(
N
,
(
A
i
)
,
(
≿
i
)
)
{\displaystyle (N,(A_{i}),(\succsim \!_{i}))}
i
∈
N
{\displaystyle i\in N}
∀
i
∈
N
∃
A
i
{\displaystyle \forall i\in N\exists A_{i}}
∀
i
∈
N
∃
≿
i
{\displaystyle \forall i\in N\exists \succsim \!_{i}}
≿
i
{\displaystyle \succsim \!_{i}}
g
:
A
⟶
C
{\displaystyle g:A\longrightarrow C}
A
i
{\displaystyle A_{i}}
a
∈
A
∧
b
∈
A
{\displaystyle a\in A\land b\in A}
a
≿
i
b
⇔
g
(
a
)
≿
i
g
(
b
)
{\displaystyle a\succsim \!_{i}b\Leftrightarrow g(a)\succsim \!_{i}g(b)}
Ω
{\displaystyle \Omega }
g
:
A
×
Ω
⟶
C
{\displaystyle g:A\times \Omega \longrightarrow C}
a
∈
A
∧
b
∈
A
∧
ω
∈
Ω
{\displaystyle \mathbf {a\in A\land b\in A\land \omega \in \Omega } }
a
≿
i
b
⇔
g
(
a
,
ω
)
≿
i
g
(
b
,
ω
)
{\displaystyle \mathbf {a\succsim \!_{i}b} \Leftrightarrow \mathbf {g(a,\omega )\succsim \!_{i}g(b,\omega )} }
u
i
:
A
⟶
P
{\displaystyle \mathbf {u_{i}:A\longrightarrow P} }
u
i
(
a
)
≥
u
i
(
b
)
⇔
a
≿
i
b
{\displaystyle \mathbf {u_{i}(a)\geq u_{i}(b)\Leftrightarrow a\succsim \!_{i}b} }
Template:Payoff matrix
Template:Payoff matrix
Player 2 chooses A
Player 2 chooses R
Player 1 chooses F
5, 5
0, 0
Player 1 chooses U
8, 2
0, 0
a
1
e
,
a
2
e
,
.
.
.
,
a
N
e
{\displaystyle \mathbf {a_{1}^{e},a_{2}^{e},...,a_{N}^{e}} }
u
i
(
a
1
e
,
a
2
e
,
.
.
.
,
a
i
e
,
.
.
.
,
a
N
e
)
≿
i
u
i
(
a
1
e
,
a
2
e
,
.
.
.
,
a
i
,
.
.
.
,
a
N
e
)
{\displaystyle \mathbf {u_{i}(a_{1}^{e},a_{2}^{e},...,a_{i}^{e},...,a_{N}^{e})\succsim \!_{i}u_{i}(a_{1}^{e},a_{2}^{e},...,a_{i},...,a_{N}^{e})} }
∀
i
∈
N
∧
∀
a
i
∗
∈
A
i
{\displaystyle \mathbf {\forall i\in N\land \forall a_{i}^{*}\in A_{i}} }
a
i
e
{\displaystyle \mathbf {a_{i}^{e}} }
a
j
e
{\displaystyle \mathbf {a_{j}^{e}} }
u
i
(
a
1
o
,
a
2
o
,
.
.
.
,
a
i
o
,
.
.
.
,
a
N
o
)
≿
i
u
i
(
a
1
e
,
a
2
e
,
.
.
.
,
a
i
e
,
.
.
.
,
a
N
e
)
{\displaystyle \mathbf {u_{i}(a_{1}^{o},a_{2}^{o},...,a_{i}^{o},...,a_{N}^{o})\succsim \!_{i}u_{i}(a_{1}^{e},a_{2}^{e},...,a_{i}^{e},...,a_{N}^{e})} }
a
1
o
,
a
2
o
,
.
.
.
