Post on 12-Feb-2016
description
1 © Cognitive Radio Technologies, 2007
Supermodular Games
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Terminology
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Partial Ordering
A relation is R is an order on a set S if the following three properties are satisfied for all x, y, z S: reflexive (x R x) transitive (x R y R z x R z) antisymmetric (x R y R x x = y)
The order is only partial order if there are some elements a, b S such that neither a R b nor b R a.
A set S taken together with a partial order is a partially-ordered set.
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Our relation of interest
Let x and y denote two vectors in KRLet x y if xk yk for all k = 1,2,…,KLet x > y if x y and there exists some k such that xk > yk
Example:x = (1,0,0,3), y = (1,2,2,3) y
xy > x
x = (1,0,4,3), y = (1,2,2,3) No relation
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Two More Relations
“meet of x and y” 1 1 2 2(min , ,min , , , min , )K Kx y x y x y x y
“join of x and y” 1 1 2 2{max , , max , , ,max , }K Kx y x y x y x y
Example:x = {1,0,0,3}, y = {1,2,2,3}x = {1,0,4,3}, y = {1,2,2,3}
1,0,0,3x y
The meet of x and y is the infimum of x and y
The join x and y is the supremum of x and y
1, 2,2,3x y
1,0,2,3x y 1,2,4,3x y
Note: if y x then x y x x y y
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SublatticeConsider Si to be a subset (maybe convex) of . Form S as imR
mi N iS S R where
1
n
kk
m m
Definition Sublattice
A set is a sublattice if it is a partially ordered () subset of and if the operations and are closed on S. (i.e. if s, s* S then s s* S and s s* S)
mR
S = {(0,0), (0,0.5), (0.5,0), (1,0), (0,1)}Sublattice?
noS = {(0,0), (0,1), (1,0), (1,1)} yes
Sublattice property – Every bounded sublattice has a greatest and least element.
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Increasing Differences
Definitionui(si, s-i) has increasing differences in (si, s-i) if, for all
, and ,i i i i i is s S s s S such that andi i i is s s s
, , , ,i i i i i i i i i i i iu s s u s s u s s u s s
In other words, an increase in the strategies of i’s rivals increasesthe value of playing a high strategy for player i.
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Supermodular Function
, , , , ,i i i i i i i i i i i i i i i i iu s s u s s u s s s u s s s s s S ui(si, s-i) is supermodular in si if for each s-i
Definition
Note if Si is single-dimensional, this is satisfied with equality.
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Supermodular games
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Equivalent Formulations
A game is supermodular if 2
0 , ,i
i j
u si j N s S
s s
A game is supermodular if Si is a sublattice of mR for all i and ,i i i iu s s u s s u s u s s s S
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Best Response Properties
Stated without proof
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Nash Equilibrium Existence
(Tarski) If A is a non-empty, compact sub-lattice of m and f : A A is non-decreasing, then f has a fixed point in A.
Note upper-semi-continuity
Would have to jump down across y=x, but that violatesUSC.
i N iBR a BR a
has a fixed point (NE)A
A
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NE Properties
(Topkis) A supermodular game for which each Ai is compact and each ui is u.s.c. in ai for each a-I, then the set of pure strategy NE is non-empty and contains greatest and least elements
and , respectivelya a
(Vives) Further the set of NE form a sub-lattice which is nonempty and complete.
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NE Uniqueness
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More NE Properties
Definition Best Response DynamicAt each stage, one player iN is permitted to deviate from ai to some randomly selected action bi Ai iff
, ,i i i i i i i i iu b a u c a c b A and ,i i i iu b a u a
(Milgrom and Roberts) A best response dynamic played on a supermodular game with compact action spaces and u.s.c. objective functions converges to a region bounded the greatest and least elements in the set of NE.
If the NE is unique, then the best response dynamic converges to theNE.
