Von Neumann, Ville, And The Minimax Theorem
AbstractVon Neumann proved the minimax theorem (existence of a saddle-point solution to 2 person, zero sum games) in 1928. While his second article on the minimax theorem, stating the proof, has long been translated from German, his first announcement of his result (communicated in French to the Academy of Sciences in Paris by Borel, who had posed the problem settled by Von Neumann's proof) is translated here for the first time. The proof presented by Von Neumann and Morgenstern (1944) is not Von Neumann's rather involved proof of 1928, but is based on what they called "The Theorem of the Alternative for Matrices" which is in essence a reformulation of an elegant and elementary result by Borel's student Jean Ville in 1938. Ville's argument was the first to bring to light the simplifying role of convexity and to highlight the connection between the existence of minimax and the solvability of systems of linear inequalities. It by-passes nontrivial topological fixed point arguments and allows the treatment of minimax by simpler geometric methods. This approach has inspired a number of seminal contributions in convex analysis including fixed point and coincidence theory for set-valued mappings. Ville's contributions are discussed briefly and von Neuman's original communication, Ville's note, and Borel's commentary on it are translated here for the first time.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
Bibliographic InfoArticle provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Game Theory Review.
Volume (Year): 12 (2010)
Issue (Month): 02 ()
Contact details of provider:
Web page: http://www.worldscinet.com/igtr/igtr.shtml
Find related papers by JEL classification:
- B4 - Schools of Economic Thought and Methodology - - Economic Methodology
- C0 - Mathematical and Quantitative Methods - - General
- C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
- C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
- D5 - Microeconomics - - General Equilibrium and Disequilibrium
- D7 - Microeconomics - - Analysis of Collective Decision-Making
- M2 - Business Administration and Business Economics; Marketing; Accounting - - Business Economics
You can help add them by filling out this form.
reading list or among the top items on IDEAS.Access and download statisticsgeneral information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Tai Tone Lim).
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.