1. Games as adversarial search problems Dynamic state space search

2. D Goforth - COSC 4117, fall 20032 Requirements of adversarial game space search  on-line search: planning cannot be completed before action multi-agent environment dynamic environment

3. D Goforth - COSC 4117, fall 20033 Features of gamesKing’s Court  deterministic/stochastic  perfect/partial information  number of agents: n>1  optimization function  interaction scheduling  deterministic  perfect  2  zero-sum  turn-taking

4. D Goforth - COSC 4117, fall 20034 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

5. D Goforth - COSC 4117, fall 20035 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

6. D Goforth - COSC 4117, fall 20036 Example game: state space Turn Max Min Max Min Max 5 4 3 2 1 0 0 2 101 0 1 00 3 2 1 0 0 1 0 0 2 10 0 0 State: (number of tokens remaining, whose turn) e.g., (2,Max)

7. D Goforth - COSC 4117, fall 20037 Example game: Max’s preferences Turn Max Min Max Min Max 5 4 3 2 1 0 0 2 101 0 1 00 3 2 1 0 0 1 0 0 2 10 0 0 evaluation function for Max: + for win (0, Min), - for loss at terminal state (0, Max) + ++ + +++ ---- --

8. D Goforth - COSC 4117, fall 20038 Example game: Max’s move, why? Turn Max Min Max Min Max 5 4 3 2 1 0 0 2 101 0 1 00 3 2 1 0 0 1 0 0 2 10 0 0 Minimax back propagation of terminal states assumption: opponent (Min) is also smart + ++ + +++ ---- -- ---- + + + + +++ -- + see p.166, Fig. 6.3

9. D Goforth - COSC 4117, fall 20039 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

10. D Goforth - COSC 4117, fall 200310 Minimax algorithm  Deterministic if environment is deterministic (no random factors) Exhaustive search to terminal states - time complexity is O(b m ) b: number of moves in a game m: number of actions per move e.g. chess b  50, m  20, b m  10 33

11. D Goforth - COSC 4117, fall 200311 Minimax search in interesting 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

12. D Goforth - COSC 4117, fall 200312 Minimax in large state space heuristic evaluation from viewpoint of Max minimization maximization

13. D Goforth - COSC 4117, fall 200313 The search-evaluate tradeoff  branching factor n  execution time for heuristic evaluation t  search to level k  total time: n k t = n k-1 (nt)  to go a level deeper in same time, evaluation function must be n times more efficient  special situations: start game, end game