On The Theory Of Games Of Strategy by John Von Neumann, 1926

On The Theory Of Games Of Strategy by John Von Neumann, 1926

Author: John Von Neumann (translated by Sonia Bargmann 1928)

Year: 1926

On The Theory of Games of Strategy. Paper by John Von Neumann, translated by Sonia Bargmann, 1928. A shortened version of this paper had been presented to the Goettingen Mathematical Society on December 7, 1926. A game of strategy is presented as a finite system of “draws” and “steps,” each with specified results, probabilities, and information conditions, and is reduced to a normal form in which each player selects a “strategy” consisting of predetermined dispositions for all possible earlier outcomes. Chance is eliminated by replacing the results of the “draw” with the “expected values” of the functions g₁, g₂, …, gₙ, yielding a simplified game in which each player chooses a number and receives g(x₁, …, xₙ). For the case n = 2, the game becomes a two‑person zero‑sum situation with payoff g(x, y), where the pure‑strategy quantities satisfy “Max Min g(x, y) ≤ Min Max g(x, y).” The introduction of probability complexes σ and η produces the bilinear form h(σ, η), within which the equality Max Min = Min Max is established. The value M and the sets Σ and H of optimal complexes follow from the relations h(σ, η) ≥ M and h(σ, η) ≤ M, and the proof is completed by the convexity property (K