Lecture 5: Adversarial Search and Games

Lecture 5: Adversarial Search and Games¶

AIMA Chapter 5 — 1 hour¶

Why Games Matter¶

• Clean testbed for search and learning

• Superhuman play: chess, Go, poker

• Same ideas: search, evaluation, learning

• Applications: negotiation, security, design

Learning Objectives¶

• Define two-player zero-sum games

• Implement minimax and alpha-beta pruning

• Design evaluation functions

• Understand Monte Carlo Tree Search

• Handle stochastic and partially observable games

Game Theory Basics¶

• Players: MAX (us), MIN (opponent)

• Zero-sum: One wins, one loses (or sum of payoffs = 0)

• Perfect information: Both see full state

• Deterministic: No chance (e.g., chess, checkers)

Game Tree¶

• Initial state: Board position

• Actions: Legal moves

• Terminal states: Game over

• Utility: +1 (MAX wins), -1 (MIN wins), 0 (draw)

Minimax Algorithm¶

• MAX chooses move that maximizes utility

• MIN chooses move that minimizes utility

• Minimax value: Best achievable outcome against optimal play

• Minimax decision: Move with highest minimax value

Minimax Algorithm (Pseudocode)¶

function MINIMAX(state) returns action
  return argmax_a MIN-VALUE(RESULT(state, a))

function MAX-VALUE(state)
  if TERMINAL-TEST(state) return UTILITY(state)
  v ← -∞
  for each a in ACTIONS(state):
    v ← max(v, MIN-VALUE(RESULT(state,a)))
  return v

Alpha-Beta Pruning¶

• Pruning: Skip branches that don’t affect final decision

• α: Best value MAX can guarantee

• β: Best value MIN can guarantee

• Cut: When α ≥ β, prune remaining siblings

Alpha-Beta: Move Ordering¶

• Best-first: Try best moves first

• Killer heuristic: Moves that caused cuts before

• Transposition table: Cache evaluated positions

• Good ordering → more pruning → faster search

Evaluation Functions¶

• Cutoff: Don’t search to terminal state

• Eval(s): Estimate utility of non-terminal state

• Requirements: Must be fast, correlate with winning chances

• Example (chess): Material + piece-square tables + mobility

Cutting Off Search¶

• Depth limit: Fixed depth (e.g., 4 ply)

• Iterative deepening: Increase depth until time runs out

• Quiescence: Search until “quiet” state (no captures)

Monte Carlo Tree Search (MCTS)¶

• Selection: Traverse tree using UCB

• Expansion: Add new node

• Simulation: Random playout to end

• Backpropagation: Update node statistics

• No evaluation function needed!

MCTS: Advantages¶

• Asymmetric: Focuses on promising branches

• Anytime: Can stop anytime, return best move

• Works for: Go, general game playing

• AlphaGo: MCTS + neural networks

Stochastic Games¶

• Chance nodes: Nature’s turn (dice, cards)

• Expectiminimax: Expectation over chance outcomes

• Evaluation: Must account for expected value

Partially Observable Games¶

• Kriegspiel: Chess where you don’t see opponent’s pieces

• Belief state: Set of possible board states

• Card games: Hidden information

Summary¶

• Minimax: Optimal play in deterministic games

• Alpha-beta: Prune branches, same result, faster

• Evaluation: Heuristic for cutoff, must be fast

• MCTS: Simulation-based, no eval needed

• Stochastic/partial: Expectiminimax, belief states

References¶

• Russell & Norvig, AIMA 4e, Ch. 5

• Chapter PDF: chapters/chapter-05.pdf

• aima-python: games4e.ipynb

Questions?¶

Next lecture: Constraint Satisfaction Problems (Chapter 6)