slides-chap6A-communication-soft

En posant b 1 10 caract eriser tous les types d

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En posant b = 1 / 10 , caract´ eriser tous les types d’´ equilibres erifier que l’´ equilibre pr´ ef´ er´ e par l’´ emetteur d´ epend en g´ en´ eral de son type Comparaison des ´ equilibres en terme d’efficience Utilit´ e esp´ er´ ee moyenne du r´ ecepteur ` a un ´ equilibre n -s´ eparateur : EU 2 = E bracketleftBig [ σ 2 ( σ 1 ( t )) t ] 2 bracketrightBig = integraldisplay 1 0 [ σ 2 ( σ 1 ( t )) t ] 2 dt = n summationdisplay k =1 integraldisplay x k x k - 1 [ σ 2 ( m k ) t ] 2 dt = n summationdisplay k =1 integraldisplay x k x k - 1 bracketleftbigg x k 1 + x k 2 t bracketrightbigg 2 dt = n summationdisplay k =1 1 3 bracketleftBigg parenleftbigg t x k 1 + x k 2 parenrightbigg 3 bracketrightBigg x k x k - 1
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Transmission strat´ egique de l’information / Information non certifiable Pour quelles valeurs de b positives existe-t-il un ´ equilibre 3-s´ eparateur ? En posant b = 1 / 10 , caract´ eriser tous les types d’´ equilibres erifier que l’´ equilibre pr´ ef´ er´ e par l’´ emetteur d´ epend en g´ en´ eral de son type Comparaison des ´ equilibres en terme d’efficience Utilit´ e esp´ er´ ee moyenne du r´ ecepteur ` a un ´ equilibre n -s´ eparateur : EU 2 = E bracketleftBig [ σ 2 ( σ 1 ( t )) t ] 2 bracketrightBig = integraldisplay 1 0 [ σ 2 ( σ 1 ( t )) t ] 2 dt = n summationdisplay k =1 integraldisplay x k x k - 1 [ σ 2 ( m k ) t ] 2 dt = n summationdisplay k =1 integraldisplay x k x k - 1 bracketleftbigg x k 1 + x k 2 t bracketrightbigg 2 dt = n summationdisplay k =1 1 3 bracketleftBigg parenleftbigg t x k 1 + x k 2 parenrightbigg 3 bracketrightBigg x k x k - 1 = 1 12 n summationdisplay k =1 ( x k x k 1 ) 3
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Transmission strat´ egique de l’information / Information non certifiable Or x k x k 1 = k/n 2 k b ( n k ) (( k 1) /n 2( k 1) b ( n ( k 1))) = 1 /n + 2 b (2 k n 1)
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Transmission strat´ egique de l’information / Information non certifiable Or x k x k 1 = k/n 2 k b ( n k ) (( k 1) /n 2( k 1) b ( n ( k 1))) = 1 /n + 2 b (2 k n 1) d’o`u EU 2 = 1 12 n k =1 ( 1 /n + 2 b (2 k n 1) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright α ) 3
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Transmission strat´ egique de l’information / Information non certifiable Or x k x k 1 = k/n 2 k b ( n k ) (( k 1) /n 2( k 1) b ( n ( k 1))) = 1 /n + 2 b (2 k n 1) d’o`u EU 2 = 1 12 n k =1 ( 1 /n + 2 b (2 k n 1) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright α ) 3 Dans α les termes en k s’annulent avec les termes en n k + 1 donc
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Transmission strat´ egique de l’information / Information non certifiable Or x k x k 1 = k/n 2 k b ( n k ) (( k 1) /n 2( k 1) b ( n ( k 1))) = 1 /n + 2 b (2 k n 1) d’o`u EU 2 = 1 12 n k =1 ( 1 /n + 2 b (2 k n 1) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright α ) 3 Dans α les termes en k s’annulent avec les termes en n k + 1 donc EU 2 = 1 12 n summationdisplay k =1 ( 1
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