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CS188:
Section
Handout
3,
Minimax
I.
Minimax
search
Consider
the
following
minimax
tree,
given
that
the
root
node
corresponds
to
the
maximizing
player
1.
What
is
the
minimax
value
of
the
root?
Name
a
future
outcome
which
achieves
the
minimax
value.
9
2.
Cross
out
all
nodes
that
will
not
be
visited
by
alpha‐beta
pruning,
assuming
children
are
visited
in
left‐to‐right
order
No
node
is
pruned
CS188:
Section
Handout
3,
Minimax
3.
Same
as
previous
question
except
children
are
now
visited
in
right‐to‐left
order.
Which
ordering
prunes
more
nodes?
>=9
9
<=6
<=3
3
6
9
<=9
9
>=9
9
4.
In
general,
how
can
one
come
up
with
a
method
for
ordering
children
of
nodes
that
tends
to
increase
the
chance
of
pruning?
In
general
we
would
want
to
use
an
evaluation
function
to
estimate
the
values
of
the
children
and
then
order
them
so
that
at
max
nodes
we
order
the
children
with
larger
estimated
values
first
and
at
min
nodes
we
order
the
children
with
smaller
estimated
values
first.
II.
Optimality
Suppose
that
you
(MAX)
are
playing
a
game
against
your
friend
(MIN).
Fortunately
for
you,
your
friend
is
very
tired
from
working
on
Project
#1
due
last
Friday,
and
as
such,
she
is
not
playing
optimally
today.
a.
Suppose
that
you
decide
to
use
minimax
decisions
in
playing
against
your
friend.
Can
the
fact
that
she
is
playing
suboptimally
hurt
the
performance
of
minimax?
In
other
words,
can
the
utility
obtained
by
using
minimax
decisions
against
a
suboptimal
player
be
lower
than
that
obtained
against
an
optimal
player?
If
so,
provide
a
game
tree
that
demonstrates
this
behaviour.
If
not,
provide
a
proof.
Proof:
MIN
playing
suboptimally
means,
by
definition,
that
MIN
selects
a
move
with
minimax
utility
greater
than
or
equal
to
the
move
predicted
by
minimax.
Since
MAX
maxes
over
these
decisions,
then
the
minimax
utility
against
a
suboptimal
is
greater
than
or
equal
to
the
minimax
utility
against
an
optimal
min.
MAX maxes over these decisions, then the minimax utility against a suboptimal is
greater than or equal to the minimax utility against an optimal min.
CS188:
Section
Handout
3,
Minimax
b. Now suppose that you are aware when your friend will make a suboptimal
b.
Now
suppose
that
you
are
aware
when
your
friend
will
make
a
suboptimal
move,
move, and which move she will make. Can you take advantage of this? In other
and
which
move
she
will
make.
Can
you
take
advantage
of
this?
In
other
words,
can
words, can a suboptimal strategy on your part achieve higher utility than a
a
suboptimal
strategy
on
your
part
achieve
higher
utility
than
a
minimax
strategy
if
minimax
strategy?
If made?
so, provide
game tree
that tree
demonstrates
this behaviour.
such
assumptions
are
If
so,
aprovide
a
game
that
demonstrates
this
If not, provide a proof that this is not possible.
behaviour.
If
not,
provide
a
proof
that
this
is
not
possible.
Here,
move
for
MAX
is
isto
toselect
the
move
that
leads
to
the
leaf
leaf
with
Here,the
theoptimal
optimal
move
for
MAX
select
the
move
that
leads
to the
with
utility
MAX
can
select
the
other
utility2.
2.But
Butif
ifMAX
MAXknows
knowsMIN
MINwill
willplay
playsuboptimally,
suboptimally,
MAX
can
select
the
other
option,
which
would
get
him
utility
1
against
an
optimal
MIN,
but
will
get
him
utility
option, which would get him utility 1 against an optimal MIN, but will get him
10
against
a
suboptimal
MIN.
utility 10 against a suboptimal MIN.z