This question has little or nothing to do with R; as usual I'm simply hoping to take advantage of the great depth of knowledge and expertise in the R community.
Anyone who is interested in replying should send email directly to me ([EMAIL PROTECTED]) and not to this list. To get to my question: In a two person zero-sum game, the value of the game to the row player is v_r = max min x'Ay x y where A is the m x n reward matrix and x and y are vectors of probabilities summing to 1 (constituting the row player's and and column player's ``mixed'' strategies, respectively). Note that x' means ``x transpose''. The value of the game to the column player is (the negative of) v_c = min max x'Ay y x These two values, i.e. v_r and v_c, are equal --- as one might hope. (One man's ceiling is another man's floor, as Paul Simon puts it.) ***Proving*** that they are equal is done (in game theory books) by setting the problem up as a linear programming problem and invoking the Duality Theorem. (The column player's LP turns out to be the dual of the row player's LP.) Initially I thought that there must/should be an easier way to prove that the two values are equal. But I can't see one. So I would like to ask: o ***Is*** there a ``simple'' direct proof that v_r = v_c? o How crucial is it that we are maximizing and minimizing over the simplices of probability vectors (summing to 1) in m-dimensional and n-dimensional space respectively? Could we optimize over some more general compact (convex?) set in R^{m+n} and preserve the equality? o Are there known ``general'' sufficient conditions (on the function phi(.,.) and the domain of optimization) so that max min phi(x,y) = min max phi(x,y) ??? x y y x o Has anyone any idea where I might look to find answers to such questions? Thanks for any insights. cheers, Rolf Turner [EMAIL PROTECTED] ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html