1. Tutorial 5 Adversary Search
CS Introduction to AI
Tutorial 5
Adversary Search

2. Intro. to AI – Tutorial 5 – By Nela Gurevich
Agenda
Introduction:
Why games?
Assumptions
Minimax algorithm
General idea
Minimax with limited depth
Alpha-Beta search
Pruning
Search routine
Example
Enhancements to the algorithm
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

3. Intro. to AI – Tutorial 5 – By Nela Gurevich
Why Games?
Games are fun
Easy to measure results
Simple moves
Big search spaces
Examples:
Chess: Deep Junior
Checkers: Chinook, Nemesis
Othello: Bill
Backgammon: TDGammon
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

4. Intro. to AI – Tutorial 5 – By Nela Gurevich
Assumptions
Two-players game
Perfect information
The knowledge available to each player is the same
Zero Sum
The move good for a player is bad for his adversary
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

5. Intro. to AI – Tutorial 5 – By Nela Gurevich
Game Trees
MIN
MAX
Win
Win
Loss
Win
Loss
Draw
Draw
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

6. Intro. to AI – Tutorial 5 – By Nela Gurevich
The Minimax Algorithm
e(v) if v is a terminal node
MM(v) = max{MM(succ)} v is a max node
min{MM(succ)} v is a min node
Where succ = successors(v) and
1 if v is a WIN node
e(v) = if v is a DRAW node
-1 if v is a LOSS node
A problem: big branching factor, deep trees
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

7. Minimax search to limited depth
Search the game tree to some search frontier d.
Compute a static evaluation function f to assess the strength values of nodes at that frontier.
Use the minimax rule to compute approximations of the strength values of the shallower nodes.
f(v) if d=0 or v is terminal MM(v,d) = max{MM(succ, d-1)} v is a max node
min{MM(succ, d-1)} v is a min node
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

8. Intro. to AI – Tutorial 5 – By Nela Gurevich
αβ Search
Shallow pruning
Deep pruning
MM 10≤ 10
MM 10≤ 10
MM ≤ 5 5
MM ≤ 5
The node will not
contribute to the max
value of the father
The node will not
contribute to the max
value of the ancestor 5
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

9. Intro. to AI – Tutorial 5 – By Nela Gurevich
αβ Procedure
α : highest max among ancestors of a node
β : lowest min among ancestors of a node
First call: αβ(v, d, min/max, -∞, ∞)
// The αβ procedure
αβ(v, d, node-type, α, β) {
If v is terminal, or d = 0 return f(v)
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

10. Intro. to AI – Tutorial 5 – By Nela Gurevich
αβ Procedure: MAX node
if node-type = MAX {
curr-max ← -infinity
loop for vi є Succ(v)
board-val ← αβ(vi, d-1, min, α, β)
curr-max ← max(board-val, curr-max)
α ← max(curr-max, α)
if (curr-max ≥ β) // Bigger than lowest min
end loop }
return curr-max
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

11. Intro. to AI – Tutorial 5 – By Nela Gurevich
αβ Procedure: MIN node
if node-type = MIN {
curr-min ← infinity
loop for vi є Succ(v)
board-val ← αβ(vi, d-1, max, α, β)
curr-min ← min(board-val, curr-min)
β ← min(curr-min, β)
if (curr-min ≤ α) // Smaller than highest max
end loop }
return curr-min
} // end of αβ
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

12. Intro. to AI – Tutorial 5 – By Nela Gurevich
Game Tree Example
MIN
MAX 12 14 21 13 15 10 11 9 20 22 3 1 4 2 5
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

13. Intro. to AI – Tutorial 5 – By Nela Gurevich
Stage 1
α = -∞ β = ∞ 13 15 14 10 11 9 ? 20 22 3 1 4 2 5
α = -∞ β = ∞
α = -∞ β = ∞
α = -∞ β = ∞ 12 14 21
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

14. Intro. to AI – Tutorial 5 – By Nela Gurevich
Stage 1 13 15 14 10 11 9 ? 20 22 3 1 4 2 5
α = β = ∞ 10
α = -∞ β = 10
α = 10 β = ∞ 10 12 14 21
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

15. Stage 2 – Shallow Pruning10 11 9
α = -∞ β = 10 10
α = β = ∞ 10
α = -∞ β = 10 9
α = 10 β = 9
α = -∞ β = 10 12 14 15 13 14 5 2 4 1 3 22 20 21
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Intro. to AI – Tutorial 5 – By Nela Gurevich

16. Game Tree example contd.12 14 21 13 15 10 11 9 20 22 3 1 4 2 5
α = β = ∞ 10 10
α = -∞ β = 10 14
α = β = 10 14
α = -∞ β = 10
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

17. Game Tree example contd.12 14 21 13 15 10 11 9 20 22 3 1 4 2 5
α = β = ∞
α = β = ∞
α = β = ∞
α = 10 β = ∞
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

18. Game Tree example contd.12 14 21 13 15 10 11 9 20 22 3 1 4 2 5
α = β = ∞ 5
α = 10 β = 5
α = 10 β = ∞
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

19. Game Tree example contd.12 14 21 13 15 10 11 9 20 22 3 1 4 2 5 10 5
α = β = 5 5
α = β = ∞ 4
α = 10 β = 4
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

20. Intro. to AI – Tutorial 5 – By Nela Gurevich
αβ Features
Correctness:
Minimax(v, d) = αβ(v, d, -∞, ∞)
Pruning: The values of the tree leaves and the search ordering determine the amount of pruning
For any given tree and any search ordering, there exists a sequence of values for the leaves, such that αβ prunes no leaf.
For any given tree there exists such ordering that αβ prunes the maximal possible number of leaves.
For randomly ordered trees αβ expands Θ(b(3/4d)) leaves
Pruning decreases the effective branching factor, and thus allows us to search game tree for greater depth
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

21. Intro. to AI – Tutorial 5 – By Nela Gurevich
Iterative αβ
Perform αβ search to increasing depth beginning from some initial depth.
Useful when the time is bounded – when the time is over, the value computed during the previous iteration can be returned
The values computed during the previous iterations may be used to perform a heuristic ordering on the nodes of the tree to increase the pruning rate
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich

22. Intro. to AI – Tutorial 5 – By Nela Gurevich
The End
27-Nov-18
Intro. to AI – Tutorial 5 – By Nela Gurevich