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Convergence of Adaptive Dynamics
The corollaries to Theorem 8 in [Milgrom_90] show that a smooth supermodular game following an adaptive dynamic process with any timing converges to a region bounded by the Nash equilibrium lattice .
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Ad-hoc power control Network description Each radio attempts to
achieve a target SINR at the receiving end of its link.
System objective is ensuring every radio achieves its target SINR
Gateway
ClusterHead
ClusterHead
Gateway
ClusterHead
ClusterHead
2ˆk kk N
J
p
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Generalized repeated gamestage game
Players – N Actions – Utility function
Action space formulation 2
ˆj j ju o
max0,j jP p
2
10 10\
ˆ 10log 10logj j jj j kj k jk N j
u p g p g p N
gjk fraction of power transmitted by j that can’t be removed by receiving end of radio j’s linkNj noise power at receiving end of radio j’s link
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Model identification & analysis Supermodular game
– Action space is a lattice– Implications
NE exists Best response converges Stable if discrete action space
Best response is also standard– Unique NE– Solvable (see prelim report)– Stable (pseudo-contraction) for infinite action
spaces
2
\
2000
ln 20
j kj
j kj kj k j
k N j
u p gp p
p g p N
ˆˆ jk
j jj
B p
p
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Validation
Noiseless Best Response Noisy Best Response
Implies all radios achieved target SINR
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Comments on Designing Networks with Supermodular Games
Scales well– Sum of supermodular functions is a supermodular function– Add additional action types, e.g., power, frequency,
routing,..., as long as action space remains a lattice and utilities are supermodular
Says nothing about desirability or stability of equilibria
Convergence is sensitive to the specific decision rule and the ability of the radios to implement it
22 © Cognitive Radio Technologies, 2007
Potential Games
time
(
)
Existence of a function (called the potential function, V), that reflects the change in utility seen by a unilaterally deviating player.
Cognitive radio interpretation:– Every time a cognitive radio
unilaterally adapts in a way that furthers its own goal, some real-valued function increases.
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Exact Potential Games Definitions, Existence, Basic Properties Examples Path Properties
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Exact Potential Games
, , , , , ,i i i i i i i i i i i i i i iP a a P b a u a a u b a a b A a A
Definition Exact Potential GameA normal form game whose objective functions are structured such that there exists some function P: A which satisfies the following property for all players:
In other words it must be possible to construct a single-dimensionalfunction whose change in value is exactly equal to the change in value of the deviating player.
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Example Potential Game (1/2)
a1
b1
a2 b2
1,1 0, 0
3, 30, 0
Coordination Game
1 2
1 2
1 2
1 2
1 ,0 ,0 ,3 ,
a a aa a b
V aa b aa b b
u1(a1,a2) - u1(b1,a2) = 1 = V(a1,a2) - V(b1,a2)
u2(a1,a2) – u2(a1,b2) = 1 = V(a1,a2) - V(a1,b2)u1(b1,b2) - u1(a1,b2) = 3 = V(b1,b2) - V(a1,b2)
u2(b1,b2) – u2(b1,a2) = 3 = V(b1,b2) - V(b1,a2)
Note: V is not unique.Consider V’ = V + c where c is a constant.
Also note the relationbetween CG Prop. 2 and V
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Example Potential Game (2/2)
a1
b1
a2 b2
4,2 -1, 1
2, 13,-2
1 2
1 2
1 2
1 2
1 ,0 ,0 ,3 ,
a a aa a b
V aa b aa b b
u1(a1,a2) - u1(b1,a2) = 1 = V(a1,a2) - V(b1,a2)
u2(a1,a2) – u2(a1,b2) = 1 = V(a1,a2) - V(a1,b2)u1(b1,b2) - u1(a1,b2) = 3 = V(b1,b2) - V(a1,b2)
u2(b1,b2) – u2(b1,a2) = 3 = V(b1,b2) - V(b1,a2)
The Same Potential!!The Same NE!
