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Few people have changed as many fields as John von Neumann. Physicists use his framework for quantum mechanics, economists use his game theory, and a computer design report he wrote in 1945 had, in the words of its later editors, a pervasive influence. This post tells his story from reliable sources and then solves a small game with his minimax idea, step by step, so you can see why mixing your choices can beat any fixed plan.
Who was John von Neumann?
According to the MacTutor biography, he was born Janos Neumann in Budapest on 28 December 1903 and died in Washington on 8 February 1957. He had a famous memory as a child. His father wanted him to have a practical career, so he studied chemistry in Berlin and Zurich, earning a chemical engineering diploma in 1926, while also taking a doctorate in mathematics at Budapest the same year.
He worked with Hilbert in Gottingen in 1926-27, moved to Princeton in 1930, and in 1933 became one of the original six mathematics professors at the new Institute for Advanced Study, alongside Einstein and Weyl. George Polya, who knew him in Zurich, said von Neumann was the only student he was ever afraid of, because he would solve an open problem straight after a lecture.
What did he do for quantum mechanics?
MacTutor records that his 1932 book, Mathematical Foundations of Quantum Mechanics, "built a solid framework for the new quantum mechanics". The physicist Leon Van Hove later wrote that the framework was worked out "by one single man in two years (1927-1929)". The algebras of operators he studied with F J Murray are now called von Neumann algebras.
What is the minimax theorem?
In Zur Theorie der Gesellschaftsspiele (On the theory of games of strategy), published in Mathematische Annalen in 1928, von Neumann proved the minimax theorem for two-player zero-sum games, where one player's gain is the other's loss. In plain terms: if both players may randomise, there is a best mixed strategy for each, and the most one player can guarantee to win equals the least the other can guarantee to lose. MacTutor notes that he expanded this, with Oskar Morgenstern, into the 1944 book Theory of Games and Economic Behavior.
Worked example: solving a small game
Two players each pick a card in secret. Row picks A or B; Column picks C or D. The number is what Column pays Row (a negative number means Row pays).
A against C: +2
A against D: −1
B against C: −1
B against D: +1
Try pure strategies. If Row always plays A, the worst case is −1. If Row always plays B, the worst case is also −1. So a fixed choice guarantees Row only −1. Column's worst cases are 2 (always C) and 1 (always D), so Column can hold the loss to 1. Since −1 and 1 differ, there is no single best fixed move.
Let Row mix. Row plays A with probability p. Against C, Row expects 2p − (1 − p) = 3p − 1. Against D, Row expects −p + (1 − p) = 1 − 2p.
Make Column indifferent. Set 3p − 1 = 1 − 2p, so p = 2/5. Row expects 3(2/5) − 1 = 1/5 whatever Column does.
Check from Column's side. If Column plays C with probability q, Row's payoffs are 3q − 1 for A and 1 − 2q for B. Setting them equal gives q = 2/5 and the same value, 1/5.
The guaranteed win for Row and the guaranteed limit on Column's loss meet at 1/5. That meeting point is exactly what the minimax theorem promises.
Common mistakes: equating the payoffs of your own two moves instead of making the opponent indifferent, forgetting the (1 − p) term, and assuming every game needs mixing. Check pure strategies first; if the worst-case numbers match, a fixed move is already best. If probability needs revising first, see our post on probability mistakes.
How did he shape the modern computer?
On 30 June 1945 von Neumann wrote the First Draft of a Report on the EDVAC for the electronic computer project at the University of Pennsylvania. It splits the machine into parts such as an arithmetic unit, a control unit and a memory, and lists the "operation instructions which govern the functioning of the device" among the things the memory must hold, alongside numbers. When the report was reprinted in IEEE Annals of the History of Computing in 1993, the editors noted that Alan Turing cited it as the definitive source on the design of a general-purpose digital computer. Our post on Alan Turing and computing picks up that side of the story.
MacTutor also notes that he worked as a consultant to the US armed forces during and after the Second World War, including at Los Alamos.
What can a student take from his way of working?
Carry tools across subjects. The same mathematics served physics, economics and engineering.
Write things down clearly. The 1945 draft spread because others could build from it.
Turn a messy situation into a small model. The four-number game above is how game theory starts.
When a probability or algebra step in a problem like this stops you, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Change the A-against-C payoff to +3 and find Row's new best mix and the value of the game.
