C&O 355: Mathematical Programming
Fall 2010
Lecture 12 Notes
Nicholas Harvey
http://www.math.uwaterloo.ca/~harvey/
1
ZeroSum Games
Let
M
be any
m
×
n
real matrix, which we use as the payoff matrix for a twoplayer, zerosum game.
Von Neumann’s theorem states that
max
x
min
y
x
T
My
=
min
y
max
x
x
T
My,
where the max and min are over distributions
x
∈
R
m
and
y
∈
R
n
. Recall that “distribution” means
that
x
≥
0,
∑
m
i
=1
x
i
= 1. Consequently, there exist distributions
x
*
∈
R
m
and
y
*
∈
R
n
such that
max
x
min
y
x
T
My
=
x
*
T
My
*
=
min
y
max
x
x
T
My.
(1)
This quantity is called the
value
of the game and is denoted by
v
.
Observation 1.
Note that for any fixed
x
, we have min
y
x
T
My
≤
v
. (In particular,
x
T
My
*
≤
v
.)
Similarly, for any particular
y
, we have max
x
x
T
My
≥
v
. (In particular,
x
*
T
My
≥
v
.)
Observation 2.
For any fixed
x
, there is a
y
achieving min
y
x
T
My
such that
y
has only one nonzero
coordinate (which must have value 1). Such a
y
corresponds to Bob choosing a single action, rather
than a randomized choice of actions.
Fix any desired error
δ
∈
(0
,
1). We will give a method to find distributions ˆ
x
and ˆ
y
such that

min
y
ˆ
x
T
My

v
 ≤
δ
and

max
x
x
T
M
ˆ
y

v
 ≤
δ.
(2)
Due to Observation 1, we see that (2) is equivalent to
min
y
ˆ
x
T
My
≥
v

δ
and
max
x
x
T
M
ˆ
y
≤
v
+
δ.
(3)
In other words, if Alice plays according to distribution ˆ
x
, then no matter how Bob plays, she is guar
anteed a payoff of at least
v

δ
. Conversely, if Bob plays according to distribution ˆ
y
, then no matter
how Alice plays, he is guaranteed to pay her at most
v
+
δ
.
2
The Multiplicative Weights Update Method
By scaling, we may assume that
M
i,j
∈
[

1
,
1] for every
i, j
.
Set
=
δ/
3.
Alice will assign some
“weights” to each of her actions, then simulate the game by herself for
T
= (ln
m
)
/
rounds, modifying
her weights between each round. These weights are essentially a probability distribution, except they
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 Winter '10
 Harvey
 Game Theory, John von Neumann, Minimax, Zerosum, Mi,j

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