Coordination Game(In Equilibriums)
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Comments on Second Example
a1
b1
a2 b2
1,1 0, 0
3, 30, 0
Coordination Game
a1
b1
a2 b2
4,2 -1, 1
2, 13,-2
Second Game
a1
b1
a2 b2
3,1 -1, 1
-1,-23,-2
Dummy Game
As we shall see, this is a property of all exact potential games.
Also a potential function for an exact potential game is always equal to the characteristic function (plus a constant) of its constituent coordination game.
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EPG Property 1
A game G = <N, {Ai}iN , {ui}iN> is an exact potential game iff there exist functions {ci}iN and {di}iN such that
(Voorneveld)
•ui = ci + di
•<N, {Ai}iN , {ci}iN> is a coordination game•<N, {Ai}iN , {di}iN> is a dummy game
Outline of proof: if: The characteristic function of the coordination game is an exact potential function of GOnly if: Let P be an exact potential of G. Clearly P forms a coordination game. Now consider a game with objective fcns given by ui – P. As the value of deviating in this game is now 0 at all points, this is a dummy game.
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EPG Property 2
The NE of an exact potential game are coincident with the NE ofits constituent coordination game.
Outline of Proof
Any unilateral deviation in a dummy game yields the same payoff. Adding a dummy game D to another game G preserves G’s NE. All exact potential games can be expressedAs the sum of a coordination game and a dummy game (EPG Property 1). Therefore the NE of the potential game mustbe the same as the NE of the coordination game.
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EPG Property 3
For an EPG, the maximizers of the EPF are NE of the EPG.
Outline of Proof
(Voorneveld)
The NE of an EPG are the NE of its coordination game (CG).By CG Property 3, the maximizers of its characteristic functions(V) are NE.All EPF can be expressed as V constant c.Since the addition of the constant does not change whichtuples yield maximum payoffs, the maximizers of the EPF are coincident with the maximizers of V, thus coincident with theNE of the CG, thus coincident with the NE of the EPG.
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EPG Property 4
(Voorneveld)Let the EPG be finite (finite action space, finite player set), then theEPG has at least one pure-strategy NE.
Note these conditions mean that the EPF must have at least onemaximum. By EPG Property 4, this must be a NE.
Outline of Proof
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Continuous Action Sets (1/2)
i
i i
P ua a
22 2ji
i j i j i j
uP ua a a a a a
If objective functions are twice differentiable then a game is a EPG iff
Let G be a game in which the strategy sets are closed intervals of . Suppose the objective functions are continuously differentiable. A function P is a potential iff P is continuously differentiable and
for every i N
for every i, j N
EPG Property 5 (Shapley)
EPG Property 6 (Shapley)
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Continuous Action Sets (2/2)
EPG Properties 1-4 also hold for continuous closed action sets.
Proofs follow in exactly the same manner.
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Vector Operations
Consider the set of EPG {EPG1, EPG2,…,EPGK}, with player setN and action space A, and objective functions , andpotential functions . Form a new game, G, with player set N, action space A, and objective functions given by
1 2, , , Ki i iu u u
1 2, , , KP P P
1 1 2 2G K Ki i i iu u u u c . Then G is an EPG with an EPF
given by 1 1 2 2 K KP P P P
Note: this means that the set of EPG formed from a particular N andA form a vector topological space (closed under addition and scalarmultiplication).
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Common EPG
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Exact Potential Game Forms
Many exact potential games can be recognized by the form of the utility function
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Coordination – Dummy Game
As previously stated, all EPG are formed from the sum of a coordination game and a dummy game so this is only here for completeness.
Consider a game G = <N, (Ai)iN , {ui}iN> such thatui = ci + di where <N, (Ai)iN , {ci}iN> is a coordination game with characteristic function V(a) and <N, (Ai)iN , {di}iN> is a dummy game.
