mark dunlop, computer and information sciences, strathclyde university mdd/ 1 algorithms &...

Mark Dunlop, Computer and Information Sciences, Strathclyde Universityhttp://www.cis.strath.ac.uk/~mdd/

1

Algorithms & Complexity 5Games

Mark D [email protected]

14.6


2

How computers play games

• Chess, draughts and naughts and crosses

• We'll do naughts and crosses– its easiest


3

Naughts and crosses

• Three X or thee O in a row to win

• Can guarantee not to lose

• Can grab a win when available


4

Minimax strategy• Use an evaluation strategy to quantify

the goodness of a position

• Terminal cases: A win for the computer gives +1, a draw 0, a loss -1

• Non-terminal: calculated by recursively assuming best play by both computer and human

• Minimax - human tries to minimise the computer goodness function, computer tries to maximise it and we take turns


5

Basic Strategy

• "If I (computer) move there, then my opponent will probably move there, then I can move there, ..., and win!"


6

Next position• The successor position of P is any position

P’ reachable from P in one move– Computer move:

• look at all next positions and calculate their score (recursively)

• pick next move with maximum score

– Guess of human move:• look at all next positions and calculate their score• assume human plays move with minimum score


7

Some Psuedo Code

public moveInfo findComputerMove()//computer plays X, human Oif board is full

return new moveInfo(DRAW)else if can_win_in_one_move

return new moveInfo(winmove,WIN)else

maxvalue = LOSE

for i=1..9if square i is empty place (i, 'X') tryvalue = findHumanMove().value

emptycell (i) if tryvalue > maxvalue maxvalue = tryvalue; bestmove = i

return new moveInfo(bestmove, maxvalue)findHumanMove is very similar


8

Problems• All this says is

– if I (computer) move there then I can win if user plays best as I expect

– If I move there the best I can do is draw– If I move there I can only lose

• Better to return some function of the probability of winning, i.e. – if I move here then for 1/6 human moves I can win

or– If I move here then for 5/6 human moves I can win


9

Complexity• Main loop cycles round 9 squares

• For each square - calculate all possible future values from that move, e.g first move– computer has 9 choices– human then has 8– computer then has 7– human then has 6– computer then has 5 and might win or not, etc...

• 9x8x7x6x5x4x3x2x1 = 362 880 combinations• Of which 97 162 are possible (stop after a win)


10

Complexity• If the computer moves first it has 97 162

positions to evaluate• If the human then picks the centre square,

the computers next turn has 5 185 evaluations (9 761 for a corner, 13 233 for a side)

• For chess it is estimated there are 10^100 positions to be examined to decide the best first move

• Standard openings help but in general, this is still too bad...


11

Complexity reducers

• Only look so far ahead– requires some function to evaluate the

strength of a current board– in chess these functions can be very

complex– still the ply or number of levels look ahead

is still the big performance factor


12

Complexity reducers

• Remember already calculated positions– Use a transposition table - almost always

a hash table– When you calculate a position you store it

in the table– Next time you face a position, see if it is

there and just read off the value (note: there are many routes to one position)

• Only look ahead where it looks good


13

Sample game tree

Note: no tree is actually created - this is the tree of recursive calls, or choices we can make.

44 max

44 36 min

44 78 40 36 max

27 44 68 78 40 27 36 36 min

42 27 86 44 68 73 78 79 73 40 27 87 68 36 55 36 max

42 25 27 7 72 86 9 44 50 68 73 31 78 17 79 37 73 23 30 40 19 27 64 87 57 68 29 36 5 55 12 36


14

Alpha-Beta Pruning• Another algorithmic design pattern

• We can ignore large parts of the tree because we can't do better than already found, e.g.

44 max

44 40 min

44 50 40 max

27 44 50 40 27 min

42 27 86 44 50 73 73 40 27 max

42 25 27 7 72 86 9 44 50 73 73 23 30 40 19 27


15

How to beat a chess program

• Most chess programs (and real games) have a time limit per move

• The programs optimise their "thinking" using alpha-beta pruning to maximise the depth in places where they are likely to be asked to play, i.e. they assume the human plays well


16

How to beat a chess program

• If you play very badly for one move, the computer will run out of time on its next shot since it will not have calculated that whole chunk of the game tree

• Grand masters didn't take long to work that one out...


17

Summary• MiniMax strategy for making best choice for

computer, and best from human too

• Look ahead level, ply, & evaluation function are both important ply usually dominates

• Transposition tables remember positions

• ALPHA-BETA pruning reduce number of nodes to calculate based on best/worst already seen

• You throw a chess program by playing badly (once...)

mark dunlop, computer and information sciences, strathclyde university mdd/ 1 algorithms &...

Documents