25 that p2 plays m and 025 that p2 plays r then p1s

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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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