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Much of the theory is due to [Topkis 79, 98], [Milgrom and Roberts 90],
[Milgrom and Shannon 94], and [Vives 90, 01].
3 Game Theory: Lecture 8 Supermodular Games Increasing Diﬀerences
Key property: Increasing diﬀerences.
Deﬁnition
Let X ⊆ R and T be some partially ordered set. A function f : X × T → R has
increasing diﬀerences in (x , t ) if for all x � ≥ x and t � ≥ t , we have
f (x � , t � ) − f (x , t � ) ≥ f (x � , t ) − f (x , t ).
Intuitively: incremental gain to choosing a higher x (i.e., x � rather than x )
is greater when t is higher, i.e., f (x � , t ) − f (x , t ) is nondecreasing in t .
You can check that the property of increasing diﬀerences is symmetric : an
equivalent statement is that if t � ≥ t , then f (x , t � ) − f (x , t ) is
nondecreasing in x .
The previous deﬁnition gives an abstract characterization. The following
result makes checking increasing diﬀerences easy in many cases.
4 Game Theory: Lecture 8 Supermodular Games Increasing Diﬀerences
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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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