Game playing

Game playing

Chapter 5, Sections 1–5

Chapter 5, Sections 1–5 1

Outline

♦ Perfect play

♦ Resource limits

♦ α–β pruning

♦ Games of chance


Games

One of the oldest areas of AI

Chess programs were especially chosen because success would be a proof ofa machine doing something intelligent.


Games vs. Search problems: Uncertainty

The presence of an opponent that introduces uncertainty makes the decisionproblem more complicated than search problems.


Games vs. search problems

“Unpredictable” opponent ⇒ solution is a strategyspecifying a move for every possible opponent reply


Games vs. Search problems: Time constraints

Real problem is that games are usually much too hard to solve:

In chess:

♦ Average branching factor: 35

♦ Games go to about 50 moves by each player

⇒ 35100 nodes! 1040 different legal positions

Time limits ⇒ unlikely to find goal, must approximate

On the other hand tic-tac-toe is boring because it is too simple to determinethe best move.


Games vs. Search problems

The complexity of games introduces a new kind of uncertainty:

not due to lack of information but because one does not have time to cal-culate the exact consequences of any move.


Types of games

deterministic chance

perfect information

imperfect information

chess, checkers,go, othello

backgammonmonopoly

bridge, poker, scrabblenuclear war


Perfect Decisions in Two-Person Games

A game can be formally defined as a kind of search problem with:

♦ initial state (of the board and whose turn it is)

♦ set of operators (which define the legal moves)

♦ terminal test (goal test )

♦ utility function: (numeric value for the outcome of a game)

Ex. backgammon (+1, -1, +2); Chess (win, lose, draw)... which is a zero-sum game.


Perfect Decisions in Two-Person Games

Players: MAX and MIN taking turns until game is over


Search Tree for the Game Tic-Tac-Toe

XXXX

XX

X

XX

MAX (X)

MIN (O)

X X

O

OOX O

OO O

O OO

MAX (X)

X OX OX O XX X

XX

X X

MIN (O)

X O X X O X X O X

. . . . . . . . . . . .

. . .

. . .

. . .

TERMINALXX

−1 0 +1Utility


Minimax

Minimax algorithm is designed to determine the optimal strategy for MAX:Perfect play for deterministic, perfect-information games

Idea: choose move to position with highest minimax value= best achievable payoff against best play


Minimax

E.g., 2-ply game:MAX

3 12 8 642 14 5 2

MIN

3

A1

A3

A2

A13A

12A

11A

21 A23

A22

A33A

32A

31

3 2 2


Minimax algorithm

function Minimax-Decision(game) returns an operator

for each op in Operators[game] do

Value[op]←Minimax-Value(Apply(op, game), game)

end

return the op with the highest Value[op]

function Minimax-Value(state, game) returns a utility value

if Terminal-Test[game](state) then

return Utility[game](state)

else if max is to move in state then

return the highest Minimax-Value of Successors(state)

else

return the lowest Minimax-Value of Successors(state)


Minimax algorithm

The optimal strategy can be determined by examining the minimax value ofeach node.

Max maximizes its worst-case outcome!

Recursive search.


Properties of minimax

Minimax: Maximizes the utility under the assumption that the opponentwill play perfectly to minimize it.

Complete??

Optimal??

Time complexity??

Space complexity??


Properties of minimax

Complete?? Yes, if tree is finite (chess has specific rules for this)

Optimal?? Yes, against an optimal opponent. Otherwise??

Time complexity?? O(bm)

Space complexity?? O(bm) (depth-first exploration)

For chess, b ≈ 35, m ≈ 100 for “reasonable” games⇒ exact solution completely infeasible


Imperfect decisions

The minimax algorithm assumes that the program has time to search all theway down to terminal states, which is usually not practical.

Shannon proposed that instead of going all the way down to terminal statesand using the utility function, the program should cut-off the search ear-lier, and apply a heuristic evaluation function to the leaves of the tree.


