Chapter 5

1. Chapter 5 Game Playing

2. Games as search • Board games (e.g., Chess, Checkers, Go) • Two player games • High branching factors • Opponent player • Contingency problem • Opponent is assumed to do their best (our worst) • Game environment • Accessible • complete access to the environment state • Deterministic • Next state determined by move and current state • Episodic: no; Static: yes (for us); Discrete: yes

3. Game search problem • Initial state • Configuration of board at start, and which player starts • Operators • Legal moves that a player can make • Goal or “terminal” test • goal states called terminal states • Utility function • quality value for terminal states from rules of game • not an estimate if game play reaches this state • Mini-max algorithm can be used to propagate values up search tree for estimates of utility of non-terminal states

4. Game trees • Root node • player 1 or “max” • trying to maximize score • Immediate children of root • player 2, or “min” • trying to minimize score • Next layer, max • Next layer, min, etc. • A layer or possible move by a player is a “ply”

5. Problems • Complexity too high for practical game playing • Could use • Depth limited search • Iterative deepening • Means that we cannot generally use minimax propagation of utility values • Use heuristic evaluation instead • Quiescent states • States that are unlikely to exhibit large changes • Next state can be a checkmate, not quiescent • Horizon problem • Failing to take action because of depth limit on search

6. Evaluation functions • Evaluation function: h(n) for games • Heuristic estimate of the expected utility of a given game state • Material advantage function in chess • Integer value for type of board piece • Quality of game state given by sum of material advantage values of all pieces • Example of a weighted linear sum EVAL(s) = w1p1 + w2p2 + w3p3 + … w’s are the weights; p’s are the counts of a piece type • E.g., p1 = 3 pawns w1 for pawn is 1; p2 = 2 knight or rook, w2 for knight or rook is 3

7. Pruning • Eliminating branches from tree in search • Alpha-beta pruning • pruning with mini-max • don’t follow branches that cannot influence decision • still accurate within evaluation function • Principle: “If you have an idea that is surely bad, do not take time to see how truly awful it is.” (P. H. Winston)

8. 8 1 1 7 2 3 5 3 2 9 3 9 7 1 9 6 2 2 16 6 2 5 4 6 1 4 2 max min max Example from P. H. Winston