On a general minimax product theorem
by
Ezio Marchi*)
Dedicated to the memory of Professor Ewald Burger
Abstract
In a recent paper by us we obtain a rather nice necessary and sufficient condition
for the product of two concave functions to be concave. Such a condition is called
Gonzi. As an application of it, in the same paper we derive a major minimax theorem
when the bilinear function is replaced by the product of two bilinear ones.
In this paper we generalize the most general minimax theorem of von Neumann
when the quotient of bilinear forms is replaced by the quotient of product of concaveconvex functions.
Key Words: concave functions, minimax theorems.
AMS Subject Classifications:
*)
Founder and First Director of the IMASL, CONICET-UNSL, Ejército de los Andes 950, San Luis,
5700, Argentina.
e-mail: emarchi@sinectis.com.ar
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Introduction
Consider four positive payoff functions being the payoff functions of a zero sum
two person matrix games with finite strategy sets I, J :
A1 , A 2 , A 3 , A 3
Let the corresponding expectation function be
E1 , E 2 , E 3 , E 4
We remind the reader that each one of them are bilinear functions. In a recent
paper we have proved a minimax theorem for the product of two bilinear functions. See
Marchi [ ].
Now following the idea of J. von Neumann, take a positive constant real number
 and consider the game with payoff function
E (x, y)  E1(x, y) E2 (x, y)   E3 (x, y) E4 (x, y)
~
~
~
~
for the mixed strategies x  I and y  J in the corresponding mixed sets I and J .
For any suitable   0 , by compactness E( x, y) is negative. On the other hand, if
~
~
E1 , E 2 are Gonzi in the first variable x  I for fixed y  J then the product E1 E 2 is
~
concave in x  I .
We remain that for two linear functions L1 ( x ) and L2 ( x ) are Gonzi is that
L (x)  L ( y) L (x)  L ( y)  0
1
1
2
2
for each x, y.
On the other hand if E 3 and E 4 are Gonzi with the reverse sign (or antiGonzi)
in the first variable for fixed the second one then the product is convex and
  E 3 ( x , y) E 4 ( x , y)
is concave in x, for fixed y.
Therefore E  ( x, y) is concave in the first variable.
In a similar way it is possible to study the behaviour of the function E  ( x, y) in
the second variable. As we mentioned before in Marchi [8] you can find the
introduction of the convexity of the function of
E1 ( x, y) E 2 ( x, y)
2
~
~
in the second variable y  J for fixed x  I . A necessary and sufficient condition is
that they are Gonzi with the reverse sign.
Assuming this and the condition to be Gonzi for the remaining functions,
  E 3 ( x , y) E 4 ( x , y)
~
~
in y  J for fixed x  I , then the product is convex. Thus the function
E (x, y)  E1(x, y) E2 (x, y)   E3 (x, y) E4 (x, y)
~
~
is convex in y  J for fixed x  I .
Therefore, using the Nikaido-Isoda theorem in a suitable way which is standard
in the game theory, see for example Burger [1] or Parthasarathy and Raghavan [9] we
have that the validity of the minimax theorem for E  , which is equivalent to the
existence of a saddle point (x  , y ) , such that
E ( x  , y )  E ( x  , y )  E (x  , y)
~ ~
for each ( x , y)  I  J . This is valid for each  .
Equivalently we have the maximin value which verifies
v  max min E  ( x, y)
x
y
 min max E  ( x, y).
y
x
For   0 since E1 E 2 is positive defined on a compact then v0  0 . On the other hand
for a given special selected   0 E   0 and therefore for the same argument v   0 .
Thus, by continuity arguments, v is continuous in  , and therefore there exists an 
such that
v  0 ,
E  ( x , y  )  0  E  ( x  , y)
for each x, y
or
E1 ( x, y ) E 2 ( x, y )   E 3 ( x, y ) E 4 ( x, y )
 0  E1 ( x  , y) E 2 ( x  , y)   E 3 ( x  , y) E 4 ( x  , y)
~ ~
for each ( x , y)  I  J .
From the first and the second inequality we get
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E1 ( x, y ) E 2 ( x, y ) E3 ( x  , y) E 4 ( x  , y)

