Download Adversarial Search

Search
Can we apply the recently discussed
search methods too all problems?


Why or why not?
How does the search change when there
are multiple actors involved?
Game Playing
A second player (or more) is an adversary

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Examples: chess, checkers, go, tic-tac-toe, connect
four, backgammon, bridge, poker
Adversarial search.
Players take turns or alternate play


Unpredictable opponent.
Solution specifies a move for every possible opponent
move.
Turn time limits

How might they affect the search process?
Types of Games
perfect
information
imperfect
information
deterministic
chance
chess, checkers,
connect four,
othello
backgammon
bridge, poker,
scrabble
Example Game: Nim
Deterministic, Opposite Turns
Two players



One pile of tokens in the middle of the
table.
At each move, the player divides a pile into
two non-empty piles of different sizes.
Player who cannot move looses the game.
Example Game: Nim
7
Max attempts to
maximize the advantage
and win.
5-2
min
Assume a pile of seven tokens.
Min always tries to
move to a state that is
the worst for Max.
6-1
5-1-1
4-2-1
4-3
3-1-1-1-1
3-3-1 min
3-2-2
3-2-1-1
4-1-1-1
max
2-2-2-1 max
2-2-1-1-1 min
2-1-1-1-1-1 max
Perfect Decision, Two person games
Two players

MAX and MIN
MAX moves first; alternate turns thereafter.
Formal definition of game




Initial State
Successor Function
Terminal Test
Utility Function
No one player has full control, must develop a
strategy.
Minimax Algorithm: Basics
3
3
3
MAX
0
9
0
MIN
2
7
2
MAX
6
MIN
2
3
5
9
0
7
4
2
1
5
6
Process of Backing up: minimax decision
Assumption: Both players are knowledgeable and play
the best possible move
Minmax Algorithm
• Generate game tree to all terminal states.
• Apply utility function to each terminal state.
• Propagate utility of terminal states up one level.
•
MAX’s turn: MAX tries to maximize utility value.
•
MIN’s turn: MIN minimizes utility value.
• Continue backing-up values toward root, one
layer at a time.
• At root node, MAX chooses the value with
highest utility.
Minimax Search
What is the time complexity of the minimax
algorithm?
Minimax decision: maximizes utility for a player
assuming that the opponent will play perfectly
and minimize its utility.

m – looking ahead m moves in game tree

Search can be done in a depth first manner
 i.e. traverse down a path till terminal node reached, then
back up value using minimax decision function.
Minimax: Applied to Nim
1
0
6-1
5-1-1
0
7
1
5-2
1 4-2-1
4-1-1-1
min
1
1
0 3-2-2
1 3-2-1-1
1
4-3
max
1 3-3-1 min
0 2-2-2-1 max
2-2-1-1-1 min
3-1-1-1-1
What do we observe from tree?
max
0 2-1-1-1-1-1
0
Partial Search
What were our concerns with the other
search algorithms we have discussed?
•
How do those concerns apply to games?
• Full game tree may be too large to
generate.
Example: Partial Search for Tic-Tac-Toe
• Generate tree to intermediate depth.
• Calculate utility/evaluation function for leaf
nodes.
• Back up values to the root node so MAX
player can make a decision.
Example: Partial Search for Tic-Tac-Toe
Position p:
win for Max, u(p) = 
loss for MAX, u(p) = –
otherwise, u(p) = (# of
complete rows, cols, &
diags still open for MAX) –
(# of complete rows, cols,
and diags still open for MIN)
Tic-Tac-Toe Move 2
MAX Player: 3rd move
Question
If we apply an evaluation function to non-terminal nodes of a
game tree, why generate a partial tree at all?
Why not just compute an evaluation function to all states
generated by applying valid operators (moves) to the current
state, and selecting the one with the highest utility value?
Assumption: An evaluation function applied to successor
nodes produces more reliable (accurate) values.
In other words, these configurations contain more
information on how the game will likely end.
Each level of game tree – termed a ply.
Complex games
What happens if Minimax is applied to large
complex games?