This game has an EPF given by V(a)
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Bilateral Symmetric Interaction Game
Introduced in Ui
, ,ij i j ji j iw a a w a a:ij i jw A A
,i j i ja a A A
\
,i ij i j i ij N i
u a w a a h a
1
1
,i
ij i j i ii N j i N
P a w a a h a
A strategic form game where each player’s objective function isa sum of bilateral symmetric interaction (BSI) terms. A BSI term
such that
. The objective function is expressed asfor every
An EPF for this game is given by
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Self-Motivated Game
Note this is not really a game (no interaction), but it is often encountered as a component of more complex games.
i i iu a h a
i ii N
P a h a
A strategic form game where each player’s objective function isa function solely of their own action, i.e.
This has an EPF given by
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Cournot Oligopoly (1/2)
i i k ik N
u b b B b cb i N
22
1ji
i j j i
uu i j Nb b b b
Cournot oligopoly characterized by real interval action sets and objective function given by
Note that
So a potential exists
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Cournot Oligopoly (2/2)
2
\i i k i i i
k N i
u b b b Bb b cb i N
Now rewrite the objective function as
Note that this is just a BSI game. So a potential can be written as
1
2
1
i
i k i i ii N j i N
P a b b Bb b cb
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Prisoners’ Dilemma
a1
b1
a2 b2
w, w x, y
z, zy, x
a1
b1
a2 b2
x-z
0
a1
b1
a2 b2
z-x+y, z-x+y z, z-x+y
z, zz-x+y, zx-z
x-z+w-y
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Ordinal Potential Games Definitions, Existence, Properties Examples Applications
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Ordinal Potential Games
, , , , , ,i i i i i i i i i i i i i i iP a a P b a u a a u b a a b A a A
Definition Ordinal Potential Game (OPG)A normal form game whose objective functions are structured such that there exists some function P: A which satisfies the following property for all players:
In other words it must be possible to construct a single-dimensionalfunction where the sign of the change in value is the same as the sign of the change in value of the deviating player.
sgn , , sgn , , , ,i i i i i i i i i i i i i i iP a a P b a u a a u b a a b A a A
Note that an EPG also satisfies this definition.
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Example Ordinal Potential Game
Not a Coordination Game
1 2
1 2
1 2
1 2
0 ,3 ,1 ,2 ,
a a aa a b
P aa b aa b b
sgn(u1(a1,a2) - u1(b1,a2)) = - = sgn(P(a1,a2) - P(b1,a2))
sgn(u2(a1,a2) – u2(a1,b2)) = - = sgn(P(a1,a2) - P(a1,b2))sgn(u1(b1,b2) - u1(a1,b2)) = - = sgn(P(b1,b2) - P(a1,b2))
sgn(u2(b1,b2) – u2(b1,a2)) = + = sgn(P(b1,b2) - P(b1,a2))
Note: P is not unique.Consider P’ = c2P + c1
a1
b1
a2 b2
0,0 1,1
0,12,0
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Properties shared with EPG
For an OPG, the maximizers of the OPF are NE of the OPG.
An OPG has at least one pure-strategy NE.
An finite OPG has FIP.
An OPG with continuous bounded action sets has AFIP.
A repeated game with the same OPG stage also converges witha better response dynamic.
(Shapley)
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Cycles
1 10
, 0k m mi m i mm
I u u a u a
Consider a cycle , the sum of the changes in value seen by the deviating players in an OPG is not always 0.
a1
b1
a2 b2
0,0 1,1
0,12,0
= ((a1, a2), (b1, a2), (b1, b2), (a1, b2), (a1, a2))
I(, u) = 2 + 1 + 1 – 1 = 3
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Weak Improvement Cycles
Non-deteriorating pathA path is an non-deteriorating path if for all k 1
1k ki k i ku a u a
Weak improvement cycleA finite non-deteriorating path = (a0, a1,…,ak) where ak = a0
No known simple necessary and sufficient condition like the secondderivative condition of EPG.
