Win Lose or Draw

• In a game where we can win, draw, or lose, we can assign utility values e.g. {1, 0, -1}.

• We could assign any values as long as the order is preserved e.g. a win is better than a draw is better than a lose {i, j, k} with i>j>k

•

Positive Linear Transformation of Utility

• x1= max(x1,x2) if x1>x2 else x2

• Consider a linear transformation of x

• E.g. f(x) = ax+b (Note a>0, b is any value)

• Max(f(x1),f(x2))– If f(x1)>f(x2) then ax1+b>ax2+b

– {b cancel} ax1>ax2

– {a cancel as a>0} x1>x2

• We can do the same for min.

• a positive linear scaling does not alter the Minmax. Any strictly monotonic function.

D E F G

= terminal position = agent = opponent

4 -5 -5 1 -7 2 -3 -8

1

MAX

MIN

4 1 2 -3

MAX

1 -3B C

A

D E F G

= terminal position = agent = opponent

4 -5 -5 1 -7 2 -3 -8

1

MAX

MIN

4 1 2 -3

MAX

1 -3B C

A

Alpha beta pruning.

Min Max

• Minmax(root) = max( min(3,12,8), min(2,x,y), min(14,5,2))

• = (3,min(2,x,y),2)

• = max(3,z,2) where z=min(2,x,y)<=2

• = 3

Procedure for converting to CNF

• (a) To eliminate ↔, – (a ↔ b) ≡ (a → b) Λ (a→ b)

• (b) To eliminate →, – (a → b) ≡ ¬ (a ν b)

• (c) Double negation ¬ (¬a) ≡ a• (d) De Morgan

– ¬ (a Λ b) ≡ (¬a ν ¬b) ¬(a ν b) ≡ (¬a Λ ¬b)

• (e) Distributivity of Λ over ν – (a Λ (b ν c )) ≡ ((a Λ b) ν (a Λ c))

• (f) Distributivity of ν over Λ – (a ν (b Λ c )) ≡ ((a ν b) Λ (a ν c))