TLEN7000/ECEN7002: Analytical Foundations of Networks

S-modular Games and Power Control

Lijun Chen 11/13/2012

2

Outline

❒  S-modular games q  Supermodular games q  Submodular games

❒  Power control q  Power control via pricing q  A general framework for distributed power control

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Supermodular games

❒  Characterized by “strategic complementarities” ❒  Supermodular games are remarkable

q  Pure strategy Nash equilibrium exists q  The equilibrium set has an order structure with extreme

elements q  Many solution concepts yield the same prediction q  Analytically appealing

•  Have nice comparative statics and behave well under various learning or adaptive algorithms

q  Encompass many applied models

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Monotone comparative statics

❒  Def: suppose and some partially ordered set. A function has increasing differ-ences (supermodular) in if for all and ,

❒  The incremental gain to choose a higher is greater when is higher.

❒  The increasing differences is symmetric, i.e., if , then is nondecreasing in .

RTXf →×:

),( tx xx ≥ʹ′ tt ≥ʹ′

).,(),(),(),( txftxftxftxf −ʹ′≥ʹ′−ʹ′ʹ′

x

tt ≥ʹ′ ),(),( txftxf −ʹ′

RX ⊆ T

t

x

5

❒  Lemma: if is twice continuously differentiable, then has increasing differences iff implies

for all , or alternatively that, for all ,

ff tt ≥ʹ′

),(),( txftxf xx ≥ʹ′

x0),( ≥txf xt

tx,

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❒  A central question: when will be increasing in ?

❒  Theorem (Topkis): Let be compact and a partially ordered set. Suppose has increasing differences in , and is upper semi-continuous in . Then, q  For all , exists and has a greatest and least

element and . q  and are increasing in .

),(maxarg)( txftxXx∈

=

tRX ⊆ T

RTXf →×:

),( tx

x

t )(tx)(tx)(tx

)(tx)(tx t

7

❒  Proof: q  Existence: is compact and is upper semicontinuous q  Take a sequence in . From compactness, there

exists a limit point . Then for all ,

Thus, and is therefore closed. It follows that has a greatest and least element.

q  Let and . Then, , which implies . By the increasing difference, . Thus maximizes . Now, pick and , it follows that . A similar argument applies to .

X f

}{ kx

)(txx∈

)(txk

kxx

∞→= lim x

).,(),(),(),( txftxftxftxf k ≥⇒≥

)(tx )(tx

)(txx∈ )(txx ʹ′∈ʹ′ 0)),,(min(),( ≥ʹ′− txxftxf0),()),,(max( ≥ʹ′−ʹ′ txftxxf

0),()),,(max( ≥ʹ′ʹ′−ʹ′ʹ′ txftxxf ),max( xx ʹ′

),( tf ʹ′⋅ )(txx = )(txx ʹ′=ʹ′

)(txxx ≥ʹ′

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Supermodular games

❒  Def: the game is a super-modular game if for all , q  is a compact subset of q  is upper semicontinuous in q  has increasing differences in

❒  Corollary: suppose is a super-modular game. Define the best response function . Then q  has greatest and least element and q  and are increasing in

},,{ NiNi uSNG ∈∈=i

iS R

iu ii ss −,

iu ),( ii ss −

},,{ NiNi uSNG ∈∈=

),(maxarg)( iiiSsii ssusBii

−∈

− =

)( ii sB − )( ii sB −)( ii sB −

)( ii sB − )( ii sB − is−

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Example: Bertrand game

❒  Two firms: form 1 and firm 2 with prices ❒  Payoff ❒  It is a supermodular game, since . ❒  Solve by iterated strict dominance

q  Let , then . •  If , then •  If , then

q  Let , then

q  (1/3, 1/3) is the only Nash equilibrium.

]1,0[, 21 ∈pp

)21(),( jiijii pppppu +−=

02

>∂∂

∂

ji

i

ppu

]1,0[0 =iS ]2/1,4/1[1 =iS

4/1<ip dominated.strictly is 410

4141 <⇒≥+⋅−>

∂

∂ij

i

i pppu

2/1>ip dominated.strictly is 210

2141 >⇒≤+⋅−<

∂

∂ij

i

i pppu

],[ kkki ssS =

kkkkk ssss 4/4/14/116/16/14/14/4/1 021 +++==++=+= −−

kkkkk ssss 4/4/14/116/16/14/14/4/1 021 +++==++=+= −−

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❒  Theorem: let be a supermodular game. Then the set of strategies surviving iterated strict dominance (ISD) has greatest and least element and , which are pure strategy Nash equilibria.

❒  Corollary: q  Pure Nash equilibrium exists. q  The largest and smallest strategies compatible with ISD,

rationalizability, correlated equilibrium and Nash equilibrium are the same.

q  If a supermodular game has a unique Nash Equilibrium, it is dominance solvable.

},,{ NiNi uSNG ∈∈=

ss

11

❒  Proof: let and be the largest element of . Let and . If , i.e., , then it is dominated by . q  By increasing differences

q  Also note that

❒  Iterate and define and . Now if , then . So, is a decreasing sequence and has a limit denoted by . Only the strategies are undominated.