(Voorneveld)All OPG lack weak improvement cycles.
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Properties not shared with an EPG
The set of OPG is not a vector space.
a1
b1
a2 b2
0,0 1,1
0,12,0
a1
b1
a2 b2
1,2 1,0
0,10,0
a1
b1
a2 b2
1,2 2,1
0,22,0
a1
b1
a2 b2
0 321
a1
b1
a2 b2
3 210
Improvement Cycle= ((a1, a2), (b1, a2), (b1, b2), (a1, b2))
Still closed under scalar multiplication though.
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Ordinal Transformations
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Ordinal Transformation
Definition Ordinal TransformationsAn ordinal transformation is a one-to-one mapping of the utility
functions {ui} to a new set of utility functions {ui’} in such a way that the ordinality of the preference and indifference relationships for all players are maintained. This can be restated as.
' ', , , , , ,i i i i i i i i i i i iu a a u b a u a a u b a a b A i N
' ', , , , , ,i i i i i i i i i i i iu a a u b a u a a u b a a b A i N
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Ordinal Transformation
Example
a1
b1
a2 b2
0,0 1,1
0,12,0
a1
b1
a2 b2
-1,3 0,8
-1,87,3
u1(a1, a2) = u1(b1, b2) < u1(a1, b2) < u1(b1, a2)
u2(a1, a2) = u2(b1, a2) < u2(a1, b2) < u2(b1, a2)
u1(a1, a2) = u1(b1, b2) < u1(a1, b2) < u1(b1, a2)
u2(a1, a2) = u2(b1, a2) < u2(a1, b2) < u2(b1, a2)
a1
b1
a2 b2
0 321
Note that both games have an OPF, and bothadmit the same OPF
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Common Ordinal Transformations
Monotonic TransformationsLinear
Logarithmic
' , 0, 0i iu au c a c
' log , 0, 0i iu au c a c
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OT Property 1
An ordinal transformation of an ordinal potential game is itself an ordinal potential game.
Since an OT preserves the ordering of all preference relationships,if the original game lacks weak improvement cycles, then thetransformed game must also lack weak improvement cycles.
Proof Outline
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OT Property 2
If an ordinal transformation of a game yields an ordinal potential game, then the original game must also be an ordinal potential game.
Since an OT preserves the ordering of all preference relationships, ifthe transformed game lacks weak improvement cycles, then so mustthe original game.
Proof Outline
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OT Property 3
An ordinal transformation of an exact potential game is an ordinal potential game (will remain an EPG if OT is linear).
Proof OutlineAn EPG is also an OPG. Apply the OT property 1.
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Generalized Ordinal Potential Games
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Optimality If ui are designed so that maximizers of V are coincident with
your design objective function, then NE are also optimal. (*) Can also introduce cost function
to utilities to move NE.
In theory, can make any action tuple the NE
– May introduce additional NE– For complicated NC, might as well
completely redesign ui
* *
0i i
V a NC a
a a
*i iu a u a NC a
V
a
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Path Properties
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EPG Property 7
(Shapley)
Consider a game with finite action sets. The following are equivalent:
(1) The game is an EPG.(2) I(, u) = 0 for every finite closed path (3) I(, u) = 0 for every finite simple closed path (4) I(, u) = 0 for every finite simple closed path of length 4
where 1 10
, k m mi m i mm
I u u a u a
and i(m) is the unique deviating player at step m
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EPG Property 8
(Shapley)Every finite EPG has FIP.
Finite Improvement Path Property (FIP) All improvement paths in the game are finite.
Proof OutlineFor any = (a0, a1,…), V(a0) < V(a1) < …Since A is finite, must be finite
Note that all NE of a finite EPG must be a terminal point in a finiteimprovement path.
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- Improvement Path
Consider a game with continuous, bounded action sets and >0.A path is an -improvement path if for all k 1
1k ki k i ku a u a
Approximate Finite Improvement Path Property (AFIP)If for every >0, every -improvement path is finite.
