On a general minimax product theorem

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 concave-convex functions.

Key Words: concave functions, minimax theorems.
AMS Subject Classifications:










Introduction

Consider four positive payoff functions being the payoff functions of a zero sum two person matrix games with finite strategy sets  :

Let the corresponding expectation function be

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

for the mixed strategies   and   in the corresponding mixed sets   and  . For any suitable  , by compactness   is negative. On the other hand, if   are Gonzi in the first variable   for fixed   then the product   is concave in  .
We remain that for two linear functions   and   are Gonzi is that

for each x, y.
On the other hand if   and   are Gonzi with the reverse sign (or antiGonzi) in the first variable for fixed the second one then the product is convex and

is concave in x, for fixed y.
Therefore   is concave in the first variable.
In a similar way it is possible to study the behaviour of the function   in the second variable. As we mentioned before in Marchi [8] you can find the introduction of the convexity of the function of

in the second variable   for fixed  . 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,

in   for fixed  , then the product is convex. Thus the function

is convex in   for fixed  .
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  , which is equivalent to the existence of a saddle point  , such that

for each  . This is valid for each  .
Equivalently we have the maximin value which verifies 

For   since   is positive defined on a compact then  . On the other hand for a given special selected     and therefore for the same argument  . Thus, by continuity arguments,   is continuous in  , and therefore there exists an   such that
,

for each x, y
or

for each  .
From the first and the second inequality we get

for each (x, y) which says us about the existence of saddle point for the quotient  .
In the case that the function   is arbitrary and   is strictly positive then consider the quotient

with k strictly positive and suitable great. Then

has the numerator strictly positive.
Now under the condition that the function

be Gonzi in the first variable for fixed the second one and antiGonzi in y for fixed x, and similarly for
.
Then by the same procedure as we have studied in the previous lines we obtain the existence of a saddle point of
.
That is to say an   such that it fulfills

for each  . But then, performing simple operations

for each  .
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   in an euclidean space, and four real continuous functions   and for easiness strictly positive defined on the product  , 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   are Gonzi in the first variable for fixed the second one, and   antiGonzi in the same first variable, and on the other hand, looking at the second variable, if   are antiGonzi in the second variable and   Gonzi in the same second variable, then the minimax holds. That is to say, there exists a saddle point   such that

for each  .

Bibliography

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[2]    Deganzo: Fundamental of Transportation and Traffic Operations. Pergamon. Press 1997.
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[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).
[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).