Resource limits

Suppose we have 100 seconds, explore 104 nodes/second⇒ 106 nodes per move

Standard approach:

• cutoff teste.g., depth limit

• evaluation function= estimated desirability of position


Resource limits

• cutoff teste.g., depth limitReplace terminal test by CUTOFF-TEST in MINIMAX-VALUE

• evaluation function= estimated desirability of positionReplace Utility by EVAL in MINIMAX-VALUE


function Minimax-Decision(game) returns an operator

for each op in Operators[game] do

Value[op]←Minimax-Value(Apply(op, game), game)

end

return the op with the highest Value[op]

function Minimax-Value(state, game) returns a utility value

if Terminal-Test[game](state) then

return Utility[game](state)

else if max is to move in state then

return the highest Minimax-Value of Successors(state)

else

return the lowest Minimax-Value of Successors(state)


Evaluation functions

Estimate of the expected utility of the game from a given position.

E.g. material value for each piece: 1 for pawn, 3 for knight or bishop,...

Performance of a game-playing program is extremely depen-dent on the quality of its evaluation function.



Black to move

White slightly better

White to move

Black winning

For chess, typically linear weighted sum of features

Eval(s) = w1f1(s) + w2f2(s) + . . . + wnfn(s)

e.g., w1 = 9 withf1(s) = (number of white queens) – (number of black queens)


Where do the weights come from?



Evaluation functions:

♦ should agree with the utility function on terminal states.

♦ must not take too long to calculate!

♦ should accurately reflect the chances of winning (if we have to cut-off,we do not know what will happen in subsequent moves)


Cutting off search

MinimaxCutoff is identical to MinimaxValue except1. Terminal? is replaced by Cutoff?2. Utility is replaced by Eval

Does it work in practice?


Cutting off search

Suppose we have 100 seconds to decide on a move and we can explore 104

nodes/second⇒ 106 nodes per move

bm = 106, b = 35 ⇒ m = 4

4-ply lookahead is a hopeless chess player!

4-ply ≈ human novice8-ply ≈ typical PC, human master12-ply ≈ Deep Blue, Kasparov


Exact values don’t matter

MIN

MAX

21

1

42

2

20

1

1 40020

20

Behaviour is preserved under any monotonic transformation of Eval

Only the order matters:payoff in deterministic games acts as an ordinal utility function


Cutting off search

Should look further:

♦ in positions where favorable captures can be made (non-quiescent posi-tions)

♦ in positions where unavoidable (but beyond the horizon moves) will affectthe situation drastically

e.g. pawn turning into queen


α–β pruning

With minimax 4-ply possible, but even average human players can makeplans 6-8 ply ahead!

How can we improve minimax search?


α–β pruning example

MAX

3 12 8

MIN 3

3



MAX

3 12 8

MIN 3

2

2

X X

3



MAX

3 12 8

MIN 3

2

2

X X14

14

3



MAX

3 12 8

MIN 3

2

2

X X14

14

5

5

3



MAX

3 12 8

MIN

3

3

2

2

X X14

14

5

5

2

2

3


Properties of α–β

Pruning does not affect final result

Good move ordering improves effectiveness of pruning (e.g. pick move withb=2 rather than b=14)

With “perfect ordering,” time complexity = O(bm/2)⇒ doubles depth of search⇒ can easily reach depth 8 and play good chess


Why is it called α–β?

..

..

..

MAX

MIN

MAX

MIN V

α is the best value (to max) found so far off the current path

If V is worse than α, max will avoid it ⇒ prune that branch

Define β similarly for min


The α–β algorithm

Basically Minimax + keep track of α, β + prune


function Max-Value(state, game,α,β) returns the minimax value of state

inputs: state, current state in game

game, game description

α, the best score for max along the path to state

β, the best score for min along the path to state

if Cutoff-Test(state) then return Eval(state)

for each s in Successors(state) do

α←Max(α,Min-Value(s, game,α,β))

if α ≥ β then return β

end

return α

function Min-Value(state, game,α,β) returns the minimax value of state

if Cutoff-Test(state) then return Eval(state)

for each s in Successors(state) do

β←Min(β,Max-Value(s, game,α,β))

if β ≤ α then return α

end

return β


Minimax algorithm

MAX

MIN

99 1000 1000 1000 100 101 102 100

10099

Minimax is an optimal method for selecting a move from a given search treeprovided that the leaf node evaluations are exactly correct.

