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This result is known as the Erdős–Ginzburg–Ziv theorem after its discoverers. It may also be deduced from the Cauchy–Davenport theorem. [2] More general results than this theorem exist, such as Olson's theorem, Kemnitz's conjecture (proved by Christian Reiher in 2003 [3]), and the weighted EGZ theorem (proved by David J. Grynkiewicz in ...
In the mathematical theory of games, in particular the study of zero-sum continuous games, not every game has a minimax value. This is the expected value to one of the players when both play a perfect strategy (which is to choose from a particular PDF). This article gives an example of a zero-sum game that has no value. It is due to Sion and ...
The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal. Generally, any game where all strategies are Pareto optimal is called a conflict game. [7] [8] Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero. [9]
In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics.It deals with problems of the following kind: given a combinatorial structure whose elements are assigned different weights (usually elements from an Abelian group), one seeks for conditions that guarantee the existence of certain substructure whose weights of its elements sum up to zero (in ).
In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, and always adds to zero (more informally, a player benefits only at the equal expense of others). [20] Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose.
Notice that a k-barycentric sequence in , with k a multiple of n, is a sequence with zero-sum. The zero-sum problem on sequences started in 1961 with the Erdős, Ginzburg and Ziv theorem: every sequence of length 2 n − 1 {\displaystyle 2n-1} in an abelian group of order n , contains an n -subsequence with zero-sum.
The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior, but their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of ...
Sion's minimax theorem is a generalization of von Neumann's minimax theorem due to Maurice Sion, [6] relaxing the requirement that It states: [6] [7] Let X {\displaystyle X} be a convex subset of a linear topological space and let Y {\displaystyle Y} be a compact convex subset of a linear topological space .