Adversarial Games

Two Flavors

Perfect Information– everything that can be known is known– Chess, Othello

Imperfect Information– Player’s have each have partial

knowledge– Poker: dispute is settled by revealing the

contents of one’s hand

Two Approaches to Perfect Information Games

Use simple heuristics and search many nodes

Use sophisticated heuristics and search few nodes

Cost of calculating the heuristics might outweigh the cost of opening many nodes

The closer h is to h*, the better informed it is. But information can be expensive

MiniMax on exhaustively searchable graphs

Two Players min: tries to achieve an outcome of 0 max: tries to achieve an outcome of

1

You are max at node A

Expand the entire search space

B

A

H

F

C D

K

G

O

E

J

N

P Q R

I

L M

MAX

Max

Min

Min

Min

Max

0

1 0

1

1

1

0 1

0

0

Wins

Max: F,J,N,Q,L Min: D,H,P,R,M

Propagating Scores: A first pass Min’s Turn a Node I

– Go to M to win– So assign I a 0

Max’s turn at node O– Go to Q to win– So assign 1 to node O

Conclusion

1. If faced with two labeled choices, you would choose 0 (if min) or 1 (if max)

2. Assume you’re opponent will play the same way

Propagating Scores

For each unlabeled node in the tree If it’s max’s turn, give it the max

score of its children If it’s min’s turn, give it the min score

of its children

Now label the treeConclusion: Max must choose C at the

first move or lose the game

Nim

7 coins 2 players Players divide coins into two piles at

each move, such that– Piles have an unequal number of coins– No pile is empty

Play ends when a player can no longer move

Start of Game

7

6,1 5,2 4,3

min

Complete the game to see that min wins only if max makes a mistake at 6,1 or 5,2

5,1,1 4,2,1

α-β Pruning

Problem For games of any complexity, you can’t

search the whole tree

Solution Look ahead a fixed number of plys (levels) Evaluate according to some heuristic

estimate Develop a criterion for pruning subtrees

that don’t require examination

Recast

Instead of representing wins, numbers represent the relative goodness of nodes.

At any junction Max chooses highest Min chooses lowestHigher means better for maxLower means better for min

Example: Max’s Turn

j

m z

g

k

n t

p q r

8

8 12

7 3 9

9

4

4

8 Max

Min

Alphabeta

Situation

Max’s turn at node g Left subtree of g has been explored If max chooses j, min will choose m So the best max can do by going left is 8. Call

this αNow Examine K and its left subtree n with a value of

4 If max chooses k, the worst min can do is 4. Because, T may be < 4. If it is min will choose

it. If not, min will choose 4 So the worst min can do, if max goes right is 4.

Call this β

Question

Should max expand the right subtree of k.

No. Because min is guaranteed 4. But if max chose j, min is only guaranteed 8.

Val(k) = min(4, val(t)) <= 4Val(g) = max(8,val(k))

= max(8, min(4,val(t)) = 8

Leads to Max Principle

Search can be stopped below any min node where β <= α of its max ancestor

Example: Min’s Turn

n

d e

k

t

p q r

4

4 3 7 3 9

9

4

4 min

Min

alphaBeta

Situation

Min’s turn at node k Left subtree of k has been explored If min chooses n, max will choose d So the best min can do by going left is 4. Call this

βNow examine T and its left subtree P with a value of

7 If min chooses T, the worst max can do is 7. Because, Q or R may be > 7. If it is Max will

choose it. If not, min will choose 7. So the worst max can do, if min goes right is 7.

Call this α

Question

Should min explore Q and R No Max is guaranteed 7 if min chooses T But if min chooses N, max gets only

4 Val(T) = max(7,val(Q), val(R)) >= 7 Val(k) = min(4,val(T)) = 4

Leads to min principle

Search can be stopped below any max node where α >= β of its min ancestor

To Summarize Max’s turn

– β is min’s guaranteed score (the worst min can do)– α is best max can do

Max principle– Search can be stopped below any min node where β <= α of its max ancestor

Min’s turn– α is max’s guaranteed score (the worst max can do)– β is best min can do

Min principle– Search can be stopped below any max node where α >= β of its min ancestor

Efficiency

Suppose a tree has depth, D, and branching factor B

D = 2, B = 2: Terminal Nodes = 22

In general, Terminal Nodes = BD

Ordering

Ordering of nodes in a tree clearly affects the number than can be pruned using alpha/beta.