SS =0 ),,( 0||

01

0Nsss =

S )( 01iii sBs −= }:{ 101

iiiii ssSsS ≤∈=1ii Ss ∉ 1

ii ss > 1is

0),(),(),(),( 0101 <−≤− −−−− iiiiiiiiiiii ssussussussu

)( 1−−= kii

ki sBs }:{ 1 k

iikii

ki ssSsS ≤∈= −

1−≤ kk ss

01 ss ≤

ki

kii

kii

ki ssBsBs =≤= −

−−+ )()( 11 }{ ks

s ii ss ≤

12

❒  Similarly, start with the smallest element of and identify .

❒  Show and are Nash equilibria. q  For all and , q  Take the limit as , . q  Similarly, prove is a Nash equilibrium

),,( 0||

01

0Nsss =

S s

s s

i is ),(),( 1 kiii

ki

kii ssussu −−+ ≥

∞→k ),(),( iiiiii ssussu −− ≥s

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Illustrative diagram

Best response

Best response

Best response

……

largest element smallest element

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Submodular games

❒  Def: suppose and some partially ordered set. A function has decreasing differ-ences (submodular) in if for all and ,

❒  A game is a submodular game if the payoff functions are submodular.

❒  More generalizations

RTXf →×:

),( tx xx ≥ʹ′ tt ≥ʹ′

).,(),(),(),( txftxftxftxf −ʹ′≤ʹ′−ʹ′ʹ′

RX ⊆ T

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Monotonicity

❒  Def: let and are two sets. We say if for any and , and .

❒  For constraint sets , if

then the set possess the descending property. The ascending property can be defined when the relation is reversed.

BA BA

Aa∈ Bb∈ Aba ∈),min( Bba ∈),max(

iii SsS ⊆− )(

),()( iiiiii sSsSss −−−− ʹ′⇒ʹ′≤

iS

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❒  Theorem: for a submodular game with descending , q  An Nash equilibrium exists. q  The best response strategy

monotonically converges to an equilibrium. ❒  Proof: Follows monotonicity of the best response.

Similar to the proof of former theorem. ❒  Similar result exists for a supermodular game

with ascending .

)(⋅iS

)},(maxmin{arg)()( iiisSsii ssusBiii

−∈

−−

=

)(⋅iS

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Power control

❒  An important component of radio resource management q  Meet target BER or SIR while limiting interference q  Increase capacity by minimizing interference q  Extend battery life

❒  Users assigned utilities that are functions of the power they consume and the signal-to-interference ratio (SIR) they attain

❒  Try to find a good balance between high SIR (or meeting target SIR) and low power consumption

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Power control via pricing

❒  Consider a single-cell network with a set of users at uplink

❒  Each user can choose a power ❒  The SIR for user

where is the channel gain from MS to BS and is the noise variance.

N

i ],[ maxminiii ppp ∈

i

2σγ

+=∑ ≠ij jj

iii ph

ph

ih2σ

1h2h 1−Nh

Nh

BS

MS1 MS2 MSN-1 MSN

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❒  Consider payoff ❒  assumed to be increasing ❒  When the utilities are supermodular?

❒  Requires q  Example: some concave functions

iiiiii pfppu αγ −=− )(),(

)(⋅f

ijffphh

ppppu

cp

fpppu

iiiii

ji

ji

iii

i

ii

i

iii

≠ʹ′+ʹ′ʹ′−=∂∂

∂

−ʹ′=∂

∂

−

−

)),()((),(

)(),(

2

22

γγγγ

γγ

0)()( <ʹ′+ʹ′ʹ′ iii ff γγγ

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❒  Power control algorithm q  At time , let . q  At each time , set user power

❒  The above algorithm converges to a Nash equilibrium that is the smallest equilibrium.

0=t min)0( pp =

kt =))}1(,(maxmin{arg)( −= − kppukp iiipi

i

i

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A general framework for distributed power control

❒  Consider a set of users and a set of base stations

❒  User uses power ❒  Denote by the gain of user at base station ❒  The SIR of user at base station is with

MN

j jp

kjh j k

j k kjjp µ

2kji kii

kjkj hp

hσ

µ+

=∑≠

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Different power control schemes

❒  Fixed assignment: the user is assigned to BS with a SIR requirement . The constraints is

❒  Minimum power assignment, Macro diversity, limited diversity and multiple reception are have the constraints of the same form

j ja

jγ

)()(

ppIp

ja

iFAj

jµγ

=≥

)( pIp j ≥

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Standard interference function

❒  The standard interference function satisfies the following properties q  Positivity: q  Monotonicity: if q  Scalability: for ,

)( pI

0)( >pI)()( pIpIpp ʹ′≥⇒ʹ′≥

1>a )()( apIpaI ≥

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❒  Define a submodular game q  Payoff q  Constraint set with a feasible solution to

❒  Theorem: if a feasible solution exists, then q  There is a fixed point to equation q  The best response strategy converges to an

equilibrium.

jj ppu =)(

}0 ),(:{)( jjjjjjj pppIpppS ʹ′≤≤≥=−

)(pIp ≥pʹ′

pʹ′)(pIp =

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Happy Thanksgiving!