What happens to the search space?
Example, chess:






decent amateur program  1000 moves /second
150 seconds /move (tournament play)
Look at approx. 150,000 moves
Chess branching factor of 35
generating trees that are 3-4 ply,
Resultant play – pure amateur
Complex Games
• Can we make search process more
intelligent, and explore more promising
parts of tree in greater depth?
MAX Player: 3rd move
Search starts
depth first
Back up value
immediately
MIN
Do not
need to
generate
these
nodes!
Why?
The alpha-beta pruning
procedure
Node A generated first
Alpha-Beta Pruning
Definitions


 value – lower bound on MAX node
 value – upper bound on MIN node
Observations


 value on MAX nodes never decrease
 values on MIN nodes never increase
Application


Search is discontinued below any MIN node with minvalue v   :  cut off
Search is discontinued below any MAX node with
max-value v   :  cut off
Alpha-Beta Search Example
MAX
a
b
c
d
f
e
MIN
g
MAX
MIN
1
2
3
4
Node
ab
d
e
c
f
5
7
Alpha
Beta
-∞
3
1
2
3
-∞
4
-∞
3
3
Completed
1
0
2
3
∞
∞
3 CUT-OFF
∞
3 CUT-OFF
∞
6
1
5
Function
Node


V
Return
Max
A
-∞, 3
+∞
-∞, 3
3, 2
Min
B
-∞
+∞, 3
+∞, 3
3, 4
Max
D
-∞,1,2,3
+∞
-∞,1,2,3
1,2,3
Min
1
-∞
+∞
Min
2
1
+∞
Min
3
2
+∞
Max
E
-∞
3
-∞, 4
4
Min
4
-∞
3
Min
C
3
+∞, 6
+∞, 6
6
Max
F
3,4,5,6
+∞
-∞,4,5,6
4,5,6
Min
4
3
+∞
Min
5
4
+∞
Min
6
5
+∞
Max
G
3
6
-∞, 6
6
Min
6
3
6
Cutoff 5 & 7
Cutoff 1 & 5
Key: -∞ = negative infinity; +∞ = positive infinity
The last value in a square is the final value assigned to the specific variable, i.e. at
the end of the search Node A’s  = 3.
Alpha-Beta Search Algorithm
Computing  and 
 value of MAX node = current largest final backedup value of its successors.
 value of MIN node = current smallest final backedup value of its successors.

Start with AB(n; -, +)
Alpha-Beta Search Algorithm
Alpha-Beta-Search(state) returns an action
v = MAX-VALUE(state,-∞, +∞)
return the action in SUCCESSORS(state) with value v
MAX-VALUE(state, , )
if TERMINAL-TEST(state) then
return UTILITY(state)
v = -∞
for a, s in SUCCESSORS(state) do
v = MAX(v, MIN-VALUE(s, , ))
if v >=  then return v
 = MAX(, v)
return v (a utility value)
MIN-VALUE(state, , )
if TERMINAL-TEST(state) then
return UTILITY(state)
v = +∞
for a, s in SUCCESSORS(state) do
v = MIN(v, MAX-VALUE(s, , ))
if v <=  then return v
 = MIN(, v)
return v (a utility value)
Alpha-Beta Search Example 2
MAX
a
b
d
h
c
f
e
j
i
l
k
MIN
g
MAX
m
n
MIN
MAX
7
4 2
5
5
1
0
2
8
7
3
0
1
2
Alpha-Beta Search Example 3
MAX
a
b
d
h
c
f
e
j
i
MIN
l
k
m
g
MAX
n
o
MIN
MAX
5
1 1
2
0
-1
-5 0
-1
0 -4
-3
3
1
1
4
Alpha-Beta Search Performance
Uses depth-first search procedure
Search effectiveness depends on node
ordering


Worst case occurs if worst moves generated
first
Best case occurs if best moves generated
first: min value first for MIN nodes, and max
value first for MAX nodes.
(Is this always possible?)
Alpha-Beta Search Performance
Search tree of depth d, b moves per node


bd tip nodes: Minimax search - O(bd)
Best case:
 look at only O(bd/2) tip nodes
 Effective branching factor reduces to
b so can double
the depth of the search.