,
a
N
o
{\displaystyle \mathbf {a_{1}^{o},a_{2}^{o},...,a_{N}^{o}} }
a
i
o
{\displaystyle \mathbf {a_{i}^{o}} }
M
i
g
l
i
o
r
R
i
s
u
l
t
a
t
o
p
e
r
i
l
G
r
u
p
p
o
⇒
M
i
g
l
i
o
r
R
i
s
u
l
t
a
t
o
I
n
d
i
v
i
d
u
a
l
e
{\displaystyle \mathbf {Miglior_{}Risultato_{}per_{}il_{}Gruppo\Rightarrow Miglior_{}Risultato_{}Individuale} }
⇒
{\displaystyle \Rightarrow }
(
(
1
,
2
)
,
(
A
i
)
,
(
≿
i
)
)
{\displaystyle \mathbf {((1,2),(A_{i}),(\succsim \!_{i}))} }
u
1
+
u
2
=
0
{\displaystyle \mathbf {u_{1}+u_{2}=0} }
m
i
n
(
u
1
(
x
∗
,
y
)
)
≥
m
i
n
(
u
1
(
x
,
y
)
)
{\displaystyle \mathbf {min(u_{1}(x^{*},y))\geq min({}u_{1}(x,y))} }
∀
x
∈
A
1
∧
∀
y
∈
A
2
{\displaystyle \mathbf {\forall x\in A_{1}\land \forall y\in A_{2}} }
m
i
n
(
u
2
(
x
,
y
∗
)
)
≥
m
i
n
(
u
2
(
x
,
y
)
)
{\displaystyle \mathbf {min(u_{2}(x,y^{*}))\geq min({}u_{2}(x,y))} }
∀
x
∈
A
1
∧
∀
y
∈
A
2
{\displaystyle \mathbf {\forall x\in A_{1}\land \forall y\in A_{2}} }
x
∗
∈
A
1
{\displaystyle \mathbf {x^{*}\in A_{1}} }
y
∗
∈
A
2
{\displaystyle \mathbf {y^{*}\in A_{2}} }
(
x
∗
,
y
∗
)
{\displaystyle \mathbf {(x^{*},y^{*})} }
m
a
x
x
m
i
n
y
u
1
(
x
,
y
)
=
m
i
n
y
m
a
x
x
u
1
(
x
,
y
)
=
u
1
(
x
∗
,
y
∗
)
{\displaystyle \mathbf {max_{x}min_{y}u_{1}(x,y)=min_{y}max_{x}u_{1}(x,y)=u_{1}(x^{*},y^{*})} }
m
a
x
x
m
i
n
y
u
1
(
x
,
y
)
=
m
i
n
y
m
a
x
x
u
1
(
x
,
y
)
{\displaystyle \mathbf {max_{x}min_{y}u_{1}(x,y)=min_{y}max_{x}u_{1}(x,y)} }
Nome del Gioco Numero di Giocatori Numero di Strategie Numero di Strategie pure in Equilibrio di Nash Cooperativo - Non Cooperativo Somma 0 - Somma Variabile Simultaneo - Sequenziale Informazione Completa - Informazione Incompleta
Dilemma del Prigioniero 2
2
1
Cooperativo - Non Cooperativo
Somma Variabile
Simultaneo
Informazione Incompleta
Caccia al Cervo 2
2
2
Cooperativo - Non Cooperativo
Somma Variabile
Simultaneo
Informazione Incompleta
Gioco del Pollo 2
2
2
Non Cooperativo
Somma Variabile
Simultaneo
Informazione Incompleta
Peace war Game N
Variabili
>2
Non Cooperativo
Somma Variabile
Sequenziale
Informazione Incompleta
Corteggiamento N
Variabili
>2
Cooperativo - Non Cooperativo
Somma Variabile
Simultaneo
Informazione Incompleta
Oligopolio di Cournot 2
