CS 188 Sp07 Discussion Note Week 6 – Game and Discrete
Probability
by Nuttapong Chentanez
Game Playing
Types: Deterministic or Stochastic? One, two or more players? Perfect
information?
Want an algorithm for calculating a strategy that recommends optimal
move at each state.
Deterministic single player – Just search eg. Freecell, 8-Puzzle, Rubik
Deterministic two players – Minimax Search, pick move that maximize
utility against
best play from the opponents
Zero-sum games – One player maximizes the result, the other minimizes
it
Exercises:
1. Consider a two-player game featuring a board with four locations,
numbered 1 through 4 and arranged in a line. Each player has a single
token. Player A starts with his token on space 1, and player B starts with
his token on space 4. Player A moves first
The two players take turns moving, and each player must move his token
to an open adjacent space in either direction. If the opponent occupies an
adjacent space, then a player may jump over the opponent to the next open
space if any. (For example, if A is on 3 and B is on 2, then A may move
back to 1.) The game ends when one player reaches the opposite end of
the board. If player A reaches space 4 first, then the value of the game
is +1; if player B reaches space 1 first, then the value of the game is 1.
May be too slow to go all the way to termination, instead search to a
limited depth of tree and replace the terminal utilities with an evaluation
function. This loses guarantee of optimal play. Depth = Ply, the more ply
the better (if utilities is reasonable).
Ideal function: Return the true utility of the position (will be optimal
play), only relative ordering is needed.
Alpha-Beta pruning : “If you have an idea which is surely bad, don’t take
time to see how truly awful it is” ~ Pat Winston
a. Draw the complete game tree, using the following conventions:
Write each state as (sA ; sB ) where sA and sB denote the token locations.
Put the terminal states in square boxes, and annotate each with its game
value in a circle.
Put loop states (states that already appear on the path to the root)
in double square boxes. Since it is not clear how to assign values
to loop states, annotate each with a “?" in a circle.
b. Now mark each node with its backed-up minimax value (also in a
circle). Explain how you handled the “?" values, and why.
How to write min-value function?
Stochastic game: This includes chance node, just take expected value on
the chance not otherwise, the same as minimax. Correct ordering for
utility function does not guarantee optimality anymore.
c. Explain why the standard minimax algorithm would fail on this game
tree and sketch how you might fix it, drawing on your answer to (b). Does
your modified algorithm give optimal decisions for all games with loops?
2. Here is the complete game tree with chance node for a very simple
game. Assume that the leaf nodes are to be evaluated in left-to-right order,
and that before a leaf node is evaluated, assume that we know nothing
about its value---the range of possible values is -∞ to ∞.
Important Rules:
Chain Rules: P(X, Y) = P(X|Y)P(Y)
Bayes’s Rule: P(X|Y) = P(Y|X) P(X) / P(Y), very useful in AI
Axioms of probability
1. 0<= P(a) <= 1, for any proposition a,
2. P(true) =1 , P(false) = 1
3. P(a  b) = P(a) + P(b) – P(a  b)
Exercise:
1. Show P(a| a  b) = 1
2.
Consider the problem of dealing 5-card poker hands from a
standard deck of 52 cards, assuming that the dealer is fair.
a. How many atomic events are there in the joint
probability distribution (how many 5-card hands are
there)?
a. Mark the value of all the internal nodes and indicate the best move at
the root with an arrow.
b.
What is the probability of each atomic event?
c.
What is the probability of dealt a royal straight flush?
Four of a kind?
b. True/False: Given the values of the first six leaves, the seventh and
eighth leaves are irrelevant and need not be evaluated.
c. True/False: Given the values of the first seven leaves, the eighth leaf is
irrelevant and need not be evaluated.
d. Now suppose all the leaf node values are known to lie between --2 and
2 inclusive. After evaluating the first two leaves, what range of values can
be deduced for the left-hand chance node?
(i) -2 to 2 (ii) 0 to 1 (iii) 0 to 2
e. Circle all the leaves that need not be evaluated under the assumption in
d.
Probability
Sample Space – Set of all possible outcome of some experiment
3.
From this table:
toothache
~ toothache
catch
~catch
catch
~catch
cavity
0.108
0.012
0.072
0.008
~cavity
0.016
0.064
0.144
0.576
Compute
a.
b.
c.
d.
P(toothache)
P(Cavity)
P(Toothache| cavity)
P(Cavity| toothache  catch)
Random Variables – Function that assign a value to each outcome in a
sample space
eg. If sample space S is the set of all students in this class, one could
define a random variable A, measuring age. If p is a person, A(p) is his/her
age.
Event - a set of outcomes that share property you are interested in. eg. For
sample space S, J may be the set of juniors. Randomly picking a person
may pr may not result in the event that he/she is a junior. P(J) denotes the
probability that the event J occurs.
Events can be union, intersects, complement to define new events.
Particular conditions on random variables such as A=6’1”, A<7’ can also
be considered an event.
Conditional Probability – P(X|Y) = P(X  Y) / P(Y) is probability that
event X occurs given that event Y occurs
Joint Distribution – P(A = a, B = b) denotes probability that A = a and B
=b
Marginal Distribution – P(A = a) =  P(A = a, B = b) This summation
b
is called “marginalization”
Conditional distribution – P(A = a| B = b) gives conditional probability
4. After your yearly checkup, the doctor has bad news and good
news. The bad news is that you tested positive for a serious disease
and the test is 99% accurate (i.e. The probability of testing positive
when you have the disease is 0.99). The good news is that this is a
rare disease, striking only 1 in 100,000 people of your age. Why is it
a good news that the disease is rare? What is the chance that you
actually have the disease?