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# Let f x t r be a twice continuously dierentiable

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Unformatted text preview: ma Let X ⊂ R and T ⊂ Rk for some k , a partially ordered set with the usual vector order. Let f : X × T → R be a twice continuously diﬀerentiable function. Then, the following statements are equivalent: The function f has increasing diﬀerences in (x , t ). For all t � ≥ t and all x ∈ X , we have ∂f (x , t � ) ∂f (x , t ) ≥ . ∂x ∂x For all x ∈ X , t ∈ T , and all i = 1, . . . , k , we have ∂2 f (x , t ) ≥ 0. ∂ x ∂ ti 5 Game Theory: Lecture 8 Supermodular Games Monotonicity of Optimal Solutions Key theorem about monotonicity of optimal solutions: Theorem (Topkis) Let X ⊂ R be a compact set and T be some partially ordered set. Assume that the function f : X × T → R is continuous [or upper semicontinuous] in x for all t ∈ T and has increasing diﬀerences in (x , t ). Deﬁne x (t ) ≡ arg maxx ∈X f (x , t ). Then, we have: 1 2 For all t ∈ T , x (t ) is nonempty and has a greatest and least element, denoted by x (t ) and x(t ) respectively. ¯ ¯ ¯ For all t � ≥ t , we have x (t � ) ≥ x (t ) and x...
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## This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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