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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
The second Ky Fan inequality is used in game theory to investigate the existence of an equilibrium.
Statement of the classical version [ edit ]
If with
0
≤
x
i
≤
1
2
{\textstyle 0\leq x_{i}\leq {\frac {1}{2}}}
for i = 1, ..., n , then
(
∏
i
=
1
n
x
i
)
1
/
n
(
∏
i
=
1
n
(
1
−
x
i
)
)
1
/
n
≤
1
n
∑
i
=
1
n
x
i
1
n
∑
i
=
1
n
(
1
−
x
i
)
{\displaystyle {\frac {{\bigl (}\prod _{i=1}^{n}x_{i}{\bigr )}^{1/n}}{{\bigl (}\prod _{i=1}^{n}(1-x_{i}){\bigr )}^{1/n}}}\leq {\frac {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}}{{\frac {1}{n}}\sum _{i=1}^{n}(1-x_{i})}}}
with equality if and only if x 1 = x 2 = ⋅ ⋅ ⋅ = x n .
Let
A
n
:=
1
n
∑
i
=
1
n
x
i
,
G
n
=
(
∏
i
=
1
n
x
i
)
1
/
n
{\displaystyle A_{n}:={\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad G_{n}={\biggl (}\prod _{i=1}^{n}x_{i}{\biggr )}^{1/n}}
denote the arithmetic and geometric mean, respectively, of x 1 , . . ., x n , and let
A
n
′
:=
1
n
∑
i
=
1
n
(
1
−
x
i
)
,
G
n
′
=
(
∏
i
=
1
n
(
1
−
x
i
)
)
1
/
n
{\displaystyle A_{n}':={\frac {1}{n}}\sum _{i=1}^{n}(1-x_{i}),\qquad G_{n}'={\biggl (}\prod _{i=1}^{n}(1-x_{i}){\biggr )}^{1/n}}
denote the arithmetic and geometric mean, respectively, of 1 − x 1 , . . ., 1 − x n . Then the Ky Fan inequality can be written as
G
n
G
n
′
≤
A
n
A
n
′
,
{\displaystyle {\frac {G_{n}}{G_{n}'}}\leq {\frac {A_{n}}{A_{n}'}},}
which shows the similarity to the inequality of arithmetic and geometric means given by G n ≤ A n .
Generalization with weights [ edit ]
If x i ∈ [0,1 / 2 ] and γi ∈ [0,1] for i = 1, . . ., n are real numbers satisfying γ 1 + . . . + γn = 1, then
∏
i
=
1
n
x
i
γ
i
∏
i
=
1
n
(
1
−
x
i
)
γ
i
≤
∑
i
=
1
n
γ
i
x
i
∑
i
=
1
n
γ
i
(
1
−
x
i
)
{\displaystyle {\frac {\prod _{i=1}^{n}x_{i}^{\gamma _{i}}}{\prod _{i=1}^{n}(1-x_{i})^{\gamma _{i}}}}\leq {\frac {\sum _{i=1}^{n}\gamma _{i}x_{i}}{\sum _{i=1}^{n}\gamma _{i}(1-x_{i})}}}
with the convention 00 := 0. Equality holds if and only if either
γi x i = 0 for all i = 1, . . ., n or
all x i > 0 and there exists x ∈ (0,1 / 2 ] such that x = x i for all i = 1, . . ., n with γi > 0.
The classical version corresponds to γi = 1/n for all i = 1, . . ., n .
Proof of the generalization [ edit ]
Idea: Apply Jensen's inequality to the strictly concave function
f
(
x
)
:=
ln
x
−
ln
(
1
−
x
)
=
ln
x
1
−
x
,
x
∈
(
0
,
1
2
]
.
{\displaystyle f(x ):=\ln x-\ln(1-x)=\ln {\frac {x}{1-x}},\qquad x\in (0,{\tfrac {1}{2}}].}
Detailed proof: (a ) If at least one x i is zero, then the left-hand side of the Ky Fan inequality is zero and the inequality is proved. Equality holds if and only if the right-hand side is also zero, which is the case when γi x i = 0 for all i = 1, . . ., n .
(b ) Assume now that all x i > 0. If there is an i with γi = 0, then the corresponding x i > 0 has no effect on either side of the inequality, hence the i th term can be omitted. Therefore, we may assume that γi > 0 for all i in the following. If x 1 = x 2 = . . . = x n , then equality holds. It remains to show strict inequality if not all x i are equal.
The function f is strictly concave on (0,1 / 2 ], because we have for its second derivative
f
″
(
x
)
=
−
1
x
2
+
1
(
1
−
x
)
2
<
0
,
x
∈
(
0
,
1
2
)
.
{\displaystyle f''(x )=-{\frac {1}{x^{2}}}+{\frac {1}{(1-x)^{2}}}<0,\qquad x\in (0,{\tfrac {1}{2}}).}
Using the functional equation for the natural logarithm and Jensen's inequality for the strictly concave f , we obtain that
ln
∏
i
=
1
n
x
i
γ
i
∏
i
=
1
n
(
1
−
x
i
)
γ
i
=
ln
∏
i
=
1
n
(
x
i
1
−
x
i
)
γ
i
=
∑
i
=
1
n
γ
i
f
(
x
i
)
<
f
(
∑
i
=
1
n
γ
i
x
i
)
=
ln
∑
i
=
1
n
γ
i
x
i
∑
i
=
1
n
γ
i
(
1
−
x
i
)
,
{\displaystyle {\begin{aligned}\ln {\frac {\prod _{i=1}^{n}x_{i}^{\gamma _{i}}}{\prod _{i=1}^{n}(1-x_{i})^{\gamma _{i}}}}&=\ln \prod _{i=1}^{n}{\Bigl (}{\frac {x_{i}}{1-x_{i}}}{\Bigr )}^{\gamma _{i}}\\&=\sum _{i=1}^{n}\gamma _{i}f(x_{i})\\&<f{\biggl (}\sum _{i=1}^{n}\gamma _{i}x_{i}{\biggr )}\\&=\ln {\frac {\sum _{i=1}^{n}\gamma _{i}x_{i}}{\sum _{i=1}^{n}\gamma _{i}(1-x_{i})}},\end{aligned}}}
where we used in the last step that the γi sum to one. Taking the exponential of both sides gives the Ky Fan inequality.
The Ky Fan inequality in game theory [ edit ]
A second inequality is also called the Ky Fan Inequality, because of a 1972 paper, "A minimax inequality and its applications". This second inequality is equivalent to the Brouwer Fixed Point Theorem , but is often more convenient. Let S be a compact convex subset of a finite-dimensional vector space V , and let
f
(
x
,
y
)
{\displaystyle f(x,y)}
be a function from
S
×
S
{\displaystyle S\times S}
to the real numbers that is lower semicontinuous in x , concave in y and has
f
(
z
,
z
)
≤
0
{\displaystyle f(z,z)\leq 0}
for all z in S . Then there exists
x
∗
∈
S
{\displaystyle x^{*}\in S}
such that
f
(
x
∗
,
y
)
≤
0
{\displaystyle f(x^{*},y)\leq 0}
for all
y
∈
S
{\displaystyle y\in S}
. This Ky Fan Inequality is used to establish the existence of equilibria in various games studied in economics.
References [ edit ]
External links [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Ky_Fan_inequality&oldid=1230019689 "
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