Unformatted text preview: lay s 1 JW, Fig 4.1
2L
M
R
1
U 8,1 0,2 4,0
C 3,3 1,2 0,0
D 5,0 2,3 8,1
I If P1 assigns prob’s 0.5 that P2 plays L, 0.25 that P2 plays M
and 0.25 that P2 plays R then P1’s expected payo↵s are:
I
I
I U: 0.5 ⇥ 8 + 0.25 ⇥ 0 + 0.25 ⇥ 4 = 5
M: 0.5 ⇥ 3 + 0.25 ⇥ 1 + 0.25 ⇥ 0 = 1.75
D: 0.5 ⇥ 5 + 0.25 ⇥ 2 + 0.25 ⇥ 8 = 5 Expected payo↵ u 1 ( s1 , ✓ 1) =
s X 1 2S 1 ✓ 1 ( s 1 ) ⇥ u 1 ( s1 , s 1 )
 {z }
 {z }
probability
that P1
thinks others
play s 1 payo↵ for
P1 if others
do play s 1 JW, Fig 4.1
2L
M
R
1
U 8,1 0,2 4,0
C 3,3 1,2 0,0
D 5,0 2,3 8,1
So if P2 assign prob. 0 that P1 plays U, 0.5 that P1 plays C and
0.5 that P1 plays D then what is the expected payo↵ for P2? Expected payo↵
u 1 ( s1 , ✓ 1) =
s X 1 2S 1 ✓ 1 ( s 1 ) ⇥ u 1 ( s1 , s 1 )
 {z }
 {z }
probability
that P1
thinks others
play s 1 payo↵ for
P1 if others
do play s 1 JW, Fig 4.1
M
R
12L
U 8,1 0,2 4,0
C 3,3 1,2 0,0
D 5,0 2,3 8,1
So if P2 assign prob. 0 that P1 plays U, 0.5 that P1 plays C and
0.5 that P1 plays D then what is the expected payo↵ for P2?
I For L it is 0 ⇥ 1 + 0.5 ⇥ 3 + 0.5 ⇥ 0 = 1.5, for M it is
0 ⇥ 2 + 0.5 ⇥ 2 + 0.5 ⇥ 3 = 2.5, and for R it is
0 ⇥ 0 + 0.5 ⇥ 0 + 0.5 ⇥ 1 = 0.5...
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 Spring '09
 KRISHNA,VIJAYXU,HAIQING
 Normal Distribution, Probability, Probability theory, probability density function, p1

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