Infinite
1
Non Cooperativo
Somma Variabile
Simultaneo
Informazione Incompleta
Matching Pennies 2
2
0
Non Cooperativo
Somma 0
Simultaneo
Informazione Incompleta
Gioco del Centipede 2
Variabili
1
Non Cooperativo
Somma Variabile
Sequenziale
Informazione Completa
Confessa
Non Confessa
Confessa
-7, -7
0, -10
Non Confessa
-10, 0
-1, -1
u
i
(
c
,
c
)
≥
u
i
(
n
,
c
)
{\displaystyle \mathbf {u_{i}(c,c)\geq u_{i}(n,c)} }
u
1
(
c
,
n
)
≥
u
i
(
n
,
n
)
{\displaystyle \mathbf {u_{1}(c,n)\geq u_{i}(n,n)} }
(
c
,
c
)
{\displaystyle \mathbf {(c,c)} }
(
c
,
n
)
{\displaystyle \mathbf {(c,n)} }
(
n
,
c
)
{\displaystyle \mathbf {(n,c)} }
(
n
,
n
)
{\displaystyle \mathbf {(n,n)} }
Cervo
Lepre
Cervo
3, 3
0, 2
Lepre
2, 0
1, 1
(
c
,
l
)
{\displaystyle \mathbf {(c,l)} }
(
c
,
c
)
{\displaystyle \mathbf {(c,c)} }
(
l
,
l
)
{\displaystyle \mathbf {(l,l)} }
(
l
,
c
)
{\displaystyle \mathbf {(l,c)} }
Cervo
Lepre
Cervo
3, 3
0, 2
Lepre
2, 0
1, 1
(
c
,
l
)
{\displaystyle \mathbf {(c,l)} }
(
c
,
c
)
{\displaystyle \mathbf {(c,c)} }
(
l
,
l
)
{\displaystyle \mathbf {(l,l)} }
(
l
,
c
)
{\displaystyle \mathbf {(l,c)} }
Pace
Guerra
Pace
2, 2
0, 3
Guerra
3, 0
1, 1
(
p
,
w
)
{\displaystyle \mathbf {(p,w)} }
(
w
,
w
)
{\displaystyle \mathbf {(w,w)} }
(
w
,
p
)
{\displaystyle \mathbf {(w,p)} }
(
p
,
p
)
{\displaystyle \mathbf {(p,p)} }
u
i
(
w
,
p
)
≥
u
i
(
p
,
p
)
{\displaystyle \mathbf {u_{i}(w,p)\geq u_{i}(p,p)} }
u
1
(
w
,
w
)
≥
u
i
(
w
,
p
)
{\displaystyle \mathbf {u_{1}(w,w)\geq u_{i}(w,p)} }
Bionda
Mora
Bionda
0, 0
5, 2
Mora
2, 5
3, 3
(
b
,
m
)
{\displaystyle \mathbf {(b,m)} }
(
m
,
m
)
{\displaystyle \mathbf {(m,m)} }
(
m
,
b
)
{\displaystyle \mathbf {(m,b)} }
(
b
,
b
)
{\displaystyle \mathbf {(b,b)} }
u
i
(
w
,
p
)
≥
u
i
(
p
,
p
)
{\displaystyle \mathbf {u_{i}(w,p)\geq u_{i}(p,p)} }
u
1
(
w
,
w
)
≥
u
i
(
w
,
p
)
{\displaystyle \mathbf {u_{1}(w,w)\geq u_{i}(w,p)} }
P
(
q
1
+
q
2
)
=
x
−
(
q
1
+
q
2
)
{\displaystyle \mathbf {P(q_{1}+q_{2})=x-(q_{1}+q_{2})} }
C
i
(
q
i
)
{\displaystyle \mathbf {C_{i}(q_{i})} }
Π
i
{\displaystyle \mathbf {\Pi _{i}} }
∂
Π
i
∂
q
i
=
0
{\displaystyle \mathbf {{\frac {\partial \Pi _{i}}{\partial q_{i}}}=0} }
Π
i
′
=
0
