Strategie Miste - W3.UniRoma1.it

Calcolo Strategie Miste per Nash, Maximin e Minimax
Maria-Augusta Miceli∗
Dipartimento di Economia e Diritto
Università di Roma "La Sapienza"
Lezioni di Economia Industriale
October 8, 2014
Supponete che il gioco seguente abbia un unico equilibrio in strategie miste (p (riga), q (colonna)).
G1 \ G2
pAlto
(1 − p)Basso
qSin
a,b
c,d
(1 − q)Des
e,f
g,h
Calcolo strategie miste
EU1 (A) = qa + (1 − q) e
EU1 (B) = qc + (1 − q) g
qa + (1 − q) e = qc + (1 − q) g
EU2 (S) = pb + (1 − p) d
EU2 (D) = pf + (1 − p) h
pb + (1 − p) d = pf + (1 − p) h
qa + (1 − q) e = qc + (1 − q) g
pb + (1 − p) d = pf + (1 − p) h
p∗
=
q∗
=
h−d
b−d−f +h
g−e
a−c+g−e
EU1 (A) = qa + (1 − q) e
µ
¶
g−e
g−e
ag − ce
=
a+ 1−
e=
a−c+g−e
a−c+g−e
a−c+g−e
EU1 (B) = qc + (1 − q) g
µ
¶
g−e
g−e
ag − ce
=
c+ 1−
g=
a−c+g−e
a−c+g−e
a−c+g−e
EU2 (S) = pb + (1 − p) d
µ
¶
h−d
h−d
bh − df
=
b+ 1−
d=
b−d−f +h
b−d−f +h
b−d−f +h
EU2 (D) = pf + (1 − p) h
µ
¶
h−d
h−d
bh − df
=
f + 1−
h=
b−d−f +h
b−d−f +h
b−d−f +h
∗ Department of Economics and Law, University of Rome "Sapienza" - 9 via del Castro Laurenziano - 00161 Roma - Italy. Email:
[email protected].
1
Per
EU1 (A) ≥ EU1 (B)
g−e
q∗ ≥
a−c+g−e
EU2 (S) ≥ EU2 (D)
h−d
p∗ ≥
b−d−f +h
p1
p2
1 − p1 − p2
q1
L
0,0
6,7
7,6
G1 \ g2
A
M
B
q2
C
7,6
0,0
6,7
1 − q1 − q2
R
6,7
7,6
0,0
EU1 (A) = 7q2 + 6 (1 − q1 − q2 )
EU1 (M) = 6q1 + 7 (1 − q1 − q2 )
EU1 (B) = 7q1 + 6q2
¤
£
, Solution is: q1 = 13 , q2 = 13
7q2 + 6 (1 − q1 − q2 ) = 6q1 + 7 (1 − q1 − q2 )
6q1 + 7 (1 − q1 − q2 ) = 7q1 + 6q2
EU2 (L) = 7p2 + 6 (1 − p1 − p2 )
EU1 (C) = 6p1 + 7 (1 − p1 − p2 )
EU1 (B) = 7p1 + 6p2
Stesso.
0.1
Maximin in MS
G1 \ G2
pAlto
(1 − p)Basso
qSin
a,b
c,d
(1 − q)Des
e,f
g,h
Calcolo strategie miste
u1(maximin) = max(min(a, e); min(c, g))
= p (q · a + (1 − q) e) + (1 − p) (q · c + (1 − q) g)
µ
µ
¶ ¶
g−e
h−d
g−e
·a+ 1−
=
e +
b−d−f +h a−c+g−e
a−c+g−e
µ
µ
¶µ
¶ ¶
g−e
h−d
g−e
·c+ 1−
+ 1−
g
b−d−f +h
a−c+g−e
a−c+g−e
ag − ce
=
a−c+g−e
YEAH!!!
2

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