It is quite likely that the value of the left brach is higher.


Deterministic games in practice

Checkers: Chinook ended 40-year-reign of human world champion MarionTinsley in 1994. Used an endgame database defining perfect play for allpositions involving 8 or fewer pieces on the board, a total of 443,748,401,247positions.

Chess: Deep Blue defeated human world champion Gary Kasparov in a six-game match in 1997. Deep Blue searches 200 million positions per second,uses very sophisticated evaluation, and undisclosed methods for extendingsome lines of search up to 40 ply. Baron von Kempelen’s ”Turk” in 1769!

Backgammon: First program to make a serious impact, BKG, used only aone-ply search but a very complicated evaluation function (1980). It plays astrong amateur level. In 1992, neural network techniques to learn evaluationfunction.

Othello/Reversi: Smaller search space (b = 5 − 15). Human championsrefuse to compete against computers, who are too good.


Go: human champions refuse to compete against computers, who are toobad. In go, b > 300, so most programs use pattern knowledge bases tosuggest plausible moves.


Nondeterministic games in practice

1000

2000

3000

0

1970 1980 1990

Mac

Hac

k (1

400)

Che

ss 3

.0 (

1500

)

Che

ss 4

.6 (

1900

)

Bel

le (

2200

)

Hite

ch (

2400

)

Dee

p T

houg

ht (

2551

)

1965 1975 1985

Fisc

her

(278

5)

Kar

pov

(270

5)

Spas

sky

(248

0)

Petr

osia

n (2

363)

Kas

paro

v (2

740)

1960

Bot

vinn

ik (

2616

)

Kor

chno

i (26

45)

Kas

paro

v (2

805)

Dee

p T

houg

ht 2

(ap

prox

. 260

0)

Extrapolate?


Nondeterministic games

E..g, in backgammon, the dice rolls determine the legal moves

Simplified example with coin-flipping instead of dice-rolling:

MIN

MAX

2

CHANCE

4 7 4 6 0 5 −2

2 4 0 −2

0.5 0.5 0.5 0.5

3 −1


Algorithm for nondeterministic games

Expectiminimax gives perfect play

Just like Minimax, except we must also handle chance nodes:

. . .if state is a chance node then

return average of ExpectiMinimax-Value of Successors(state)


Nondeterministic games

DICE

MIN

MAX

DICE

MAX

. . .

. . .

. . .

B

2 1 −1 1−1

. . .6,66,51,1

1/361,21/18

TERMINAL

6,66,51,11/36

1,21/18

......

.........

.........

...... ......

...C

Rest?



Dice rolls increase b: 21 possible rolls with 2 dice

Backgammon ≈ 20 legal moves

depth 4 = 20× (21× 20)3 ≈ 1.2× 109

As depth increases, probability of reaching a given node shrinks⇒ value of lookahead is diminished

TDGammon uses depth-2 search + very good Eval≈ world-champion level



α–β pruning is much less effective:

The advantage of α–β pruning is that it ignores future developments thatjust are not going to happen, given best play.

Thus in concentrates on likely moves.

In games with dice, there are no likely sequences of moves, because for thosmoves to take place, the dice would have to come out the right way to makethem legal.


Digression: Exact values DO matter

DICE

MIN

MAX

2 2 3 3 1 1 4 4

2 3 1 4

.9 .1 .9 .1

2.1 1.3

20 20 30 30 1 1 400 400

20 30 1 400

.9 .1 .9 .1

21 40.9

Behaviour is preserved only by positive linear transformation of Eval

Hence Eval should be proportional to the expected payoff


Summary

Games are fun to work on! (and dangerous)

They illustrate several important points about AI

♦ perfection is unattainable ⇒ must approximate

♦ good idea to think about what to think about

♦ uncertainty constrains the assignment of values to states

Games are to AI as grand prix racing is to automobile design


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Game playing