Game Playing 2

transcript

GAME PLAYING 2

THIS LECTURE Alpha-beta pruning Games with chance Partially observable games

NONDETERMINISM Uncertainty is caused by the actions of

another agent (MIN), who competes with our agent (MAX)

MAX’s play

MAX cannot tell what move will be played

MIN’s play

NONDETERMINISM Uncertainty is caused by the actions of

another agent (MIN), who competes with our agent (MAX)

MAX’s play

MAX must decide what to play for BOTH these outcomes

MIN’s playInstead of a single path, the agent must construct an entire plan

MINIMAX BACKUP

MIN’s turn

MAX’s turn

DEPTH-FIRST MINIMAX ALGORITHM MAX-Value(S)

1. If Terminal?(S) return Result(S)2. Return maxS’SUCC(S) MIN-Value(S’)

MIN-Value(S)1. If Terminal?(S) return Result(S)2. Return minS’SUCC(S) MAX-Value(S’)

MINIMAX-Decision(S) Return action leading to state S’SUCC(S) that

maximizes MIN-Value(S’)

REAL-TIME GAME PLAYING WITH EVALUATION FUNCTION e(s): function indicating estimated

favorability of a state to MAX Keep track of depth, and add line:

If(depth(s) = cutoff) return e(s) After terminal test

CAN WE DO BETTER? Yes ! Much better !

Pruning

This part of the tree can’t have any effect on the value that will be backed up to the root

EXAMPLE

The beta value of a MINnode is an upper bound onthe final backed-up value.It can never increase

EXAMPLE

The beta value of a MINnode is an upper bound onthe final backed-up value.It can never increase

EXAMPLE

The alpha value of a MAXnode is a lower bound onthe final backed-up value.It can never decrease

EXAMPLE

b = -1

EXAMPLE

b = -1

Search can be discontinued belowany MIN node whose beta value is less than or equal to the alpha valueof one of its MAX ancestors

ALPHA-BETA PRUNING Explore the game tree to depth h in depth-

first manner Back up alpha and beta values whenever

possible Prune branches that can’t lead to changing

the final decision

ALPHA-BETA ALGORITHM Update the alpha/beta value of the parent of

a node N when the search below N has been completed or discontinued

Discontinue the search below a MAX node N if its alpha value is the beta value of a MIN ancestor of N

Discontinue the search below a MIN node N if its beta value is the alpha value of a MAX ancestor of N

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

EXAMPLE

0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35

0 -3 3

HOW MUCH DO WE GAIN? Consider these two cases:

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

The 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)

ALPHA-BETA IMPLEMENTATION MAX-Value(S,a,b)

1. If Terminal?(S) return Result(S)2. For all S’SUCC(S)3. a max(a,MIN-Value(S’,a,b))4. If a b, then return a5. Return a

MIN-Value(S,a,b)1. If Terminal?(S) return Result(S)2. For all S’SUCC(S)3. b min(b,MAX-Value(S’,a,b))4. If a b, then return b5. Return b

Alpha-Beta-Decision(S) Return action leading to state S’SUCC(S) that maximizes MIN-

Value(S’,-,+)

HEURISTIC ORDERING OF NODES Order the nodes below the root according to

the values backed-up at the previous iteration

OTHER IMPROVEMENTS Adaptive horizon + iterative deepening Extended search: Retain k>1 best paths,

instead of just one, and extend the tree at greater depth below their leaf nodes (to help dealing with the “horizon effect”)

Singular extension: If a move is obviously better than the others in a node at horizon h, then expand this node along this move

Use transposition tables to deal with repeated states

Null-move search

GAMES OF CHANCE

GAMES OF CHANCE Dice games: backgammon, Yahtzee, craps, … Card games: poker, blackjack, …

Is there a fundamental difference between the nondeterminism in chess-playing vs. the nondeterminism in a dice roll?

CHANCE

EXPECTED VALUES The utility of a MAX/MIN node in the game

tree is the max/min of the utility values of its successors

The expected utility of a CHANCE node in the game tree is the average of the utility values of its successors

ExpectedValue(s) = s’SUCC(s) ExpectedValue(s’) P(s’)

MinimaxValue(s) = max s’SUCC(s) MinimaxValue(s’)Compare to

MinimaxValue(s) = min s’SUCC(s) MinimaxValue(s’)

CHANCE nodes

MAX nodes

MIN nodes

ADVERSARIAL GAMES OF CHANCE E.g., Backgammon MAX nodes, MIN nodes, CHANCE nodes Expectiminimax search Backup step:

MAX = maximum of children CHANCE = average of children MIN = minimum of children CHANCE = average of children

4 levels of the game tree separate each of MAX’s turns!

Evaluation function? Pruning?