{\displaystyle \mathbf {{\Pi _{i}^{'}}=0} }
Π
i
′
=
D
[
P
(
q
1
+
q
2
)
⋅
q
i
−
c
i
⋅
q
i
]
{\displaystyle \mathbf {{\Pi _{i}^{'}}=D[P(q_{1}+q_{2})\cdot q_{i}-c_{i}\cdot q_{i}]} }
x
−
(
q
1
+
q
2
)
{\displaystyle \mathbf {x-(q_{1}+q_{2})} }
Π
i
′
=
D
[
x
⋅
q
i
−
q
i
2
−
q
i
⋅
q
i
¯
−
c
i
⋅
q
i
]
{\displaystyle \mathbf {{\Pi _{i}^{'}}=D[x\cdot q_{i}-q_{i}^{2}-q_{i}\cdot q_{\overline {i}}-c_{i}\cdot q_{i}]} }
Π
i
′
=
D
[
x
⋅
q
i
]
−
D
[
q
i
2
]
−
D
[
q
i
⋅
q
i
¯
]
−
D
[
c
i
⋅
q
i
]
{\displaystyle \mathbf {{\Pi _{i}^{'}}=D[x\cdot q_{i}]-D[q_{i}^{2}]-D[q_{i}\cdot q_{\overline {i}}]-D[c_{i}\cdot q_{i}]} }
Π
i
′
=
x
−
2
q
i
−
q
i
¯
−
c
i
{\displaystyle \mathbf {{\Pi _{i}^{'}}=x-2q_{i}-q_{\overline {i}}-c_{i}} }
x
−
2
q
i
−
q
i
¯
−
c
i
=
0
{\displaystyle \mathbf {x-2q_{i}-q_{\overline {i}}-c_{i}=0} }
q
i
=
x
−
q
i
¯
−
c
i
2
{\displaystyle \mathbf {q_{i}={\frac {x-q_{\overline {i}}-c_{i}}{2}}} }
q
i
¯
+
2
q
i
=
x
−
c
i
{\displaystyle \mathbf {q_{\overline {i}}+2q_{i}=x-c_{i}} }
q
i
+
c
i
=
x
−
(
q
i
+
q
i
¯
)
{\displaystyle \mathbf {q_{i}+c_{i}=x-(q_{i}+q_{\overline {i}})} }
q
i
2
+
c
i
⋅
q
i
=
P
(
q
1
+
q
2
)
⋅
q
i
{\displaystyle \mathbf {q_{i}^{2}+c_{i}\cdot q_{i}=P(q_{1}+q_{2})\cdot q_{i}} }
q
i
2
=
P
(
q
1
+
q
2
)
⋅
q
i
−
c
i
⋅
q
i
{\displaystyle \mathbf {q_{i}^{2}=P(q_{1}+q_{2})\cdot q_{i}-c_{i}\cdot q_{i}} }
Π
i
e
(
q
1
;
q
2
)
=
q
i
2
{\displaystyle \mathbf {\Pi _{i}^{e}(q_{1};q_{2})=q_{i}^{2}} }
P
=
α
−
β
(
x
1
+
x
2
)
{\displaystyle \mathbf {P=\alpha -\beta (x_{1}+x_{2})} }
C
i
(
x
i
)
{\displaystyle \mathbf {C_{i}(x_{i})} }
x
1
{\displaystyle \mathbf {x_{1}} }
x
2
{\displaystyle \mathbf {x_{2}} }
α
>
0
∧
β
>
0
{\displaystyle \mathbf {\alpha >0\land \beta >0} }
(
α
−
β
(
x
i
+
x
i
¯
)
)
x
i
{\displaystyle \mathbf {(\alpha -\beta (x_{i}+x_{\overline {i}}))x_{i}} }
α
x
i
−
β
x
i
2
−
β
x
i
¯
x
i
{\displaystyle \mathbf {\alpha x_{i}-\beta x_{i}^{2}-\beta x_{\overline {i}}x_{i}} }
G
u
a
d
a
g
n
o
=
R
i
c
a
v
o
−
C
o
s
t
o
{\displaystyle \mathbf {Guadagno=Ricavo-Costo} }
Π
i
=
α
x
i
−
β
x
i
2
−
β
x
i
¯
x
i
−
C
i
x
i
{\displaystyle \mathbf {\Pi _{i}=\alpha x_{i}-\beta x_{i}^{2}-\beta x_{\overline {i}}x_{i}-C_{i}x_{i}} }
∂
Π
i
∂
x
i
=
0