Average case:
 Number of nodes examined:
3d
O (b 4 )
 Search depth increase 4/3
Pruning does not
affect final result
Games with chance
Example, Games with dice

Backgammon: white can decide what his/her
legal moves are, but can’t determine black’s
because that depends on what black rolls.
Game tree must include chance nodes.
How to pick best move?

Cannot apply minimax directly.
Games with Chance
For each possible move, compute an
average or expected value.
Expectiminimax (n)  UTILITY ( n )
max
min

sSuccessors( n )
sSuccessors( n )
sSuccessors(n)
if n is a terminal state
Expectiminimax ( s )
if n is a Max node
Expectiminimax ( s )
if n is a Min node
P(s)*Expectiminimax( s )
if n is a chance node
Complexity – O(bmnm) where n is the number of distinct
values (e.g. dice rolls)
Expectimax example
Types of Games
perfect
information
imperfect
information
deterministic
chance
chess, checkers,
connect four,
othello
backgammon
bridge, poker,
scrabble
Game AI Model
AI is given processor time
Execution Management
World Interface
AI receives
information
Group AI
Content Creation
Strategy
Scripting
Character AI
Decision Making
Movement
Animation
Physics
AI is turned into on-screen action
Artificial Intelligence for Games – Ian Millington 2006
AI has implications for
Related technologies
Game AI Schematic
Content Construction
Main Game Engine
AI specific tools
AI data is used to
construct characters
Modeling package
Level Loader
World interface
Extracts relevant
Game data
Game engine
Calls AI each frame
Package level data
Artificial Intelligence for Games – Ian Millington 2006
Behavior database
World
Interface
Level design tool
Per-frame
processing
AI Engine
Results of AI
Are written back
To game data
AI behavior
manager
AI receives data
from the game and
from its
internal information
AI in Games
Types of AI in games


Hacks
Heuristics
 Common Heuristics




Most Constrained
Do the most difficult thing first
Try the most promising thing first
Algorithms
Artificial Intelligence for Games – Ian Millington 2006
AI in Games
Path following
Collision avoidance
Finding paths
World
Representations
Movement planning
Decision making
Tactical analysis
Terrain analysis
Artificial Intelligence for Games – Ian Millington 2006
Coordinating action
Learning
Board Games
Best AI agents for Chess, Backgammon
and Reversi employ dedicated hardware,
algorithms and optimizations.
Basic underlying algorithms are common
across games.


Most popular AI for board games is minimax
family.
Recently Memory-enhanced test driver (MTD)
algorithms are gaining favor.
Artificial Intelligence for Games – Ian Millington 2006
Transposition Tables
Transposition Tables (memory)


Multiple board positions that are identical.
Table contains board positions and results
of a search from that position.
Artificial Intelligence for Games – Ian Millington 2006
Transposition Tables
Hash game states


An hashing algorithm can work but…
Zobrist Keys: a set of fixed-length random
bit patterns stored for each possible state
of each possible location on the board.
 Chess: 64 x 2 x 6 = 768 entries
 For each non-empty square, the Zobrist key is
located and XORed with a running hash total.
Artificial Intelligence for Games – Ian Millington 2006
Transposition Tables
What to store


Minimax: Store value and the depth used
to calculate the value.
Alpha-Beta: store the value, if the value is
accurate or “fail-soft” (because of pruning).
 Alpha pruning: fail-low value
 Beta pruning: fail-high value
Artificial Intelligence for Games – Ian Millington 2006
Transposition Tables
Issues

Path Dependency
 Some games have scores dependent upon the
sequence of moves.
 Repetitions will be identically scored, so
algorithm may disregard a winning move.
 Fix by using a Zorbist key for “number of
repeats”.
Artificial Intelligence for Games – Ian Millington 2006
Transposition Tables
Issues

Instability
 The stored values fluctuate during the same
search since values may be over written.

Not guaranteed to get the same value on each
lookup.
 In rare cases, could have an oscillating value
situation.
Artificial Intelligence for Games – Ian Millington 2006
Memory-Enhanced Test (MT)
Algorithm
Require an efficient transposition table
that acts as memory.