GENERALIZING MINIMAX VALUES Utilities can be continuous numerical values,

rather than +1,0,-1 Allows maximizing the amount of “points” (e.g.,

$) rewarded instead of just achieving a win Rewards associated with terminal states Costs can be associated with certain

decisions at non-terminal states (e.g., placing a bet)

ROULETTE “Game tree” only has depth 2

Place a bet Observe the roulette wheel

No bet

Bet: Red, $5

Red Not red

Chance node

18/38 20/38Probabilities

CHANCE NODE BACKUP Expected value:

For k children, with backed up values v1,…,vk

Chance node value =p1 * v1 + p2 * v2 + … + pk * vk

Red Not red

Chance node

18/38 20/38Probabilities

Bet: Red, $5

Value:18/38 * 10 + 20/38 * 0= 4.74

MAX/CHANCE NODES

Red Not red

18/38 20/38

Bet: Red, $5

Chance

Bet: 17, $5

3.95 = 150/38

17 Not 17

1/38 37/38

Max should pick the action leading to the node with the highest value

A SLIGHTLY MORE COMPLEX EXAMPLE

Two fair coins Pay $1 to start, at

which point both are flipped

Can flip up to two coins again, at a cost of $1 each

Payout: $5 for HH, $1 for HT or TH, $0 for TT

1/2 1/2

HT Flip T Flip H

HT HH TTHT

1/2 1/2 1/2 1/2

Flip TFlip HHT

DoneFlip T

1/2 1/2

HT Flip T Flip H

HT HH TTHT

1/2 1/2 1/2 1/2

Flip TFlip HHT

DoneFlip T

1/2 1/2

1-2=-1 5-2=3 -1 -2 -1 -2

1/2 1/2

HT Flip T Flip H

HT HH TTHT

1/2 1/2 1/2 1/2

Flip TFlip HHT

DoneFlip T

1/2 1/2

-1 3 -1 -2 -1 -2

-11 -3/2

1/2 1/2

HT Flip T Flip H

HT HH TTHT

1/2 1/2 1/2 1/2

Flip TFlip HHT

DoneFlip T

1/2 1/2

-1 -2 -1 -2

-11 -3/2

1/2 1/2

HT Flip T Flip H

HT HH TTHT

1/2 1/2 1/2 1/2

Flip TFlip HHT

DoneFlip T

1/2 1/2

-1 -2 -1 -2

-12 -3/2

1/2 1/2

HT Flip T Flip H

HT HH TTHT

1/2 1/2 1/2 1/2

Flip TFlip HHT

DoneFlip T

1/2 1/2

-1 -2 -1 -2

-12 -3/2

1/2 1/2

HT Flip T Flip H

HT HH TTHT

1/2 1/2 1/2 1/2

Flip TFlip HHT

DoneFlip T

1/2 1/2

-1 -2 -1 -2

-12 -3/2

1/2 1/2

HT Flip T Flip H

HT HH TTHT

1/2 1/2 1/2 1/2

Flip TFlip HHT

DoneFlip T

1/2 1/2

-1 -2 -1 -2

-12 -3/2

CARD GAMES Blackjack (6-deck), video poker: similar to

coin-flipping game But in many card games, need to keep track

of history of dealt cards in state because it affects future probabilities One-deck blackjack Bridge Poker

PARTIALLY OBSERVABLE GAMES Partial observability

Don’t see entire state (e.g., other players’ hands) “Fog of war”

Examples: Kriegspiel (see R&N) Battleship Stratego

OBSERVATION OF THE REAL WORLDRealworldin some state

Percepts

On(A,B)

On(B,Table)Handempty

Interpretation of the percepts in the representation language

Percepts can be user’s inputs, sensory data (e.g., image pixels), information received from other agents, ...

SECOND SOURCE OF UNCERTAINTY:IMPERFECT OBSERVATION OF THE WORLD Observation of the world can be: Partial, e.g., a vision sensor can’t see through

obstacles (lack of percepts)

The robot may not know whether there is dust in room R2

obstacles Ambiguous, e.g., percepts have multiple

possible interpretations

On(A,B) On(A,C)

obstacles Ambiguous, e.g., percepts have multiple

possible interpretations Incorrect

PARTIALLY-OBSERVABLE CARD GAMES One possible strategy:

Consider all possible deals given observed information

Solve each deal as a fully-observable problem Choose the move that has the best average

minimax value “Averaging over clairvoyance” [Why doesn’t this always work?]

BELIEF STATE A belief state is the set of all states that an

agent think are possible at any given time or at any stage of planning a course of actions, e.g.:

To plan a course of actions, the agent searches a space of belief states, instead of a space of states

SENSOR MODEL State space S The sensor model is a function

SENSE: S 2S

that maps each state s S to a belief state (the set of all states that the agent would think possible if it were actually observing state s)

Example: Assume our vacuum robot can perfectly sense the room it is in and if there is dust in it. But it can’t sense if there is dust in the other roomSENSE( ) =

SENSE( ) =

VACUUM ROBOT ACTION MODEL

Right either moves the robot right, or does nothing

Left always moves the robot to the left, but it may occasionally deposit dust in the right room

Suck picks up the dirt in the room, if any, and always does the right thing

• The robot perfectly senses the room it is in and whether there is dust in it

• But it can’t sense if there is dust in the other room

TRANSITION BETWEEN BELIEF STATES Suppose the robot is initially in state:

After sensing this state, its belief state is:

Just after executing Left, its belief state will be:

After sensing the new state, its belief state will be:

or if there is no dust if there is dust in R1 in R1

TRANSITION BETWEEN BELIEF STATES Playing a “game against nature”

LeftClean(R1) Clean(R1)

After receiving an observation, the robot will have one of these two belief states

AND/OR TREE OF BELIEF STATES

A goal belief state is one in which all states are goal states

An action is applicable to a belief state B if its preconditions are achieved in all states in B

loop goal

RECAP Alpha-beta pruning: reduce complexity of

minimax to O(bh/2) ideally, O(b3h/4) typically Games with chance

Expected values: averaging over probabilities A 2nd source of uncertainty: partial

observability Reason about sets of states: belief state

Much more on latter 2 topics later

HOMEWORK Reading: R&N 6.1-3

Game Playing 2

Documents