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 ( a 1 * ,.., a n * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGHb Waa0baaSqaaiaaigdaaeaacaGGQaaaaOGaaiilaiaac6cacaGGUaGa aiilaiaadggadaqhaaWcbaGaamOBaaqaaiaacQcaaaGccaGGPaaaaa@3F4C@ such that u i ( a i * , a i * ) u i ( a i , a i 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaamyAaaqabaGccaGGOaGaamyyamaaDaaaleaacaWGPbaabaGa aiOkaaaakiaacYcacaWGHbWaa0baaSqaaiabgkHiTiaadMgaaeaaca GGQaaaaOGaaiykaiabgwMiZkaadwhadaWgaaWcbaGaamyAaaqabaGc caGGOaGaamyyamaaBaaaleaacaWGPbaabeaakiaacYcacaWGHbWaa0 baaSqaaiabgkHiTiaadMgaaeaacaaIXaaaaOGaaiykaaaa@4C1B@ 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 ( a 1 * , a 2 * ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWGHb Waa0baaSqaaiaaigdaaeaacaGGQaaaaOGaaiilaiaadggadaqhaaWc baGaaGOmaaqaaiaacQcaaaGccaGGPaaaaa@3D01@ is a Nash equilibrium if the following two conditions hold

a 1 * arg max a 1 A 1 u 1 ( a 1 , a 2 * ) a 2 * arg max a 2 A 2 u 2 ( a 1 * , a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyyam aaDaaaleaacaaIXaaabaGaaiOkaaaakiabgIGiolaaykW7caaMc8Ua ciyyaiaackhacaGGNbGaaGjbVpaayaaabaGaciyBaiaacggacaGG4b aaleaacaWGHbWaaSbaaWqaaiaaigdaaeqaaSGaeyicI4Saamyqamaa BaaameaacaaIXaaabeaaaOGaayjo+dGaamyDamaaBaaaleaacaaIXa aabeaakiaacIcacaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa dggadaqhaaWcbaGaaGOmaaqaaiaacQcaaaGccaGGPaaabaGaamyyam aaDaaaleaacaaIYaaabaGaaiOkaaaakiabgIGiolaaykW7caaMc8Ua ciyyaiaackhacaGGNbGaaGjbVpaayaaabaGaciyBaiaacggacaGG4b aaleaacaWGHbWaaSbaaWqaaiaaikdaaeqaaSGaeyicI4Saamyqamaa BaaameaacaaIYaaabeaaaOGaayjo+dGaamyDamaaBaaaleaacaaIYa aabeaakiaacIcacaWGHbWaa0baaSqaaiaaigdaaeaacaGGQaaaaOGa aiilaiaadggadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaaa@7142@

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

B i : A i A i g i v e n b y B i ( a i ) = { a i A i : u i ( a i , a i ) u i ( b i , a i ) f o r a l l b i A i } = arg max a i A i u i ( a i , a i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamOqam aaBaaaleaacaWGPbaabeaakiaacQdacaaMc8UaamyqamaaBaaaleaa cqGHsislcaWGPbaabeaakiabgkziUkaadgeadaWgaaWcbaGaamyAai aaykW7aeqaaOGaaGPaVlaadEgacaWGPbGaamODaiaadwgacaWGUbGa aGPaVlaadkgacaWG5baabaGaamOqamaaBaaaleaacaWGPbaabeaaki aacIcacaWGHbWaaSbaaSqaaiabgkHiTiaadMgaaeqaaOGaaiykaiab g2da9iaacUhacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaeyicI4Saam yqamaaBaaaleaacaWGPbaabeaakiaacQdacaWG1bWaaSbaaSqaaiaa dMgaaeqaaOGaaiikaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGSa GaamyyamaaBaaaleaacqGHsislcaWGPbaabeaakiaacMcacqGHLjYS caWG1bWaaSbaaSqaaiaadMgaaeqaaOGaaiikaiaadkgadaWgaaWcba GaamyAaaqabaGccaGGSaGaamyyamaaBaaaleaacqGHsislcaWGPbaa beaakiaacMcacaaMc8UaamOzaiaad+gacaWGYbGaaGPaVlaadggaca WGSbGaamiBaiaaykW7caWGIbWaaSbaaSqaaiaadMgaaeqaaOGaeyic I4SaamyqamaaBaaaleaacaWGPbaabeaakiaac2haaeaacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeyypa0Jaci yyaiaackhacaGGNbWaaGbaaeaaciGGTbGaaiyyaiaacIhaaSqaaiaa dggadaWgaaadbaGaamyAaaqabaWccqGHiiIZcaWGbbWaaSbaaWqaai aadMgaaeqaaaGccaGL44pacaWG1bWaaSbaaSqaaiaadMgaaeqaaOGa aiikaiaadggadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamyyamaaBa aaleaacqGHsislcaWGPbaabeaakiaacMcaaeaaaaaa@AF3F@