EPG Property 9Every EPG with continuous bounded action sets has AFIP.
EPG Property 10Every EPG with bounded action sets possesses an -equilibrium point. (A point from which there are no improvement deviations greater than or equal to )
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Convergence Properties of EPG
EPG Property 11Because they satisfy FIP, all repeated games where each stage is the same finite EPG and all players are myopic converge to a NE of the EPG when play follows a better response dynamic.
(Shapley)
EPG Property 12Because they satisfy AFIP, all repeated games where each stage is the same bounded infinite EPG and players are myopic convergeto a -equilibrium point a when play follows a better response dynamic.
(Shapley)
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Implications of Monotonicity Monotonicity implies
– Existence of steady-states (maximizers of V)– Convergence to maximizers of V for numerous combinations of
decision timings decision rules – all self-interested adaptations Does not mean that that we get good performance
– Only if V is a function we want to maximize
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Token Economies Pairs of cognitive radios exchange tokens for services
rendered or bandwidth rented Example:
– Primary users leasing spectrum to secondary users D. Grandblaise, K. Moessner, G. Vivier and R. Tafazolli, “Credit
Token based Rental Protocol for Dynamic Channel Allocation,” CrownCom06.
– Node participation in peer-to-peer networks T. Moreton, “Trading in Trust, Tokens, and Stamps,” Workshop on
the Economics of Peer-to-Peer Systems, Berkeley, CA June 2003. Why it works – it’s a potential game when there’s no
externality to the trade– Ordinal potential function given by sum of utilities
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Lyapunov Stability
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Direct Method for Continuous Systems
(from potential game chapter/report)
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Application of Continuous Direct Method
[Slade1994] – directional updates
For differentiable exact potential games, negative of potential function is the Lyapunov function
This is more concisely shown in [Anderson2004]
[Slade94] M. E. Slade, “What Does an Oligopoly Maximize?” Journal of Industrial Economics 58, pp. 45–61. 1994.
[Anderson2004] S. Anderson, J. Goeree, C. Holt, “Noisy Directional Learning and the Logit Equilibrium,” Scandinavian Journal of Economics, (106) September 2004, pp. 581-602.
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Direct Method for Discrete Time Systems
Theorem 3.4 in A. Medio, M. Lines, Nonlinear Dynamics: A Primer, Cambridge University Press, Cambridge, UK, 2001.
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Better Response Stability Assume f is any better response process
implemented on a generalized ordinal potential game with potential V.
Any fixed point of f is Lyapunov stable. Proof: Lyapunov function
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Continuous Better Response Stability
Assume f is any better response process that converges to an isolated potential maximzer for a potential game then f is Lyapunov stable
Proof: Lyapunov function
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Best Response Stability Assume f is any best response process
implemented on a generalized best response potential game with potential V.
Any NE s Lyapunov stable. Proof: Lyapunov function
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How potential games handle the shortcomings
Steady-states– Finite game NE can be found from maximizers of V.
Optimality– Can adjust exact potential games with additive cost function
(that is also an exact potential game)– Sometimes little better than redesigning utility functions
Game convergence – Potential game assures us of FIP (and weak FIP)– DV satisfy Zangwill’s (if closed)
Noise/Stability– Isolated maximizers of V have a Lyapunov function for
decision rules in DV
Remaining issue:– Can we design a CRN such that it is a potential game for
the convergence, stability, and steady-state identification properties
– AND ensure steady-states are desirable?
74 © Cognitive Radio Technologies, 2007
Comments on Network Design Options Approaches can be combined
– Policy + potential– Punishment + cost adjustment– Cost adjustment + token economies
Mix of centralized and distributed is likely best approach Potential game approach has lowest complexity, but cannot be
extended to every problem Token economies requires strong property rights to ensure
proper behavior Punishment can also be implemented at a choke point in the
network