{\displaystyle \mathbf {{\frac {\partial \Pi _{i}}{\partial x_{i}}}=0} }
Π
i
′
=
0
{\displaystyle \mathbf {{\Pi _{i}^{'}}=0} }
D
[
α
x
i
−
β
x
i
2
−
β
x
i
¯
x
i
−
C
i
x
i
]
{\displaystyle \mathbf {D[\alpha x_{i}-\beta x_{i}^{2}-\beta x_{\overline {i}}x_{i}-C_{i}x_{i}]} }
Π
i
′
=
α
−
2
β
x
i
¯
−
C
i
=
0
{\displaystyle \mathbf {{\Pi _{i}^{'}}=\alpha -2\beta x_{\overline {i}}-C_{i}=0} }
α
−
C
i
−
β
x
i
¯
2
β
=
x
i
{\displaystyle \mathbf {{\frac {\alpha -C_{i}-\beta x_{\overline {i}}}{2\beta }}=x_{i}} }
y
=
m
x
+
q
{\displaystyle \mathbf {y=mx+q} }
x
i
=
−
1
2
x
i
¯
+
α
−
C
i
2
β
{\displaystyle \mathbf {x_{i}=-{\frac {1}{2}}x_{\overline {i}}+{\frac {\alpha -C_{i}}{2\beta }}} }
x
i
=
x
i
¯
{\displaystyle \mathbf {x_{i}=x_{\overline {i}}} }
2
x
i
=
−
x
i
¯
+
α
−
C
i
β
{\displaystyle \mathbf {2x_{i}=-x_{\overline {i}}+{\frac {\alpha -C_{i}}{\beta }}} }
2
x
i
+
x
i
¯
=
α
−
C
i
β
{\displaystyle \mathbf {2x_{i}+x_{\overline {i}}={\frac {\alpha -C_{i}}{\beta }}} }
x
i
=
x
i
¯
=
1
3
α
−
C
i
β
{\displaystyle \mathbf {x_{i}=x_{\overline {i}}={\frac {1}{3}}{\frac {\alpha -C_{i}}{\beta }}} }
Π
i
N
E
=
(
α
−
C
i
)
2
9
β
{\displaystyle \mathbf {\Pi _{i}^{NE}={\frac {(\alpha -C_{i})^{2}}{9\beta }}} }
x
i
=
k
{\displaystyle \mathbf {x_{i}=k} }
x
i
¯
=
0
{\displaystyle \mathbf {x_{\overline {i}}=0} }
x
i
=
α
−
C
i
2
β
{\displaystyle \mathbf {x_{i}={\frac {\alpha -C_{i}}{2\beta }}} }
x
i
C
O
=
x
i
¯
C
O
=
1
2
(
α
−
C
i
)
2
β
{\displaystyle \mathbf {x_{i}^{CO}=x_{\overline {i}}^{CO}={\frac {1}{2}}{\frac {(\alpha -C_{i})}{2\beta }}} }
Π
i
C
O
=
(
α
−
C
i
2
α
−
C
i
4
β
)
−
C
i
(
α
−
C
i
4
β
)
{\displaystyle \mathbf {\Pi _{i}^{CO}=\left({\frac {\alpha -C_{i}}{2}}{\frac {\alpha -C_{i}}{4\beta }}\right)-C_{i}\left({\frac {\alpha -C_{i}}{4\beta }}\right)} }
Π
i
C
O
=
(
α
−
β
(
2
x
i
)
)
x
i
−
C
i
x
i
{\displaystyle \mathbf {\Pi _{i}^{CO}=(\alpha -\beta (2x_{i}))x_{i}-C_{i}x_{i}} }
Π
i
C
O
=
α
x
i
−
2
β
x
i
2
−
C
i
x
i
{\displaystyle \mathbf {\Pi _{i}^{CO}=\alpha x_{i}-2\beta x_{i}^{2}-C_{i}x_{i}} }
Π
i
C
O
=
x
i
(
α
−
2
β
x
i
−
C
i
)
{\displaystyle \mathbf {\Pi _{i}^{CO}=x_{i}(\alpha -2\beta x_{i}-C_{i})} }
Π
i
C
O
=
x
i
(
α
−
C
i
)
−
2
β
x
i
2
{\displaystyle \mathbf {\Pi _{i}^{CO}=x_{i}(\alpha -C_{i})-2\beta x_{i}^{2}} }
