EA C461 – Artificial Intelligence
Adversarial Search
S.P.Vimal
http://discovery.bits-pilani.ac.in/~vimalsp/1910AI/
To discuss…
• Game Playing
• Optimal Decisions in Games
– Minimax Algorithm
– Multiplayer Games
• Alpha-Beta Pruning
Vimal
EA C461- Artificial Intelligence
Game playing
• One of the very first tasks AI undertook
• Chess, with average branching factor 35, with
each player taking 50 steps, the number of nodes
generated is 35100.
• Make decisions, apparently no optimal choice
• Inefficiencies penalized severely
• "Unpredictable" opponent  specifying a move
for every possible opponent reply
• Time limits  unlikely to find goal, must
approximate
Vimal
EA C461- Artificial Intelligence
Problems
• Two players
– MAX, MIN
• Components of the problem
– Initial State
– Successor function
– Terminal Test
• Test of states where the game ends
– Utility function
• +1, 0, -1
Vimal
EA C461- Artificial Intelligence
Game tree (2-player, deterministic, turns)
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EA C461- Artificial Intelligence
Game strategy
• An optimal strategy leads to an outcome which is
at least as good as playing against an infallible
opponent
One move
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EA C461- Artificial Intelligence
Minimax Value
Minimax-Value (n) =
Utility(n)
, if n is a terminal node
MaxsεSuccessors(n) Minimax-Value(s) , if n is a Max Node
MinsεSuccessors(n) Minimax-Value(s) , if n is a Min Node
Optimal play for MAX assumes, MIN also plays optimally.
Vimal
EA C461- Artificial Intelligence
Minimax algorithm
Vimal
EA C461- Artificial Intelligence
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
Vimal
EA C461- Artificial Intelligence
Vimal
EA C461- Artificial Intelligence
α-β Pruning
• Prune away those nodes which cannot possibly
influence the final outcome
Vimal
EA C461- Artificial Intelligence
α-β pruning example
Vimal
EA C461- Artificial Intelligence
α-β pruning example
Vimal
EA C461- Artificial Intelligence
α-β pruning example
Vimal
EA C461- Artificial Intelligence
α-β pruning example
Vimal
EA C461- Artificial Intelligence
α-β pruning example
What if successors are examined in a different order…
Vimal
EA C461- Artificial Intelligence
α-β Pruning
Minimax (root) = max( min(3,12,8), min(2,x,y),
min(14,5,2) )
= max( 3, min(2,x,y), 2 )
= max( 3, z, 2 ) ,
z≤2
=3
Vimal
EA C461- Artificial Intelligence
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)
Vimal
EA C461- Artificial Intelligence
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
Vimal
EA C461- Artificial Intelligence
Implementation
• The game tree is traversed in depth-first
order.
• Each non-leaf node will have a beta value
and an alpha value stored.
• For each max node, the minimum beta value for
all its min node ancestors is stored as beta.
• For each min node, the maximum alpha value for
all its max node ancestors is stored as alpha.
• Initially, the root node is assigned an alpha value
of negative infinity and a beta value of infinity.
Vimal
EA C461- Artificial Intelligence
Implementation
• The variable children is used to represent all
of the children of the current node
• The following call means
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EA C461- Artificial Intelligence
A better beta is already found…
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EA C461- Artificial Intelligence
A better alpha is already found…
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EA C461- Artificial Intelligence
Trace alpha-beta
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EA C461- Artificial Intelligence
Resource limits
• Replace utility function with heuristic
function
– Gives an estimate of utility
• Replace terminal test by cur-off test
Vimal
EA C461- Artificial Intelligence
Evaluation functions
• Evaluation function should
– Order the terminal state by it’s true utility
– Not take long time
– Correlate strongly with the actual chances of winning,
for every non terminal states
• Works by calculating features of the state
– No of pawns possessed by each players, etc
– Groups features to make classes/categories of the state.
• Mostly evaluation function identifies the category
in which the current state falls, and give an
ordering of its successor with it’s expected utility.
Vimal
EA C461- Artificial Intelligence
Evaluation functions
• Categorizing may be difficult
– Material value of each piece can be taken into account
•
•
•
•
Pawn
Knight/bishop
Rook
Queen
1
3
5
9
– Specific features like
• Good pawn structure
• King safty
• For chess, typically linear weighted sum of
features
Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s)
Vimal
EA C461- Artificial Intelligence
Evaluation functions
• Weighted linear combination of features may be
poor considering the fact
– Bishops are more powerful in the endgame
– Pair of bishops slightly more worthies than a single
bishop
– Importantly, the features are not the part of the rules of
the game 
• They come out of experience
• Experience suggests that, secured material
advantage has more chance of winning, when all
other things being equal…
– A 3 point advantage near the end of the game is
sufficient to ensure victory
• Bishop is indeed worth 3 pawns
Vimal
EA C461- Artificial Intelligence
Cutting off search
• Replace terminal test with the cutoff
function
– If CUTOFF-TEST (state, depth) then return EVAL (state)
• Apply evaluation functions only to
quiescent positions, positions unlikely to
exhibit a wild swing
• Generate around million nodes/second in
latest PC
Vimal
– Given 3 mins to make a move approximately
200 million nodes can be generated
– With b=35, this implies 5 levels of look ahead
(minimax)
– With alpha-beta we get around 10 ply look
EA C461- Artificial Intelligence
ahead
Cutting off search
• Minimax
MAX
– Selects optimal move,
given the leave
evaluations are exactly
correct.
– Evaluations at leave
are mostly estimations
MIN
100
99
MAX
• Evaluation functions
can be probability
distribution over a set
of possible values
Vimal
99
1000 1000
1000
EA C461- Artificial Intelligence
100
101
102
100
Chess in Gaming Literature
• 1957 – Herbert Simon – 10 years computer will
beat human world champion
• 1997 – Deep Blue – defeated – Gary Kasparov –
Six-Game Exhibition Match
• Deep Blue
– Running as parallel computer, 30 IBM RS/6000 running
software search, 480 custom VLSI chess processors
generates moves and orders them
– Hardware search for last few levels of the tree …
– 126 million nodes/second average with the peak speed
of 300 million nodes/sec
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EA C461- Artificial Intelligence
Chess in Gaming Literature
– Used standard iterative deepening alpha-beta
search with transposition tables
– Max depth approx 40 plies
– Evaluation function with 8000 features
– About 4000 opening positions, consensus
recommendations from the database of
700,000 grandmaster games
– Large solved end game database (all positions
with 5 pieces, as many as 6 pieces)
• 2002 – FRITZ (on ordinary PC) – Vladimir
Kramnik – drawn
Vimal
EA C461- Artificial Intelligence
Summary
•
•
•
•
Vimal
Games are fun to work on!
They illustrate several important points about AI
perfection is unattainable  must approximate
good idea to think about what to think about
EA C461- Artificial Intelligence