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A_Course_in_Game_Theory_-_Martin_J._Osborne 23

# A_Course_in_Game_Theory_-_Martin_J._Osborne 23 - Let X be a...

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Page 7 vectors of n nonnegative real numbers by . For and we use to mean . for i = 1,..., n and x > y to mean x i > y i for i = 1,..., n . We say that a function is increasing if f ( x ) > f ( y ) whenever x > y and is nondecreasing if whenever x > y . A function is concave if for all , all , and all . Given a function we denote by arg the set of maximizers of f ; for any we denote by f ( Y ) the set . Throughout we use N to denote the set of players. We refer to a collection of values of some variable, one for each player, as a profile ; we denote such a profile by , or, if the qualifier " " is clear, simply ( x i ). For any profile and any we let x -i be the list of elements of the profile x for all players except i . Given a list and an element x i we denote by ( x -i , x i ) the profile . If X i is a set for each then we denote by X -i the set . A binary relation .gif"> is convex; it is strictly quasi-concave if every such set is strictly convex.
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Unformatted text preview: Let X be a set. We denote by | X | the number of members of X . A partition of X is a collection of disjoint subsets of X whose union is X . Let N be a finite set and let be a set. Then is Pareto efficient if there is no for which y i > x i for all is strongly Pareto efficient if there is no for which for all and y i > x i for some . A probability measure μ on a finite (or countable) set X is an additive function that associates a nonnegative real number with every subset of X (that is, whenever B and C are disjoint) and satisfies μ ( X ) = 1. In some cases we work with probability measures over spaces that are not necessarily finite. If you are unfamiliar with such measures, little is lost by restricting attention to the finite case; for a definition of more general measures see, for example, Chung (1974, Ch. 2)....
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