games as adversarial search problems dynamic state space search
TRANSCRIPT
Games as adversarialsearch problems
Dynamic state space search
D Goforth - COSC 4117, fall 2003 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 spaceTurn
Max
Min
Max
Min
Max
5
4
3
2
1
0
0
2
1 01
0
1
00
3
2
1
0
0
1
0
0
2
1 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 preferencesTurn
Max
Min
Max
Min
Max
5
4
3
2
1
0
0
2
1 01
0
1
00
3
2
1
0
0
1
0
0
2
1 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
1
0
0
2
1 01
0
1
00
3
2
1
0
0
1
0
0
2
1 0
0
0
Minimax back propagation of terminal statesassumption: opponent (Min) is also smart
+
+ + + + + +
- - - -
- -
- - - -
+
+
+ + + + +
--
+
see p.166, Fig. 6.3
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 gamem: 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
heuristic evaluation from viewpoint of Max
minimization
maximization
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