"The notion of an equilibrium point is the basic ingredient of our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing 'good strategies'." (John F Nash, "Non-Cooperative Games", 1950)
"One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely. The two approaches to the problem, via the negotiation model or via the axioms, are complementary; each helps to justify and clarify the other." (John F Nash, "Non-cooperative Games", Annals of Mathematics Vol. 54 (2), 1951)
"Rather than solve the two-person cooperative game by analyzing the bargaining process, one can attack the problem axiomatically by stating general properties that 'any reasonable solution' should possess. By specifying enough such properties one excludes all but one solution. " (John F Nash, "Two-Person Cooperative Games", 1953)
"The information obtained by discovering dominances for one player may be of relevance to the others, insofar as the elimination of classes of mixed strategies as possible components of an equilibrium point is concerned. For the t's whose components are all undominated are all that need be considered and thus eliminating some of the strategies of one player may make possible the elimination of a new class of strategies for another player." (John F Nash, "Two-Person Cooperative Games", 1953)
"Instead of having a single control unit sequencing the
operations of the machine in series (except for certain subsidiary operations as
certain input and output functions) as is now done, the idea is to decentralize
control with several different control units capable of directing various
simultaneous operations and interrelating them when appropriate."
"It is interesting to consider what a thinking machine will be like. It seems clear that as soon as the machines become able to solve intellectual problems of the highest difficulty which can be solved by humans they will be able to solve most of the problems enormously faster than a human." (John F Nash, "Parallel Control", 1954)
"We could define the intelligence of a machine in terms of
the time needed to do a typical problem and the time needed for the programmer to
instruct the machine to do it."
"Successful treatment of non-linear partial differential
equations generally depends on 'a priori' estimates controlling the
behavior of solutions. These estimates are themselves theorems about linear
equations with variable coefficients, and they can give a certain compactness
to the class of possible solutions." (John F Nash, "Continuity of Solutions of
Parabolic and Elliptic Equations", 1958)
"If you're going to develop exceptional ideas, it
requires a type of thinking that is not simply practical thinking [...]"
"What truly is logic? Who decides reason? […] It's only in the mysterious equations of love that any logical reasons can be found." (John F Nash)
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