Tutorial 5 Adversary Search CS 236501 Introduction to AI Tutorial 5 Adversary Search

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

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

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

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

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) = 0 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

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

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

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

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

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

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

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

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

Stage 2 – Shallow Pruning 10 11 9 α = -∞ β = 10 10 α = 10 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 27-Nov-18 Intro. to AI – Tutorial 5 – By Nela Gurevich

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

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

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

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

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

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

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