Implemented as a zero-width AB negamax
with a transition table.
 Negamax swaps and inverts the alpha and beta
values and checks and prunes only against the
Beta value.
Artificial Intelligence for Games – Ian Millington 2006
MTD Algorithm
The driver routine repeatedly calls the MT
Algorithm to “zoom in” on the correct minimax
value and determine the next move. – Memoryenhanced Test Drivers.
Artificial Intelligence for Games – Ian Millington 2006
MTD Algorithm
Track the upper bound on the score value – Gamma
Let Gamma be the first guess of the score.
Calculate another guess by calling the MT algorithm for
the current board position, the maximum depth, zero for
the current depth, and the gamma value.
If the returned guess is not the same as gamma, then
calculate another guess in order to confirm that the
guess is correct. (usually limit the number of iterations)
Return the guess as the score.
Artificial Intelligence for Games – Ian Millington 2006
Opening Books
An opening book is a set of moves
combined with information regarding how
good the average outcome is when using
the moves.

Typically at the start of the game.
Play books – certain games have stages
where plays can be employed.
State of the Art: Games
Traditional games




Chess: HITECH (Berliner, CMU) – alpha-beta, horizon effect, 10
million positions per move, accurate evaluation functions: defeated
human grandmaster in 1987
Deep Thought: Hsu, Ananthraman, and others at CMU: generated
720,000 chess positions per second – won World Computer Chess
Championship in 1989
Hsu, Campbell and others, CMU to IBM: more focus on fast
computation – parallel processing to solve complex problems:
evolution from Deep Thought to Deep Blue (1993)
Deep Blue – massively parallel 32 node RISC system + distributed
memory built by IBM. Each node has 8 VLSI chess processors –
therefore, 256 processors working in tandem.
First played Kasparov in 1996, won game 1, but eventually lost to
Kasparov.
State of the Art: Games

May 1997 (Deeper Blue) rematch. Processor speed
doubled, evaluation functions improved with help from a
multitude of grandmasters (chief advisor: Joel Benjamin,
US chess champ)
Deep(er) Blue was generating and evaluating up to 200
million chess positions per sec, i.e., 200 billion moves
every 3 minutes. Typical grandmaster computes only
500 moves in 3 minutes.
Result: Deep Blue: won 2, lost 1, drew 3.
Ques: Was Deep Blue really intelligent?
State of the Art: Games

Chess: (continued) … Quote from Feng-hsuing Hsu:
“The previous version of Deep Blue, lost a match to Gary
Kasparov in Philadelphia in 1996. But 2/3rds of the way into
that match, we had played to a tie with Kasparov. That old
version of Deep Blue was already faster than the machine I
had conjectured in 1985, and yet it was not enough. There
was more to solving the Computer Chess Problem than just
increasing the hardware speed. Since that match, we rebuilt
Deep Blue from scratch, going through every match
problem, and engaging grandmasters extensively in our
preparations. Somehow, all that work cause Grandmaster
Joel Benjamin, our chess advisor, to say, “You know,
sometimes Deep Blue plays chess.” Joel could not distinguish
with certainty Deep Blue’s moves from the moves of top
grandmasters.”
State of the Art: Games
Checkers: Big breakthrough in AI: Samuels in 1952 at IBM; played
itself thousands of times and learned its own evaluation functions.
Evaluation function was a weighted sum of a set of functions.
Best checkers programs now – as good as human champions
Othello: human players refuse to play computer games, who can
play perfectly.
Go: branching factor of game > 300, so straight search doesn’t
work. Have to use pattern knowledge bases.
Does not match up to human champions
20  (21 20)3  1.2 109
State of the Art Games
Backgammon: 2 dice  20 legal moves (can
be 6000 with 1-1 roll)



depth 4 = as depth increases, probability of
reaching nodes decreases, so value of look ahead
is diminished. (alpha-beta not a great idea).
best backgammon game: uses depth 2 search +
very good evaluation function.
plays at world championship level.