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:
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 NikaidoIsoda 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 nonempty, 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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