Call ND the number of terminal nodes Can be shown that with alpha/beta best case

performance is:ND = 2BD/2 – 1 for even D

ND = 2B(D+1)/2 + B(D-1)/2 for odd D

Example

Suppose B = 5, D = 6w/out alpha/beta ND= 56 = 15625

w/alpha/beta ND= 2* 53 - 1 = 249

Approx. 1.6% of worst case

Reduction in Branching Factor

Without alpha/beta pruning: ND = BD

With alpha/beta pruning: ND = 2BD/2 - 1 for even D

Reducing the branching factor used in computing ND from B to B1/2

(since: 1/2= BD/2

Average Performance

On average, the peformance reduces B to B3/4

Suppose B = 5, Bab = 3.34

Suppose D = 6Then ND = 56 = 15625

NDAB = 3.346 = 1388 ≈ 8.8% of worst case

Combinatorial Explosion Key Idea: branching factor makes optimal solution

intranctable– Sum of Subsets Problem

2– Traveling Salesperson

(N + 1)/2– 8 puzzle

2.67– 15 puzzle

Approximately 4Let B = average branching factorLet T = total nodesLet D = depth of searchThenT = B + B2 + B3 + … + BD

= B(BD – 1)/(B – 1) + 1

Sum of Subsets

Given a set, S, of positive integers, find all subsets whose sum is m.

E.G.S = {7,11,13,24}m = 31Solutions S-1 = {7,11,13}S-2 = {7,24}

Problem Representation

Solution is a sequence of 1s and 0s, indicating that elements of S have been chosen or not.

Rep of S-1 (1,1,1,0)Rep of S-2 (1,0,0,1)State space is a tree where left turn

indicates a 1 and right turn indicates a 0

Partial Space

S-1

Three left turns and a right turn to get to S-1 is the sequence (1,1,1,0)

Clearly the branching factor is 2

TSP

Can also be represented as a state space search.

Suppose 4 cities4 Choices at level 03 Choices at level 12 Choices at level 21 Choice at level 3

T = 4*3*2*1 = 4P4 = 4!/(4-4)! = 4!

Branching Factor

B_F = (sum of choices at each level)/#of levels

= (4+3+2+1)/4 = 2.5

Clearly this increases with the size of the tour:(1+2+3+…+n)/n = (n(n+1)/2)/n = (n+1)/2

Relationship between branching factor and nodes in the tree

Whenever the branching factor >= 2, we have an exponentially complex problem

Let T = number of nodes in a full binary tree

T = 20 + 21 + 22 + … + 2d-1 = 2d – 1 Easily proved through induction

Replace 2 by branching factor, B

T = B0 +B1 + B2 + … + BL = B(BL -1)/(B-1) + 1Where L is d-1, d being the depth of the treeProofBasis: T = B(B0 – 1)/(B-1) + 1 = 1Inductive hypothesis:T = B0 +B1 + B2 + … + BL = B(BL -1)/(B-1) + 1

Show that this is true at level L+1That is,Show= B0 +B1 + B2 + … + BL+1 = B(BL+1 -1)/(B-1) + 1

B0 +B1 + B2 + … + BL + BL+1 = B(BL -1)/(B-1) + BL+1 + 1= (B(BL -1) + BL+1(B-1))/(B-1) + 1 %common D= (B( (BL -1) + BL(B-1))/(B-1) + 1 %factor B out= (B(BL -1 + BL+1 –BL )/(B-1) + 1 %multiply b= B(BL+1 -1)/(B-1) + 1 %subtract

Which is what we were trying to prove

Two Concepts

1. B – average number of descendents that emerge from any state in the space

2. Total nodes = B(BL -1)/(B-1) + 1

Where L is the deepest level (or, the depth of the search)

Another Problem: 8 Puzzle

A B C1 2 3 1 2 1 28 4 3 4 5 3 4 57 6 5 6 7 8 6 7 8A: 4 moves for blank * 1 position = 4B: 2 moves for blank * 4 positions = 8C: 3 moves for blank * 4 positions = 12

B = (4 + 8 + 12) /(1 + 4 + 4) = 2.67

Does the branching factor of larger (15, 24) puzzles approach 4 as the puzzles get larger?