E3 ( x, y ) E 4 ( x, y ) E3 ( x  , y) E 4 ( x  , y)
for each (x, y) which says us about the existence of saddle point for the quotient
E1 E 2
.
E3 E 4
In the case that the function E1 E 2 is arbitrary and E3 E 4 is strictly positive then
consider the quotient
E1 E 2
k
E3 E 4
with k strictly positive and suitable great. Then
E1 E 2  k E 3 E 4
E3 E 4
has the numerator strictly positive.
Now under the condition that the function
E1 ( x, y) E 2 ( x, y)  k E 3 ( x, y) E 4 ( x, y)
be Gonzi in the first variable for fixed the second one and antiGonzi in y for fixed x,
and similarly for
E 3 ( x , y) E 4 ( x , y) .
Then by the same procedure as we have studied in the previous lines we obtain the
existence of a saddle point of
E1 E 2  k E 3 E 4
.
E3 E 4
That is to say an ( x , y) such that it fulfills
E1 ( x, y) E 2 ( x, y)  k E3 ( x, y) E 4 ( x, y) E1 ( x, y) E 2 ( x, y)  k E3 ( x, y) E 4 ( x, y)

E 3 ( x , y) E 4 ( x , y)
E 3 ( x , y) E 4 ( x , y)
E1 ( x, y) E 2 ( x, y)  k E3 ( x, y) E 4 ( x, y)

E 3 ( x , y) E 4 ( x , y)
~ ~
for each ( x , y)  I  J . But then, performing simple operations
E1 ( x, y) E 2 ( x , y) E1 ( x, y) E 2 ( x , y) E1 ( x, y) E 2 ( x , y)


E 3 ( x , y) E 4 ( x , y) E 3 ( x , y) E 4 ( x , y) E 3 ( x , y) E 4 ( x , y)
~ ~
for each ( x , y)  I  J .
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Thus, we have proved a general theorem of the minimax generalizing widely
von Neumann’s result.
Developing the same method as we did in the previous parts it is possible to
derive a general minimax theorem.
Consider two non-empty, compact, convex subsets K1, K 2 in an euclidean
space, and four real continuous functions f 1 , f 2 , f 3 , f 4 and for easiness strictly positive
defined on the product K1  K 2 , such that they are in the first variable for any of the
second one concave and convex in the second one for fixed the first one. Then
Theorem: Under these conditions, if f 1 , f 2 are Gonzi in the first variable for fixed the
second one, and f 3 , f 4 antiGonzi in the same first variable, and on the other hand,
looking at the second variable, if f 1 , f 2 are antiGonzi in the second variable and f 3 , f 4
Gonzi in the same second variable, then the minimax holds. That is to say, there exists a
saddle point (x, y)  K1  K2 such that
f 1 ( x , y) f 2 ( x , y) f 1 ( x , y) f 2 ( x , y) f 1 ( x , y) f 2 ( x , y)


f 3 ( x , y) f 4 ( x , y) f 3 ( x , y) f 4 ( x , y) f 3 ( x , y) f 4 ( x , y)
for each (x, y)  K1  K2 .
Bibliography
[1]
Burger, E.: Introduction to the Theory of Games. Prentice Hall. (1972).
[2]
Deganzo: Fundamental of Transportation and Traffic Operations. Pergamon. Press
1997.
[3]
Kolmogorov, A.N., Fomin, S.V.: Elementos de la Teoría de Funciones y Análisis
Funcional. Ed. MIR. (1972).
[4]
Lay, S.R: Convex Sets and their Applications. John Wiley. (1982).
[5]
Marchi, E.: Equilibrium Points of Rational n-person Games. Journal Math.
Analysis and Application, Vol. 54, No 1, pp. 1-4 (1976).
[6]
Marchi, E.: On the Decomposition of General Extensive Games Journal. London
Math. Soc. (2), 7, pp. 561- 567. (1973).
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[7]
Marchi, E.: On the theorem of minimax of the Theory of Games. Annali di
Matematica Pura ed Appl. Serie IV, Tomo 77, pp. 202-282 (1967) (MR 38, 1914).
[8]
Marchi, E.: When is the product of two concave functions concave? (to appear)
[9]
Parthasarathy, T. and T.E.S. Raghavan: Some Topics in Two Person Games.
Elseiner. (1971).
[10] Rudin, W.: Real and Complex Analysis. Second Ed. Mac Graw Hill. (1979).
[11] Rockaffeller, T: Convex Analysis. Princeton University. Press 1967.
[12] v. Neumann, J.: Über ein ökonomisches Gleichurgs system und eine
Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergeb. Eines. Math. Koll.
8, 73-83 (1935/36).
[13] v. Neumann, J.: Zür Theorie der Gesellschaftsspielle Math. Annalen 100, 295-320
(1928).
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