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Etaires incertaines l esp erance math ematique ou

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etaires incertaines : l’ esp´ erance math´ ematique ou valeur actuarielle : summationdisplay i p i x i 20 38 10 18 38 10 0 0 L 0 10 0 10 1 0 L E( L ) = 18 38 10 20 38 10 = 20 38 E( L ) = 0 26/ Exemple : Assurance . Une maison d’une valeur de 1000 peut brˆuler (´ etat b ) avec probabilit´ e 1/10 et ne pas brˆuler (´ etat n ) avec probabilit´ e 9/10 : Ω = { b, n } , π ( b ) = 1 / 10 , π ( n ) = 9 / 10 Consid´ erons les trois alternatives (actes) a , a et a ′′ suivantes : Ne pas s’assurer : a ( ω ) = 0 si ω = b 1000 si ω = n Assurance totale (prime = 100 ) : a ( ω ) = 900 pour tout ω Ω Assurance avec franchise (prime = 70 , franchise = 300 ) : a ′′ ( ω ) = 630 si ω = b 930 si ω = n Cons´ equences C = { 0 , 630 , 900 , 930 , 1000 } Loteries induites par les actes a , a et a ′′ : L = parenleftbigg 1 10 , 0 , 0 , 0 , 9 10 parenrightbigg L = (0 , 0 , 1 , 0 , 0) L ′′ = parenleftbigg 0 , 1 10 , 0 , 9 10 , 0 parenrightbigg
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Th´ eorie des jeux Introduction 27/ Inconv´ enients de l’esp´ erance math´ ematique : – Pas de prise en compte de l’attitude vis-` a-vis du risque du d´ ecideur – Cons´ equences mon´ etaires uniquement – Paradoxe de Saint-P´ etersbourg Paradoxe de Saint-P´ etersbourg Une pi` ece de monnaie ´ equilibr´ ee est lanc´ ee ` a r´ ep´ etition tant que pile se r´ ealise D` es que face se r´ ealise au k -i` eme jet le gain est de 2 k euros Esp´ erance math´ ematique de gain pour ce pari : summationdisplay k =1 1 2 k 2 k = 1 + 1 + 1 + · · · = Pourtant la valeur attribu´ ee ` a ce pari par la plupart des gens est bien en-dessous de 100 et mˆ eme de 10 euros . . . 28/ En 1738 Daniel Bernoulli (1700–1782) propose d’int´ egrer le fait que les agents ont une utilit´ e (satisfaction) marginale d´ ecroissante pour la monnaie et ´ evaluent un pari par l’ esp´ erance de l’utilit´ e des diff´ erentes cons´ equences Par exemple, l’esp´ erance math´ ematique du logarithme du gain : summationdisplay k =1 1 2 k ln(2 k ) = (ln 2) summationdisplay k =1 k parenleftbigg 1 2 parenrightbigg k = (ln 2) bracketleftBigg 2 summationdisplay k =1 k parenleftbigg 1 2 parenrightbigg k summationdisplay k =1 k parenleftbigg 1 2 parenrightbigg k bracketrightBigg = (ln 2) bracketleftBigg summationdisplay k =0 ( k + 1) parenleftbigg 1 2 parenrightbigg k summationdisplay k =1 k parenleftbigg 1 2 parenrightbigg k bracketrightBigg = (ln 2) bracketleftBigg 1 + summationdisplay k =1 parenleftbigg 1 2 parenrightbigg k bracketrightBigg = ln 4 Valeur d’un montant mon´ etaire certain de 4 euros 1944 : von Neumann et Morgenstern fournissent une axiomatique rigoureuse pour la solution propos´ ee par Bernoulli
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Th´ eorie des jeux Introduction 29/ Fig. 1 – John von Neumann (1903–1957) 30/ Hypoth` eses de von Neumann et Morgenstern : Rationalit´ e , ou pr´ eordre complet . Compl´ etude. Pour tout L , L ∈ L , on a L followsequal L ou L followsequal L (ou les deux) Transitivit´ e. Pour tout L , L , L ′′ ∈ L , si L followsequal L et L followsequal L ′′ , alors L followsequal L ′′ Continuit´ e. Pour tout L , L , L ′′ ∈ L , les ensembles { α [0 , 1] : αL + (1 α ) L
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