80
GAME PLAYING 2
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
+1
+10
-1
MAX’s turn
0
+10 0
0 -1
+1
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 !
3
-1
Pruning
-1
3
This part of the tree can’t have any effect on the value that will be backed up to the root
EXAMPLE
EXAMPLE
b = 2
2
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
1
b = 1
2
EXAMPLE
a = 1
The alpha value of a MAXnode is a lower bound onthe final backed-up value.It can never decrease
1
b = 1
2
EXAMPLE
a = 1
1
b = 1
2 -1
b = -1
EXAMPLE
a = 1
1
b = 1
2 -1
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
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0 -3
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0 -3
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0 -3
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0 -3 3
3
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0 -3 3
3
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
5
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
5
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
0MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
0
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
0
1
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
2
2
2
2
1
1
MAX
MIN
MAX
MIN
MAX
MIN
EXAMPLE
0 5 -3 25-2 32-3 033 -501 -350 1-55 3 2-35
0
0
0
0 -3 3
3
0
2
2
2
2
1
1
-3
1
1
-5
-5
-5
1
2
2
2
2
1MAX
MIN
MAX
MIN
MAX
MIN
HOW MUCH DO WE GAIN? Consider these two cases:
3
a = 3
-1
b=-1
(4)
3
a = 3
4
b=4
-1
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?
MAX
CHANCE
MIN
CHANCE
MAX
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
+10 0
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
+10 0
Bet: Red, $5
Value:18/38 * 10 + 20/38 * 0= 4.74
MAX/CHANCE NODES
Red Not red
18/38 20/38
+10 0
Bet: Red, $5
4.74
MAX
Chance
Bet: 17, $5
3.95 = 150/38
17 Not 17
1/38 37/38
+150
0
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
HT
HT HH
1/2 1/2
TTHT
1/2 1/2
HT Flip T Flip H
Done
HT HH TTHT
1/2 1/2 1/2 1/2
Flip TFlip HHT
Done
TT
DoneFlip T
HT TT
1/2 1/2
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
HT
HT HH
1/2 1/2
TTHT
1/2 1/2
HT Flip T Flip H
Done
HT HH TTHT
1/2 1/2 1/2 1/2
Flip TFlip HHT
Done
TT
DoneFlip T
HT TT
1/2 1/2
1
5-1=4
1-1=0
1-2=-1 5-2=3 -1 -2 -1 -2
-1
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
HT
HT HH
1/2 1/2
TTHT
1/2 1/2
HT Flip T Flip H
Done
HT HH TTHT
1/2 1/2 1/2 1/2
Flip TFlip HHT
Done
TT
DoneFlip T
HT TT
1/2 1/2
1
4
0
-1 3 -1 -2 -1 -2
-11 -3/2
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
HT
HT HH
1/2 1/2
TTHT
1/2 1/2
HT Flip T Flip H
Done
HT HH TTHT
1/2 1/2 1/2 1/2
Flip TFlip HHT
Done
TT
DoneFlip T
HT TT
1/2 1/2
1
-1 -2 -1 -2
-11 -3/2
1 4
0
-1 3
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
HT
HT HH
1/2 1/2
TTHT
1/2 1/2
HT Flip T Flip H
Done
HT HH TTHT
1/2 1/2 1/2 1/2
Flip TFlip HHT
Done
TT
DoneFlip T
HT TT
1/2 1/2
1
-1 -2 -1 -2
-12 -3/2
2
3
4
0
-1 3
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
HT
HT HH
1/2 1/2
TTHT
1/2 1/2
HT Flip T Flip H
Done
HT HH TTHT
1/2 1/2 1/2 1/2
Flip TFlip HHT
Done
TT
DoneFlip T
HT TT
1/2 1/2
1
-1 -2 -1 -2
-12 -3/2
2
3
-3/2
4
0
-1 3
TTHT
1/2 1/2
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
HT
HT HH
1/2 1/2
HT Flip T Flip H
Done
HT HH TTHT
1/2 1/2 1/2 1/2
Flip TFlip HHT
Done
TT
DoneFlip T
HT TT
1/2 1/2
1
-1 -2 -1 -2
-12 -3/2
2
3
2
-3/2
-1
1/2
4
0
-1 3
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
HT
HT HH
1/2 1/2
TTHT
1/2 1/2
HT Flip T Flip H
Done
HT HH TTHT
1/2 1/2 1/2 1/2
Flip TFlip HHT
Done
TT
DoneFlip T
HT TT
1/2 1/2
1
-1 -2 -1 -2
-12 -3/2
2
3
2
-3/2
-1
1/2
3
4
0
-1 3
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
68
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, ...
69
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)
R1 R2
The robot may not know whether there is dust in room R2
70
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 Ambiguous, e.g., percepts have multiple
possible interpretations
ABC
On(A,B) On(A,C)
71
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 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
Left
Suck
Suck
goal
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
Right
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
