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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.
 Spring '10
 AsuOzdaglar

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