Games as adversarial search problems

1. Games as adversarialsearch problems Dynamic state space search

2. Requirements ofadversarial game space search • on-line search: planning cannot be completed before action • multi-agent environment • dynamic environment D Goforth - COSC 4117, fall 2003

3. Features of games King’s Court • deterministic/stochastic • perfect/partial information • number of agents: n>1 • optimization function • interaction scheduling • deterministic • perfect • 2 • zero-sum • turn-taking D Goforth - COSC 4117, fall 2003

4. Games as state spaces • state space variables describe relevant features of game • start state(s) define initial conditions for play • any legal state of the game is a state in the space • transition edges in the space define legal moves by players • two player turn-taking games define bi-partite state spaces • terminal states (no out-edges) are determined by a ‘terminal test’ and define end-of-game D Goforth - COSC 4117, fall 2003

5. Example game: • turn-taking zero-sum game: • two players: Max (plays first), Min • n tokens • rules: take 1, 2 or 3 tokens • start state: 5 tokens, Max to play • goal: take last token D Goforth - COSC 4117, fall 2003

6. Example game: state space Turn Max Min Max Min Max 5 4 3 2 3 2 1 2 1 0 1 0 2 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 State: (number of tokens remaining, whose turn) e.g., (2,Max) D Goforth - COSC 4117, fall 2003

7. Example game: Max’s preferences Turn Max Min Max Min Max 5 4 3 2 - - 3 2 1 2 1 0 1 0 + + + + + + 2 1 0 1 0 0 1 0 0 0 - - - - 1 0 0 0 0 + 0 evaluation function for Max: + for win (0, Min), - for loss at terminal state (0, Max) D Goforth - COSC 4117, fall 2003

8. Example game: Max’s move, why? Turn Max Min Max Min Max 5 - - + 4 3 2 - - + + + + + + 3 2 1 2 1 0 1 0 - - - - + + + + + + 2 1 0 1 0 0 1 0 0 0 - - - - + 1 0 0 0 0 + 0 see p.166, Fig. 6.3 Minimaxback propagation of terminal states assumption: opponent (Min) is also smart D Goforth - COSC 4117, fall 2003

9. Minimax algorithm • Back propagation in dynamic environment • evaluate state space to decide one move • attempt to find move that is best for all possible reactions • Minimax assumption • worst case assumption about dynamic aspect of environment (opponent’s choice) • if assumption wrong, situation is better than assumed D Goforth - COSC 4117, fall 2003

10. Minimax algorithm • Deterministic if • environment is deterministic (no random factors) • Exhaustive search to terminal states - time complexity is O(bm) b: number of moves in a game m: number of actions per move e.g. chess b  50, m  20, bm  1033 D Goforth - COSC 4117, fall 2003

11. Minimax search ininteresting games • space is too large to search to terminal states (except possibly in endgames) • use of heuristic functions to evaluate partial paths • deeper search evaluates ‘closer’ to terminal states D Goforth - COSC 4117, fall 2003

12. Minimax in large state space maximization minimization heuristic evaluation from viewpoint of Max D Goforth - COSC 4117, fall 2003

13. The search-evaluate tradeoff • branching factor n • execution time for heuristic evaluation t • search to level k • total time: nkt = nk-1(nt)  to go a level deeper in same time, evaluation function must be n times more efficient • special situations: start game, end game D Goforth - COSC 4117, fall 2003