OR 4350 Game Theory
February 21, 2012
Prof. Bland
Linear Programming and ZeroSum TwoPerson Games
These notes (revised from notes of Prof Todd) show how linear programming can
be used to prove von Neumann’s Minimax Theorem and to obtain the value and the
maximin and minimax strategies for the players in a twoperson zerosum game with
a finite number of pure strategies for each player.
We suppose that an
m
×
n
matrix
A
gives the payoffs to player I (and hence

A
gives II’s payoffs). We’ll illustrate with an example in which
A
=

1
1
3
0
2

4

2
6

1
3
2

3
.
Before we proceed to the LP formulation, a little notation. Suppose we are given
an
m
×
n
matrix
A
. For each
j
= 1
, . . . , n
denote by
A
j
the
jth
column of
A
. For each
i
= 1
, . . . , m
denote by
A
i
the
ith
row of
A
. Let
P
=
{
p
∈
R
I
m
:
p
≥
0
,
∑
p
i
= 1
}
and
let
Q
=
{
q
∈
R
I
n
:
q
≥
0
,
∑
q
j
= 1
}
. We will assume, for now, that each
p
∈
P
is a
row vector and each
q
∈
Q
is a column vector, so that the product
pAq
is welldefined.
(Mea culpa: toward the end of this handout, when we get to the matrix notation for
an LP formulation of player I’s optimal mixed strategy, we will treat
p
as though it
is a column vector.)
LP formulation
Put yourself in player I’s place. Each
p
∈
P
is a mixed strategy, a randomization
over I’s
m
pure strategies. For any fixed
p
∈
P
and any 1
≤
j
≤
n
the inner product
pA
j
is the expected value of I’s payoff if I uses the mixed strategy
p
and II uses the
pure strategy
j
. So min
j
pA
j
is a lower bound on I’s expected payoff, if she uses mixed
strategy
p
. (Note that for fixed
p
, min
j
pA
j
= min
q
∈
Q
pAq
.) Therefore player I would like
to find among all
p
∈
P
one that makes min
j
pA
j
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 Spring '08
 SHMOYS
 Game Theory, PAJ, min pAq

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