Win Lose or Draw

1. 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

2. 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.

3. B C 1 -3 D E F G 4 1 2 -3 4 -5 -5 1 -7 2 -3 -8 MAX 1 A MIN MAX = terminal position = agent = opponent

4. B C 1 -3 D E F G 4 1 2 -3 4 -5 -5 1 -7 2 -3 -8 MAX 1 A MIN MAX = terminal position = agent = opponent

5. Alpha beta pruning.

6. 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

7. 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 Λover ν • (a Λ(b ν c )) ≡((aΛ b) ν (a Λ c)) • (f) Distributivity of νover Λ • (a ν(b Λ c )) ≡((a ν b) Λ(a ν c))