Π
i
C
O
=
x
i
=
α
−
C
i
4
β
−
(
α
−
C
i
)
2
8
β
{\displaystyle \mathbf {\Pi _{i}^{CO}=x_{i}={\frac {\alpha -C_{i}}{4\beta }}-{\frac {(\alpha -C_{i})^{2}}{8\beta }}} }
Π
i
C
O
=
(
α
−
C
i
)
2
8
β
{\displaystyle \mathbf {\Pi _{i}^{CO}={\frac {(\alpha -C_{i})^{2}}{8\beta }}} }
x
i
¯
=
α
−
C
i
2
β
−
1
2
x
i
{\displaystyle \mathbf {x_{\overline {i}}={\frac {\alpha -C_{i}}{2\beta }}-{\frac {1}{2}}x_{i}} }
x
i
¯
=
α
−
C
i
2
β
−
1
2
α
−
C
i
4
β
{\displaystyle \mathbf {x_{\overline {i}}={\frac {\alpha -C_{i}}{2\beta }}-{\frac {1}{2}}{\frac {\alpha -C_{i}}{4\beta }}} }
x
i
¯
=
3
8
α
−
C
i
8
β
{\displaystyle \mathbf {x_{\overline {i}}={\frac {3}{8}}{\frac {\alpha -C_{i}}{8\beta }}} }
P
D
T
=
α
−
β
(
x
i
+
x
i
¯
)
=
α
−
β
5
(
α
−
C
i
)
8
β
{\displaystyle \mathbf {P^{DT}=\alpha -\beta (x_{i}+x_{\overline {i}})=\alpha -\beta {\frac {5(\alpha -C_{i})}{8\beta }}} }
P
D
T
=
3
α
+
5
C
i
8
{\displaystyle \mathbf {P^{DT}={\frac {3\alpha +5C_{i}}{8}}} }
Π
i
¯
D
=
P
D
T
x
i
¯
D
−
C
i
¯
D
(
x
i
¯
D
)
{\displaystyle \mathbf {\Pi _{\overline {i}}^{D}=P^{DT}x_{\overline {i}}^{D}-C_{\overline {i}}^{D}(x_{\overline {i}}^{D})} }
Π
i
¯
D
=
3
8
α
−
C
i
¯
β
(
3
α
+
5
C
i
¯
8
−
C
i
¯
)
{\displaystyle \mathbf {\Pi _{\overline {i}}^{D}={\frac {3}{8}}{\frac {\alpha -C_{\overline {i}}}{\beta }}\left({\frac {3\alpha +5C_{\overline {i}}}{8}}-C_{\overline {i}}\right)} }
Π
i
¯
D
=
9
64
(
α
−
C
i
¯
)
2
β
{\displaystyle \mathbf {\Pi _{\overline {i}}^{D}={\frac {9}{64}}{\frac {(\alpha -C_{\overline {i}})^{2}}{\beta }}} }
Π
i
T
=
P
D
T
x
i
D
−
C
i
T
(
x
i
T
)
{\displaystyle \mathbf {\Pi _{i}^{T}=P^{DT}x_{i}^{D}-C_{i}^{T}(x_{i}^{T})} }
Π
i
T
=
6
64
(
α
−
C
i
)
2
β
{\displaystyle \mathbf {\Pi _{i}^{T}={\frac {6}{64}}{\frac {(\alpha -C_{i})^{2}}{\beta }}} }
Collude
Non Collude
Collude
(
α
−
C
i
)
2
8
β
{\displaystyle \mathbf {\frac {(\alpha -C_{i})^{2}}{8\beta }} }
(
α
−
C
i
)
2
8
β
{\displaystyle \mathbf {\frac {(\alpha -C_{i})^{2}}{8\beta }} }
6
64
(
α
−
C
i
)
2
β
{\displaystyle \mathbf {{\frac {6}{64}}{\frac {(\alpha -C_{i})^{2}}{\beta }}} }
9
64
(
α
−
C
i
)
2
β
{\displaystyle \mathbf {{\frac {9}{64}}{\frac {(\alpha -C_{i})^{2}}{\beta }}} }
