minimax inequality

minimax inequality

The

minimax inequality

was first proved by

John von Neumann

. It is the starting point to discuss two-players zero-sum static

games theory

.

𝐓𝐡𝐞𝐨𝐫𝐞𝐦

⁢

𝟏

:

minimax

⁢

inequality

,

simple

⁢

strategies

¯

For any

m

×

n

matrix

A

i

,

j

, we have

(1)

max

1

≤

i

≤

m

⁡

min

1

≤

j

≤

n

⁡

A

i

,

j

≤

min

1

≤

j

≤

n

⁡

max

1

≤

i

≤

m

⁡

A

i

,

j

(2)

max

1

≤

i

≤

m

⁡

min

1

≤

j

≤

n

⁡

A

i

,

j

=

min

1

≤

j

≤

n

⁡

max

1

≤

i

≤

m

⁡

A

i

,

j

if and only if

A

i

,

j

~

≤

A

i

~

,

j

~

≤

A

i

~

,

j

∀

i

,

j

is valid for some

(

i

~

,

j

~

)

For a 2 players

zero-sum game

, the entries of

A

i

,

j

is interpreted as the payoff when player 1 has chosen the

i

t

⁢

h

strategy

while player 2 has chosen the

j

t

⁢

h

strategy. The value

A

i

~

,

j

~

is known as the

value

of the game.

𝐏𝐫𝐨𝐨𝐟

Since

min

1

≤

j

≤

n

⁡

A

i

,

j

≤

max

1

≤

i

≤

m

⁡

A

i

,

j

∀

i

,

j

.
The LHS is

independent

of

j

while the RHS is independent of

i

, therefore we obtain

max

1

≤

i

≤

m

⁡

min

1

≤

j

≤

n

⁡

A

i

,

j

≤

min

1

≤

j

≤

n

⁡

max

1

≤

i

≤

m

⁡

A

i

,

j

𝐓𝐡𝐞𝐨𝐫𝐞𝐦

⁢

𝟐

:

minimax

⁢

inequality

,

mixed

⁢

strategies

¯

Let

S

m

=

{

x

∈

ℝ

m

|

x

i

≥

0

⁢

∀

i

,

∑

i

=

1

m

x

i

=

1

}

⊆

ℝ

m

.
For any

m

×

n

matrix

A

i

,

j

, we have

max

x

∈

S

m

⁡

min

y

∈

S

n

⁢

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

=

min

y

∈

S

n

⁡

max

x

∈

S

m

⁢

∑

i

=

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

Here

0

≤

x

i

≤

1

is interpreted as the

probability

that Player 1 will choose strategy

i

while

0

≤

y

j

≤

1

is the probability that Player 2 will choose strategy

j

.

𝐏𝐫𝐨𝐨𝐟

For any

x

∈

S

m

and any

y

∈

S

n

we have

max

x

∈

S

m

⁡

min

y

∈

S

n

⁢

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

≤

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

Taking maximum for

x

∈

S

m

on both sides, we have

max

x

∈

S

m

⁡

min

x

∈

S

n

⁢

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

≤

max

s

∈

S

m

⁢

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

Taking minimum for

y

∈

S

n

on both sides, we have

v

1

=

max

x

∈

S

m

⁡

min

y

∈

S

n

⁢

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

≤

v

2

=

min

y

∈

S

n

⁡

max

x

∈

S

m

⁢

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

The prove of other half of the inequality takes two steps:

Step 1

Suppose there is a

y

∈

S

n

such that

∑

j

=

1

n

A

i

,

j

⁢

y

j

≤

0

⇒

There is some

x

~

∈

S

m

such that

∑

i

=

1

m

(

∑

j

=

1

n

A

i

,

j

⁢

y

j

)

⁢

x

~

i

≤

0

⇒

max

x

∈

S

n

⁢

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

≤

0

⇒

v

2

=

min

y

∈

S

n

max

x

∈

S

m

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

x

i

y

j

≤

0

(

*

1

)

Step 2

Suppose there is a

x

∈

S

m

such that

∑

i

=

1

m

A

i

,

j

⁢

x

i

⁢

y

j

>

0

⇒

There is some

y

~

∈

S

n

such that

∑

j

=

1

n

(

∑

i

=

1

m

A

i

,

j

⁢

x

i

)

⁢

y

~

j

≥

0

⇒

min

y

∈

S

n

⁢

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

⁢

x

i

⁢

y

j

≥

0

⇒

v

1

=

max

x

∈

S

m

min

y

∈

S

n

∑

i

=

1

m

∑

j

=

1

n

A

i

,

j

x

i

y

j

≥

0

(

*

2

)

Combining (*1) and (*2) we see that either

0

≤

v

1

or

v

2

≤

0

is the case and

v

1

<

0

<

v

2

cannot be valid. Repeat the same procedure to the matrix

A

~

i

,

j

=

A

i

,

j

-

λ

and we see that

v

1

-

λ

<

0

<

v

2

-

λ

is invalid, i.e.

v

1

<

λ

<

v

2

is not valid for any

λ

. Since

λ

is arbitrary, we conclude that

v

2

≤

v

1

.

An entire theory on minimax has already been developed and is one of the major research area in optimization theory. The following contains some good sources for further reference:

References

1

V.F.Demyanov and V.N.Malozemov,

Introduction to Minimax

, Keter Publishing House Jerusalem Ltd, 1974.

Title

minimax inequality

Canonical name

MinimaxInequality

Date of creation

2013-03-22 16:57:16

Last modified on

2013-03-22 16:57:16

Owner

bchui (10427)

Last modified by

bchui (10427)

Numerical id

32

Author

bchui (10427)

Entry type

Theorem

Classification

msc 91A99

Classification

msc 91A05

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