Game Theory for Economic Analysis by Tatsuro Ichiishi and Karl Shell (Auth.)

By Tatsuro Ichiishi and Karl Shell (Auth.)

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Here player j passively expects the others not to change their strategies. Another static) interpretation is that the jth j (more j J player's utility level (x, £ ) when 1 J is uJ+1 w he chooses £ and J all of the others choose ( x , . . , xj ~ \ x , . . , x ), in which case u is assumed to be independent of x . Thus player i influences player j through fs utility function u\ and vice versa. A game in normal form is now defined as a j list of specified data {X\ u }jeN . The nature of a game in normal form constrained by the second interpretation, may be best understood if it is reformulated as a game in strategic form.

A rule F that assigns to each point x o f l a subset F(x) of Y. Sometimes a correspondence is called a multivalued function. Given metrics on X and Y (or more generally, topologies on X and 7), one can define the continuity of a correspondence F by endowing 0>(Y) with a "natural" topology. Two of such topologies on @>{Y) are introduced: the Vietoris topologies. 2). 3). 1). For a metric space Y one can introduce the HausdorfT distance on the family &'(Y) : = £P(Y)\{0}. Two further continuities of F are defined, based on the HausdorfT distance, and their relationship with upper and lower semicontinuities is presented (Exercise 2).

C. at (p, w ). J Let $ : = {X\1

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