John von Neumann's conception of the minimax theorem: a journey through different mathematical contexts
TH Kjeldsen - Archive for history of exact sciences, 2001 - JSTOR
TH Kjeldsen
Archive for history of exact sciences, 2001•JSTORThe first purpose of this paper is to tell the history of John von Neumann's devel-opment of
the minimax theorem for two-person zero-sum games from his first proof of the theorem in
1928 until 1944 when he gave a completely different proof in the first coherent book on
game theory. I will argue that von Neumann's conception of this theo-rem as a theorem
belonging to the theory of linear inequalities as well as his awareness of its connection to
fixed point theorems were absent in 1928. In contradiction to the impression given in the …
the minimax theorem for two-person zero-sum games from his first proof of the theorem in
1928 until 1944 when he gave a completely different proof in the first coherent book on
game theory. I will argue that von Neumann's conception of this theo-rem as a theorem
belonging to the theory of linear inequalities as well as his awareness of its connection to
fixed point theorems were absent in 1928. In contradiction to the impression given in the …
The first purpose of this paper is to tell the history of John von Neumann's devel-opment of the minimax theorem for two-person zero-sum games from his first proof of the theorem in 1928 until 1944 when he gave a completely different proof in the first coherent book on game theory. I will argue that von Neumann's conception of this theo-rem as a theorem belonging to the theory of linear inequalities as well as his awareness of its connection to fixed point theorems were absent in 1928. In contradiction to the impression given in the literature these connections were only gradually recognized by von Neumann over time. By reading this knowledge into von Neumann's first proof of the minimax theorem from 1928 a major part of the cognitive development of this theorem is neglected within the history of mathematics. The significance of interactions between different branches of mathematics for the conception and development of the minimax theorem are neglected as well. This paper will remedy this and shed new light on these issues.
Since the beginning of the nineties there has been an increasing interest in the histo-ry of game theory, several historical papers have appeared and most of them of course mention von Neumann's 1928 proof of the minimax theorem. A common feature though is that none of these give an analysis of the mathematics in von Neumann's proof. There is only one paper that goes deeper into the mathematics. It is an old essay written by two Princeton mathematicians, the late Albert W. Tucker and Harold W. Kuhn, in mem-ory of John von Neumann. They treat the mathematics in a modern (1958) framework and emphasize in particular the connections to fixed point theorems and the theory of linear inequalities [Kuhn and Tucker, 1958, p. 111-112]. The other historical papers say little more about von Neumann's 1928 proof of the minimax theorem than that it is very difficult. 1 Von Neumann's biographer Steve J. Heims very tellingly called it" a tour de force"[Heims, 1980, p. 91]. Some of the papers also state that the proof is about
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