Equilibrium Points in N-Person Games

Equilibrium Points in N-Person Games John F. Nash

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Equilibrium Points in N-Person Games 1950 Proceedings of the National Academy of Sciences 10.1073/pnas.36.1.48 nash1950-equilibrium nash1950-equilibrium.txt, the project's staged copy 3c3a0720edc88884f58d9cadda4940970ae43160 open-access economics

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them existence Quotation withheld because this record is not cleared for redistribution. equilibrium points in n-person games Quotation withheld because this record is not cleared for redistribution. mathematical-proof mathematical-proof Quotation withheld because this record is not cleared for redistribution. mechanisms Because the countering correspondence is convex-valued and has a closed graph, a fixed-point theorem applies to it, and any fixed point of that correspondence is by definition a self-countering n-tuple, i.e. an equilibrium. Quotation withheld because this record is not cleared for redistribution. Equilibrium points are not interchangeable in general: outside the two-person zero-sum case, distinct equilibrium points need not give the players the same expectations. Quotation withheld because this record is not cleared for redistribution. The existence proof imports Kakutani's fixed-point theorem for convex-valued closed-graph set correspondences and applies it to the game's countering correspondence. Quotation withheld because this record is not cleared for redistribution. builds-on → Kakutani's fixed point theorem Quotation withheld because this record is not cleared for redistribution. Every finite n-person game has at least one equilibrium point. Quotation withheld because this record is not cleared for redistribution. The set of n-tuples that counter a given n-tuple is convex. Quotation withheld because this record is not cleared for redistribution. Continuity of the pay-off functions implies that the graph of the countering correspondence is closed. Quotation withheld because this record is not cleared for redistribution. Outside the two-person zero-sum case, different equilibrium points need not give the players the same expectations. about this quote: THE QUOTE'S OPENING WAS MOVED BY THE TEXT EXTRACTION: the paper's sentence begins 'In this case any two equilibrium' (printed earlier in this copy, inside the sentence before), and 'this case' is the two-person zero-sum case. Quotation withheld because this record is not cleared for redistribution. In the two-person zero-sum case, existence of an equilibrium point is equivalent to von Neumann's main theorem. Quotation withheld because this record is not cleared for redistribution. n-person game n-person game Quotation withheld because this record is not cleared for redistribution. equilibrium point equilibrium point Quotation withheld because this record is not cleared for redistribution. pay-off functions pay-off functions Quotation withheld because this record is not cleared for redistribution. mapping of the product space mapping of the product space Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1073/pnas.36.1.48.