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# Much of the theory is due to topkis 79 98 milgrom and

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Unformatted text preview: rules. 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 Lem...
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## This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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