Minimax theorems are beautiful saddle-point results regarding conditions on a function $f$ under which $\max_x \min_y f(x,y) = \min_y \max_x f(x,y)$. In the common "normal form" game case, $x$ and $y$ are probability distributions corresponding to player strategies and $f$ is bilinear. However, the result is true for more general quasiconcave-quasiconvex functions $f$, and that distinction is what this question is all about!

My question is: What did von Neumann (1928) prove and did Sion (1956) actually generalize it?


von Neumann first published a minimax theorem in “Zur Theorie der Gesellschaftsspiele” (1928), possibly available here. If I could read German then I would not need to ask this question, however it seems an English translation does not exist.

In 1956, Sion wrote "On General Minimax Theorems", available here, in which he claims that von Neumann's minimax theorem was stated and proved for bilinear forms $f(x,y)$. Sion claims that several others have generalized the theorem since, including Shiffman who "seems to have been the first to have considered concave-convex functions." Sion proves the result for quasiconcave-quasiconvex $f$. His result has a wikipedia page claiming it generalizes von Neumann's.

This seems final. But if I am correctly reading the very nice historical paper "John von Neumann's conception of the minimax theorem" (Kieldsen, 2001), available here, Section 2.3, then von Neumann's original 1928 paper originally already proved the generalization to the quasiconvex-quasiconcave case! The survey directly quotes the German passage from von Neumann's paper and gives a summary (though not complete translation).

Was Sion mistaken or unaware of what von Neumann proved? If so, were the references he cited mistaken as well?

One possible reason for confusion is the presentation of the theorem in von Neumann and Morgenstern's famous 1944 English-language book, "Theory of Games and Economic Behavior" (which I find very unpleasant to read, but never mind). Although saddle points are discussed in 13.4 and 13.5, I think that the only minimax theorem presented in the book is for bilinear forms, in Section 17.6. So if you only read the book, you might think the minimax theorem was only proven for bilinear $f$. By the way, that section has a footnote almost a page long concerning the history of the minimax theorem. It is mentioned that a generalization was proved in "tlber ein okonomisches Gleichungssystem und eine Verallgemeinerung des Brouwer'schen Fixpunktsatzes," von Neumann 1937. This uses fixed-point theorems, as investigated by Kieldsen 2001, but I don't see from that survey anything about whether $f$ was only bilinear, or more general, in the 1937 paper. So I would love to know

Exactly what are the translated statements of the minimax theorems proven in von Neumann (1928) and von Neumann (1937)?


This is too long for a comment, but I was not able to access the translations, so I can not provide details. Kieldsen (p.48ff) does say "Actually von Neumann proved a generalized version of the minimax theorem. Heconsidered a broader class of functions than the bilinear formsh... Today a function with the property (K) is called quasiconcave in ξ and quasiconvex in η".

The 1928 paper was translated by Bargman as On the Theory of Games of Strategy, published in Contributions to the Theory of Games, volume 4, edited by A. W. Tucker & R. D. Luce, 1959.

The note announcing it was translated just recently by Ben-EL-Mechaiekh and Dimand, International Game Theory Review, Vol. 12, No. 2, pp. 115-137 (2010).

The 1937 paper was translated by Morgenstern as A Model of General Economic Equilibrium, Review of Economic Studies 13 (1945–46), 1–9.


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