Game playing

Outline

• Optimal decisions• α-β pruning• Imperfect, real-time decisions

Games vs. search problems

• "Unpredictable" opponent specifying a move for every possible opponent repl

• Time limits unlikely to find goal, must approximate

Game tree (2-player, deterministic, turns)

Minimax

• Perfect play for deterministic games• Idea: choose move to position with highest

minimax value = best achievable payoff against best play

• E.g., 2-ply game:

•

Minimax algorithm

Properties of minimax

• Complete? Yes (if tree is finite)• Optimal? Yes (against an optimal opponent)• Time complexity? O(bm)• Space complexity? O(bm) (depth-first exploration)

• For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible

α-β pruning example

α-β pruning example

α-β pruning example

α-β pruning example

α-β pruning example

Properties of α-β

• Pruning does not affect final result

• Good move ordering improves effectiveness of pruning

• With "perfect ordering," time complexity = O(bm/2) doubles depth of search

• A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)

Why is it called α-β?

• α is the value of the best (i.e., highest-value) choice found so far at any choice point along the path for max

• If v is worse than α, max will avoid it prune that branch

• Define β similarly for min

The α-β algorithm

The α-β algorithm

How much do we gain?

Assume a game tree of uniform branching factor b Minimax examines O(bh) nodes, so does alpha-beta

in the worst-case The gain for alpha-beta is maximum when:

• The MIN children of a MAX node are ordered in decreasing backed up values

• The MAX children of a MIN node are ordered in increasing backed up values

Then alpha-beta examines O(bh/2) nodes [Knuth and Moore, 1975]

But this requires an oracle (if we knew how to order nodes perfectly, we would not need to search the game tree)

If nodes are ordered at random, then the average number of nodes examined by alpha-beta is ~O(b3h/4)

Heuristic Ordering of Nodes

Order the nodes below the root according to the values backed-up at the previous iteration

Order MAX (resp. MIN) nodes in decreasing (increasing) values of the evaluation function computed at these nodes

Games of imperfect information

• Minimax and alpha-beta pruning require too much leaf-node evaluations.

• May be impractical within a reasonable amount of time.

• SHANNON (1950):– Cut off search earlier (replace

TERMINAL-TEST by CUTOFF-TEST)– Apply heuristic evaluation function EVAL

(replacing utility function of alpha-beta)

Cutting off search

• Change:– if TERMINAL-TEST(state) then return

UTILITY(state)into– if CUTOFF-TEST(state,depth) then return EVAL(state)

• Introduces a fixed-depth limit depth– Is selected so that the amount of time will not exceed

what the rules of the game allow.

• When cuttoff occurs, the evaluation is performed.

Heuristic EVAL

• Idea: produce an estimate of the expected utility of the game from a given position.

• Performance depends on quality of EVAL.• Requirements:

– EVAL should order terminal-nodes in the same way as UTILITY.

– Computation may not take too long.– For non-terminal states the EVAL should be

strongly correlated with the actual chance of winning.

• Only useful for quiescent (no wild swings in value in near future) states

Heuristic EVAL example

Eval(s) = w1 f1(s) + w2 f2(s) + … + wnfn(s)

Heuristic EVAL example

Eval(s) = w1 f1(s) + w2 f2(s) + … + wnfn(s)

Addition assumes independence

Heuristic difficulties

Heuristic counts pieces won

Horizon effectFixed depth search thinks it can avoidthe queening move

Games that include chance

• Possible moves (5-10,5-11), (5-11,19-24),(5-10,10-16) and (5-11,11-16)

Games that include chance

• Possible moves (5-10,5-11), (5-11,19-24),(5-10,10-16) and (5-11,11-16)

• [1,1], [6,6] chance 1/36, all other chance 1/18

chance nodes

Games that include chance

• [1,1], [6,6] chance 1/36, all other chance 1/18 • Can not calculate definite minimax value, only

expected value

Expected minimax value

EXPECTED-MINIMAX-VALUE(n)=UTILITY(n) if n is a terminalmaxs successors(n) MINIMAX-VALUE(s) if n is a max node

mins successors(n) MINIMAX-VALUE(s) if n is a max node

s successors(n) P(s) . EXPECTEDMINIMAX(s) if n is a chance node

These equations can be backed-up recursively all the way to the root of the game tree.

Position evaluation with chance nodes

• Left, A1 wins• Right A2 wins• Outcome of evaluation function may not change

when values are scaled differently.• Behavior is preserved only by a positive linear

transformation of EVAL.

State-of-the-Art

Checkers: Tinsley vs. Chinook

Name: Marion TinsleyProfession:Teach mathematicsHobby: CheckersRecord: Over 42 years loses only 3 games of checkersWorld champion for over 40 years

Mr. Tinsley suffered his 4th and 5th losses against Chinook

Chinook

First computer to become official world champion of Checkers!Has all endgame table for 10 pieces or less: over 39 trillion entries.

Chess: Kasparov vs. Deep Blue

Kasparov

5’10” 176 lbs 34 years50 billion neurons

2 pos/secExtensiveElectrical/chemicalEnormous

HeightWeight

AgeComputers

SpeedKnowledge

Power SourceEgo

Deep Blue

6’ 5”2,400 lbs

4 years32 RISC processors

+ 256 VLSI chess engines200,000,000 pos/sec

PrimitiveElectrical

None

1997: Deep Blue wins by 3 wins, 1 loss, and 2 draws

Chess: Kasparov vs. Deep Junior

August 2, 2003: Match ends in a 3/3 tie!

Deep Junior

8 CPU, 8 GB RAM, Win 2000

2,000,000 pos/secAvailable at $100

Othello: Murakami vs. Logistello

Takeshi MurakamiWorld Othello Champion

1997: The Logistello software crushed Murakami by 6 games to 0

Go: Goemate vs. ??

Name: Chen ZhixingProfession: RetiredComputer skills:

self-taught programmerAuthor of Goemate (arguably the

best Go program available today)

Gave Goemate a 9 stonehandicap and still easilybeat the program,thereby winning $15,000

Go: Goemate vs. ??

Name: Chen ZhixingProfession: RetiredComputer skills:

self-taught programmerAuthor of Goemate (arguably the

strongest Go programs)

Gave Goemate a 9 stonehandicap and still easilybeat the program,thereby winning $15,000

Jonathan Schaeffer

Go has too high a branching factor for existing search techniques

Current and future software must rely on huge databases and pattern-recognition techniques

Go has too high a branching factor for existing search techniques

Current and future software must rely on huge databases and pattern-recognition techniques

Backgammon

• 1995 TD-Gammon by Gerald Thesauro won world championship on 1995

• BGBlitz won 2008 computer backgammon olympiad

Secrets Many game programs are based on alpha-beta +

iterative deepening + extended/singular search + transposition tables + huge databases + ...

For instance, Chinook searched all checkers configurations with 8 pieces or less and created an endgame database of 444 billion board configurations

The methods are general, but their implementation is dramatically improved by many specifically tuned-up enhancements (e.g., the evaluation functions) like an F1 racing car

Perspective on Games: Con and Pro

Chess is the Drosophila of artificial intelligence. However, computer chess has developed much as genetics might have if the geneticists had concentrated their efforts starting in 1910 on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies.

John McCarthy

Saying Deep Blue doesn’t really think about chess is like saying an airplane

doesn't really fly because it doesn't flap

its wings.

Drew McDermott

Other Types of Games

Multi-player games, with alliances or not

Games with randomness in successor function (e.g., rolling a dice) Expectminimax algorithm

Games with partially observable states (e.g., card games) Search of belief state spaces

See R&N p. 175-180