(Notice that, for each a i A i , B i ( a i ) i s a s e t w h i c h m a y o r m a y n o t b e a sin g l e t o n ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaeyOeI0IaamyAaaqabaGccqGHiiIZcaWGbbWaaSbaaSqaaiab gkHiTiaadMgaaeqaaOGaaiilaiaadkeadaWgaaWcbaGaamyAaaqaba GccaGGOaGaamyyamaaBaaaleaacqGHsislcaWGPbaabeaakiaacMca caaMc8UaamyAaiaadohacaaMc8UaamyyaiaaykW7caWGZbGaamyzai aadshacaaMc8Uaam4DaiaadIgacaWGPbGaam4yaiaadIgacaaMc8Ua amyBaiaadggacaWG5bGaaGPaVlaad+gacaWGYbGaaGPaVlaad2gaca WGHbGaamyEaiaaykW7caWGUbGaam4BaiaadshacaaMc8UaamOyaiaa dwgacaaMc8UaamyyaiaaykW7ciGGZbGaaiyAaiaac6gacaaMb8Uaam 4zaiaadYgacaWGLbGaamiDaiaad+gacaWGUbGaaiykaiaac6caaaa@780A@ So, for example, in a 2-person game, if player 2 plays a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaaGOmaaqabaaaaa@37B9@ player 1s best choice is to play some action in B 1 ( a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeadaWgaa WcbaGaaGymaaqabaGccaGGOaGaamyyamaaBaaaleaacaaIYaaabeaa kiaacMcaaaa@3AD4@ B 1 ( a 2 ) = { a 1 A 1 : u 1 ( a 1 , a 2 ) u 2 ( b 1 , a 2 ) f o r a l l b 1 A 1 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamOqam aaBaaaleaacaaIXaaabeaakiaacIcacaWGHbWaaSbaaSqaaiaaikda aeqaaOGaaiykaiabg2da9iaacUhacaWGHbWaaSbaaSqaaiaaigdaae qaaOGaeyicI4SaamyqamaaBaaaleaacaaIXaaabeaakiaacQdacaWG 1bWaaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadggadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaamyyamaaBaaaleaacaaIYaaabeaakiaacMca cqGHLjYScaWG1bWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadkgada WgaaWcbaGaaGymaaqabaGccaGGSaGaamyyamaaBaaaleaacaaIYaaa beaakiaacMcacaaMc8UaamOzaiaad+gacaWGYbGaaGPaVlaadggaca WGSbGaamiBaiaaykW7caWGIbWaaSbaaSqaaiaaigdaaeqaaOGaeyic I4SaamyqamaaBaaaleaacaaIXaaabeaakiaac2haaeaacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7aaaa@7695@

For instance, in the game :

L M R U 1,0 1,2 0,2 D 0,3 1,1 2,0 W e h a v e B 1 ( L ) = { U } , B 1 ( M ) = { U , D } a n d B 1 ( R ) = { D } , w h i l e B 2 ( U ) = { M , R } a n d B 2 ( D ) = { L } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaGPaVl aaykW7caaMc8UaaGPaVxaabeqabmaaaeaacaaMe8Uaamitaaqaaiaa d2eaaeaacaWGsbaaaaqaaiaadwfacaaMc8UaaGzaVpaaL4babaGaaG ymaiaacYcacaaIWaaaamaaL4babaGaaGymaiaacYcacaaIYaaaamaa L4babaGaaGimaiaacYcacaaIYaaaaaqaaiaadseacaaMc8+aauIhae aacaaIWaGaaiilaiaaiodaaaGaaGzaVlaaykW7daqjEaqaaiaaigda caGGSaGaaGymaaaadaqjEaqaaiaaikdacaGGSaGaaGimaaaaaeaaca WGxbGaamyzaiaaykW7caWGObGaamyyaiaadAhacaWGLbGaaGPaVlaa dkeadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamitaiaacMcacqGH9a qpcaGG7bGaaiyvaiaac2hacaGGSaGaaGPaVlaackeadaWgaaWcbaGa aGymaaqabaGccaGGOaGaamytaiaacMcacqGH9aqpcaGG7bGaaiyvai aacYcacaGGebGaaiyFaiaaykW7caGGHbGaaiOBaiaacsgacaaMc8Ua aiOqamaaBaaaleaacaaIXaaabeaakiaacIcacaWGsbGaaiykaiabg2 da9iaacUhacaGGebGaaiyFaiaacYcacaGG3bGaaiiAaiaacMgacaGG SbGaaiyzaiaaykW7caGGcbWaaSbaaSqaaiaaikdaaeqaaOGaaiikai aacwfacaGGPaGaeyypa0Jaai4Eaiaac2eacaGGSaGaaiOuaiaac2ha caaMc8Uaaiyyaiaac6gacaGGKbGaaGPaVlaackeadaWgaaWcbaGaaG OmaaqabaGccaGGOaGaamiraiaacMcacqGH9aqpcaGG7bGaaiitaiaa c2haaaaa@9F24@

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