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Unformatted text preview: ete lattice and f : S → S an increasing function.
Then, the set of ﬁxed points of f , denoted by E , is nonempty and (E , ≥)
is a complete lattice.
f(s) f(s) s s
19 Game Theory: Lecture 7 Supermodular Games Supermodularity of a Function
Let (X , ≥) be a lattice. A function f : X → R is supermodular on S if for
all x , y ∈ X
f (x ) + f (y ) ≤ f (x ∧ y ) + f (x ∨ y ).
Note that supermodularity is automatically satisﬁed if X is single
dimensional. 20 Game Theory: Lecture 7 Supermodular Games Monotonicity of Optimal Solutions
From now on, we will assume that X ⊆ R. The following analysis and theory extends to the case where X is a lattice. We ﬁrst study the monotonicity properties of optimal solutions of
parametric optimization problems. Consider a problem
x (t ) = arg max f (x , t ),
x ∈X where f : X × T → R, X ⊆ R, and T is some partially ordered set.
We will mostly focus on T ⊆ RK with the usual vector order, i.e.,
for some x , y ∈ T , x ≥ y if and only if xi ≥ yi for all i = 1, . . . , k .
We are interested in conditions under which we can establish that
x (t ) is a nondecreasing function of t . 21 Game Theory: Lecture 7 Supermodular Games Increasing Diﬀerences
Key property: Increasing diﬀerences.
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.
22 Game Theory: Lecture 7 Supermodular Games Increasing Diﬀerences
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 )
For all x ∈ X , t ∈ T , and all i = 1, . . . , k , we have
∂2 f (x , t )
∂ x ∂ ti
23 Game Theory: Lecture 7 Supermodular Games Example I – Network eﬀects (positive externalities)
A set I of users can use one of two products X and Y (e.g., Blu-ray and
Bi (J , k ) denotes payoﬀ to i when a subset J of users use k and i ∈ J .
There exists a p ositive externality if
Bi (J , k ) ≤ Bi (J � , k ), when J ⊂ J � , i.e., player i better oﬀ if more users use the same technology as him.
This leads to a strategic form game with actions Si =...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.
- Spring '10