({a}_{1},{a}_{2}\right)\ge {u}_{2}\left({b}_{1},{a}_{2}\right)\text{\hspace{0.17em}}for\text{\hspace{0.17em}}all\text{\hspace{0.17em}}{b}_{1}\in {A}_{1}\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

For instance, in the game :

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{ccc}\text{\hspace{0.17em}}L& M& R\end{array}\\ U\text{\hspace{0.17em}}\text{​}\overline{)1,0}\overline{)1,2}\overline{)0,2}\\ D\text{\hspace{0.17em}}\overline{)0,3}\text{​}\text{\hspace{0.17em}}\overline{)1,1}\overline{)2,0}\\ We\text{\hspace{0.17em}}have\text{\hspace{0.17em}}{B}_{1}\left(L\right)=\left\{U\right\},\text{\hspace{0.17em}}{B}_{1}\left(M\right)=\left\{U,D\right\}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{B}_{1}\left(R\right)=\left\{D\right\},while\text{\hspace{0.17em}}{B}_{2}\left(U\right)=\left\{M,R\right\}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{B}_{2}\left(D\right)=\left\{L\right\}\end{array}$