Non Collude
9
64
(
α
−
C
i
)
2
β
{\displaystyle \mathbf {{\frac {9}{64}}{\frac {(\alpha -C_{i})^{2}}{\beta }}} }
6
64
(
α
−
C
i
)
2
β
{\displaystyle \mathbf {{\frac {6}{64}}{\frac {(\alpha -C_{i})^{2}}{\beta }}} }
(
α
−
C
i
)
2
9
β
{\displaystyle \mathbf {\frac {(\alpha -C_{i})^{2}}{9\beta }} }
(
α
−
C
i
)
2
9
β
{\displaystyle \mathbf {\frac {(\alpha -C_{i})^{2}}{9\beta }} }
(
c
,
c
)
{\displaystyle \mathbf {(c,c)} }
(
c
,
n
)
{\displaystyle \mathbf {(c,n)} }
(
n
,
c
)
{\displaystyle \mathbf {(n,c)} }
(
n
,
n
)
{\displaystyle \mathbf {(n,n)} }
Π
i
=
a
x
i
2
+
b
x
i
+
c
{\displaystyle \mathbf {\Pi _{i}=ax_{i}^{2}+bx_{i}+c} }
Π
i
=
−
β
x
i
2
+
(
α
−
C
i
−
β
x
i
¯
)
x
i
{\displaystyle \mathbf {\Pi _{i}=-\beta x_{i}^{2}+(\alpha -C_{i}-\beta x_{\overline {i}})x_{i}} }
L
:
(
α
−
C
i
−
β
x
i
¯
2
β
;
(
α
−
C
i
−
β
x
i
¯
)
2
4
β
)
{\displaystyle \mathbf {L:\left({\frac {\alpha -C_{i}-\beta x_{\overline {i}}}{2\beta }};{\frac {(\alpha -C_{i}-\beta x_{\overline {i}})^{2}}{4\beta }}\right)} }
Π
i
=
β
4
x
i
¯
2
−
α
−
C
i
2
x
i
¯
+
(
α
−
C
i
)
2
4
β
{\displaystyle \mathbf {\Pi _{i}={\frac {\beta }{4}}x_{\overline {i}}^{2}-{\frac {\alpha -C_{i}}{2}}x_{\overline {i}}+{\frac {(\alpha -C_{i})^{2}}{4\beta }}} }
Testa
Croce
Testa
1, -1
-1, 1
Croce
-1, 1
1, -1
a
j
e
{\displaystyle \mathbf {a_{j}^{e}} }
a
j
o
{\displaystyle \mathbf {a_{j}^{o}} }
a
o
{\displaystyle \mathbf {a^{o}} }
![{\displaystyle \mathbf {{\Pi _{i}^{'}}=D[P(q_{1}+q_{2})\cdot q_{i}-c_{i}\cdot q_{i}]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2052ff27381bc2cf64f7e31a96c57b5b98821a)
![{\displaystyle \mathbf {{\Pi _{i}^{'}}=D[x\cdot q_{i}-q_{i}^{2}-q_{i}\cdot q_{\overline {i}}-c_{i}\cdot q_{i}]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/41d038baf4ccaff74fd27943e6a4ed3d2741afde)
![{\displaystyle \mathbf {{\Pi _{i}^{'}}=D[x\cdot q_{i}]-D[q_{i}^{2}]-D[q_{i}\cdot q_{\overline {i}}]-D[c_{i}\cdot q_{i}]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9dbf84a1fc58f4e135c24f69b8b1512bf5ce1fd)
![{\displaystyle \mathbf {D[\alpha x_{i}-\beta x_{i}^{2}-\beta x_{\overline {i}}x_{i}-C_{i}x_{i}]} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b044117e913d472b252bc781dbbf554887e4865)
