asgn3_sol

asgn3_sol - CSC 5350 Assignment 3 1(a This game can be...

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Unformatted text preview: CSC 5350 Assignment 3 1. (a) This game can be modeled as an extensive game with incomplete in- formation as follows: A ch ch B C B C R Y p B p C p B p C R Y R R R Y Y Y (0,0,1) (1,1,0) (0,1,0) (1,0,1) (1,1,0) (0,0,1) (1,0,1) (0,1,0) i. N = { A, B, C } ii. H = {∅ , R, Y, ( R, B ) , ( R, C ) , ( Y, B ) , ( Y, C ) , ( R, B, R ) , ( R, B, Y ) , ( R, C, R ) , ( R, C, Y ) , ( Y, B, R ) , ( Y, B, Y ) , ( Y, C, R ) , ( Y, C, Y ) } iii. P ( ∅ ) = A, P ( R ) = P ( Y ) = ch, P ( R, B ) = B, P ( R, C ) = C, P ( Y, B ) = B, P ( Y, C ) = C iv. I A = {{∅}} I B = {{ ( R, B ) , ( Y, B ) }} I C = {{ ( R, C ) , ( Y, C ) }} (b) X A ( ∅ ) = (), X B ( R, B ) = X B ( Y, B ) = (), X C ( R, C ) = X C ( Y, C ) = (), so this game is a game with perfect recall. (c) i. Assume that the belief system is consistent with Amy’s behavioral strat- egy. if P R > 1 / 2, the best behavioral strategy of B is (R(1),Y(0)) 1 if P R < 1 / 2 the best behavioral strategy of B is (R(0),Y(1)) if P R = 1 / 2, the best behavioral strategy of B is (R(p),Y(q)) where p+q=1. ii. Assume that the belief system is consistent with Amy’s behavioral strat- egy. if P R > 1 / 2, the best behavioral strategy of C is (R(1),Y(0)) if P R < 1 / 2 the best behavioral strategy of C is (R(0),Y(1)) if P R = 1 / 2, the best behavioral strategy of C is (R(p),Y(q)) where p+q=1. iii. if Betty and Cindy use the strategy in i. and ii. The best behavioural strategy of Amy is (R(1), Y(0)) or (R(0), Y(1)). (d) According to (c), we define β as: β A = ( R (1) , Y (0)) , β B = ( R (1) , Y (0)) , β C = ( R (1) , Y (0)) , Then the belief system is: μ = {{∅} 7-→ ∅ (1), { R } 7-→ R (1) , { Y } 7-→ Y (1) , { ( R, B ) , ( Y, B ) } 7-→ (( R, B )(1) , ( Y, B )(0)), { ( R, C ) , ( Y, C ) } 7-→ (( R, C )(1) , ( Y, C )(0)), { R, B, R } 7-→ ( R, B, R )(1) , { R, B, Y } 7-→ ( R, B, Y )(1) , { R, C, R } 7-→ ( R, C, R )(1) , { R, C, Y } 7-→ ( R, C, Y )(1) , { Y, B, R } 7-→ ( Y, B, R )(1) , { Y, B, Y } 7-→ ( Y, B, Y )(1) , { Y, C, R } 7-→ ( Y, C, R )(1) , { Y, C, Y } 7-→ ( Y, C, Y )(1) } Define: β ε A = ( R (1- ε ) , Y ( ε )) , β ε B = ( R (1- ε ) , Y ( ε )) , β ε C = ( R (1- ε ) , Y ( ε )) , Then: μ ε = {{∅} 7-→ ∅ (1),...
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asgn3_sol - CSC 5350 Assignment 3 1(a This game can be...

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