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Minimax Theorems for VectorValued
"... We present a Ky Fan type inequality of mixed kind for vectorvalued multifunctions. We use it for proving our first type minimax theorem for vectorvalued multifunctions. It is a generalization of the classical Sion minimax theorem for scalar functions (in the compact case), as well as, a generaliza ..."
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We present a Ky Fan type inequality of mixed kind for vectorvalued multifunctions. We use it for proving our first type minimax theorem for vectorvalued multifunctions. It is a generalization of the classical Sion minimax theorem for scalar functions (in the compact case), as well as, a
GAME THEORY AND THE MINIMAX THEOREM
"... Abstract. Game theory is a very important branch of applied mathematics with many uses in the social sciences, biological sciences, and philosophy. Game theory attempts to mathematically explain behavior in situations in which an individual’s outcome depends on the actions of others. Arguably the m ..."
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the most important result in game theory, the Minimax Theorem was stated in 1928 by mathematician John von Neumann in his paper Zur Theorie Der Gesellschaftsspiele, and forms the basis for all subsequent findings in the subject. This paper will provide a brief introduction to zerosum games and the
Proof of the Minimax Theorem
, 2007
"... The proof of the minimax theorem follows the format given in Luce and Raiffa [2], but has been modified to use our class terminology. Changes in the tutorial since it was first posted in Fall 2007 are highlighted in red. Formalisms We first present the properties of a twoperson, zerosum game. To h ..."
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The proof of the minimax theorem follows the format given in Luce and Raiffa [2], but has been modified to use our class terminology. Changes in the tutorial since it was first posted in Fall 2007 are highlighted in red. Formalisms We first present the properties of a twoperson, zerosum game
AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS
"... 1. Introduction * The von Neumann minimax theorem [2] for finite games asserts that for every rxs matrix M=\\m(i, j)\ \ with real elements there exist a number v and vectors such that P=(Pi, •••, P ..."
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1. Introduction * The von Neumann minimax theorem [2] for finite games asserts that for every rxs matrix M=\\m(i, j)\ \ with real elements there exist a number v and vectors such that P=(Pi, •••, P
An Ordinal Minimax Theorem∗
, 2015
"... It is well known that Nash equilibria in twoplayer zerosum games are interchangeable and equivalent. We show that the same properties also hold for the weak saddle, a setvalued and ordinal solution concept proposed by Lloyd Shapley in the 1950s. Consider a zerosum game represented by a matrix A. ..."
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It is well known that Nash equilibria in twoplayer zerosum games are interchangeable and equivalent. We show that the same properties also hold for the weak saddle, a setvalued and ordinal solution concept proposed by Lloyd Shapley in the 1950s. Consider a zerosum game represented by a matrix A. The set R of rows represents the row player’s actions, and the set C of columns represents the column player’s actions. If the row player chooses action r ∈ R, and the column player chooses action c ∈ C, then the payoff of the row player is given by the entry u(r, c) ∶ = Ar,c of the matrix, while the payoff of the column player is given by the negative −Ar,c. A set R1 of rows is said to (weakly) dominate a set R2 of rows with respect to a set C of columns if for all rows r2 ∈ R2, there exists a row r1 ∈ R1 such that u(r1, c) ≥ u(r2, c) for all c ∈ C. Similarly, a set C1 of columns is said to (weakly) dominate a set C2 of columns if for all columns c2 ∈ C2, there exists a column c1 ∈ C1 such that u(r, c1) ≤ u(r, c2) for all r ∈ R. Given subsets R1 ⊆ R2 ⊆ R of rows and subsets C1 ⊆ C2 ⊆ C of columns, the product R1 ×C1 is said to be a externally stable in R2 ×C2 if R1 dominates R2/R1 with respect to C1, and C1 dominates C2/C1 with respect to R1. The product R1 ×C1 is said to be a (weak) saddle in R2×C2 if it is externally stable in R2×C2 and no proper subset of it is externally stable in R2 ×C2 (Shapley, 1953a,b, 1964). As an example, one can check that for the matrix
A PARAMETRIC MINIMAX THEOREM WITH APPLICATION
"... Abstract. We discuss a parametric minimax problem which arises from the theoretical foundation of certain decomposition methods in global optimization. Specifically, conditions for lower semicontinuity of the saddle value of a quasiconvexquasiconcave function depending on a parameter are developed ..."
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are developed and a general parametric minimax theorem is proved. Key words. Parametric minimax problem. Continuity of the saddle value. Stability of the minimax. Lagrange relaxation and decomposition in global optimization. 1
Minimax theorems with applications to convex metric spaces
, 1995
"... A minimax theorem is proved which contains a recent result of Pinelis and a version of the classical minimax theorem of Ky Fan as special cases. Some applications to the theory of convex metric spaces (farthest points, rendezvous value) are presented. ..."
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Cited by 1 (0 self)
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A minimax theorem is proved which contains a recent result of Pinelis and a version of the classical minimax theorem of Ky Fan as special cases. Some applications to the theory of convex metric spaces (farthest points, rendezvous value) are presented.
A minimax theorem for directed graphs
 J. London Math. Soc
, 1978
"... THEOREM. For a finite directed graph G, a minimum transversal of the directed cuts of G is equal in cardinality to a maximum disjoint collection of directed cuts. This minimax equality was conjectured about a decade ago by one of the authors ([7; page 43], [8], [9]) and, independently, by Neil Rober ..."
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Cited by 43 (1 self)
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THEOREM. For a finite directed graph G, a minimum transversal of the directed cuts of G is equal in cardinality to a maximum disjoint collection of directed cuts. This minimax equality was conjectured about a decade ago by one of the authors ([7; page 43], [8], [9]) and, independently, by Neil
Results 1  10
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