Lower Bounds on the Complexity of Solving Two Classes of Non-cooperative Games

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by  Ehsan Nekouei

Lower Bounds on the Complexity of Solving Two
Classes of Non-cooperative Games
Ehsan Nekouei, Member, IEEE, Girish N. Nair, Member, IEEE, Tansu Alpcan, Senior Member, IEEE, Robin
J. Evans, Life Fellow, IEEE
Abstract
This paper studies the complexity of solving two classes of non-cooperative games in a distributed manner in which the
players communicate with a set of system nodes over noisy communication channels. The complexity of solving each game
7 class is defined as the minimum number of iterations required to find a Nash equilibrium (NE) of any game in that class with ǫ
1 accuracy. First,weconsider theclassG of allN-player non-cooperative gameswithacontinuous actionspacethatadmit atleast
0 one NE. Using information-theoretic inequalities, we derive a lower bound on the complexity of solving G that depends on the
2 Kolmogorov2ǫ-capacityoftheconstraintsetandthetotalcapacityofthecommunicationchannels.Wealsoderivealowerbound
on the complexity of solving games in G which depends on the volume and surface area of the constraint set. We next consider
n
the class of all N-player non-cooperative games with at least one NE such that the players’ utility functions satisfy a certain
a
J (differential) constraint. We derive lower bounds on the complexity of solving this game class under both Gaussian and non-
Gaussiannoisemodels.Ourresultinthenon-Gaussian caseisderivedbyestablishingaconnectionbetweentheKullback-Leibler
4
distance and Fisher information.
2

Index Terms
]
T
Non-cooperative games, Nash seeking algorithms, information-based complexity, minimax analysis, Fano’s inequality
I
.
s
c I. INTRODUCTION
[
A. Motivation

1
Game theory offers a suite of analytical frameworks for investigating the interaction between rational decision-makers,
v
hereafter called players. In the past decade, game theory has found diverse applications across engineeringdisciplines ranging
7
1 from power control in wireless networks to modeling the behavior of travelers in a transport system. The Nash Equilibrium
7 (NE) is the fundamental solution concept for non-cooperative games, in which a number of players compete to maximize
6 conflictingutilityfunctionsthatareinfluencedbytheactionofothers.AttheNE,noplayerbenefitsfromaunilateraldeviation
0 from its NE strategy.
.
1 Findingthe NEof a non-cooperativegameisa fundamentalresearchproblemthatliesatthe heartofgametheoryliterature.
0 Fornon-cooperativegameswithcontinuousactionspaces,variousNashseekingalgorithmshavebeenproposedintheliterature,
7
e.g.see[1],[2].Inthispaper,weinvestigatetheintrinsicdifficultyoffindingaNEinsuchgames.Usingthenotionofcomplexity
1
fromtheconvexoptimizationliterature,andinformation-theoreticinequalities,wederivelowerboundsontheminimumnumber
:
v of iterations required to find a NE within a desired accuracy, for any N-player, non-cooperativegame in a given class.
i
X
r B. Related Work
a
The book by [3] pioneered the investigation of complexity in convex optimization problems. In this model, an algorithm
sequentiallyqueriesanoracleabouttheobjectivefunctionofaconvexoptimizationproblem,andtheoraclerespondsaccording
to the queries and the objective function. They derive bounds on the minimum number of queries required to find the global
optimizerof any functionin a givenfunctionclass. In [4], information-theoreticlower boundswere derivedon the complexity
ofconvexoptimizationproblemswith a stochastic firstorderoracleforthe classof functionswith a knownLipschitzconstant.
In a stochastic first order oracle model, the algorithm receives randomized information about the objective function and its
subgradient. These results were extended to different function classes in [5].
The paper [6] considered a model in which the algorithm observes noisy versions of the oracle’s response and established
lower bounds on the complexity of convex optimization problems under first order as well as gradient-only oracles. In [7],
complexity lower bounds were obtained for convex optimization problems with a stochastic zero-order oracle. The paper
[8] studied the complexity of convex optimization problems under a zero-order stochastic oracle in which the optimization
algorithm submits two queries at each iteration and the oracle responds to both queries. These results were extended to the
case in which the algorithm makes queries about multiple points at each iteration in [9]. In [10], the complexity of convex
optimization problems was studied under an erroneous oracle model wherein the oracle’s responses to queries are subject to
absolute/relative errors.
This work was supported by the Australian Research Council’s Discovery Projects funding scheme (DP140100819). Department of Electrical and
Electronic Engineering, TheUniversityofMelbourne, VIC3010,Australia. E-mails:{ehsan.nekouei,gnair,tansu.alpcan,robinje}@unimelb.edu.au. Thispaper
waspresented inpartatNecSys2016workshop, Tokyo,September2016.
2
Fig.1. A pictorial representation of the communication graph between system nodes and players. Solid arrows denote the uplink channels
and dashed arrows denote the downlink channels.
C. Contributions
This paper studies the complexity of solving two classes of non-cooperativegames in a distributed setting in which players
communicate,not with an oracle, but with a set of utility system nodes (USNs) and constraintsystem nodes (CSNs) to obtain
the required informationfor updating their actions. Each USN computes the utility-related information for a subset of players
whereas a CSN evaluates a subset of constraint functions. The communication between players and system nodes is subject
to noise, i.e., the system nodes will receive noisy versions of players’ actions, and the players will receive noisy information
from the system nodes.
First, we consider the game class which consists of all N-playernon-cooperativegames with a joint action space defined
G
by L convex constraints such that all the games in admit at least one Nash equilibrium (NE). We derive lower bounds on
G
the minimum number of iterations requiredto get within ǫ of a NE of any game in with confidence1 δ without imposing
G −
any particular structure on the computationmodelat USNs. Our results indicate that the complexityof solving the game class
is limited by the Kolmogorov 2ǫ-capacity of the constraint set and the total capacity of communication channels from the
G
USNs to the players. We also derive a lower bound on the complexity of solving the game class in terms of the volume
G
and surface area of the constraint set. We note that, in a precursor conference paper [11], we have studied the complexity of
solving the game class under a slightly different setting than that in the current manuscript.
G
We next consider the game class which consists of all non-cooperative games with a joint action space defined by L
γ
G
constraintssuchthat(i)allthegamesin admitatleastoneNE,(ii)thenormoftheJacobianmatrixofthepseudo-gradient
γ
G
vector, induced by utility functions of players, is more than γ. We study the complexity of solving the game class under
γ
G
the partial-derivative computation model at USNs and various noise models. Under the partial-derivative computation model,
each player receives a noisy version of the partial derivative of its utility function, with respect to its action, in each iteration.
Our results show that the complexity of solving the game class up to ǫ accuracy is at least of order 1 , as ǫ tends to
Gγ γ2ǫ2
zero, with Gaussian communication channels. We also consider a setting in which the channels from system nodes to players
are non-Gaussian and the channels from players to system nodes are noiseless. In the non-Gaussian setting, our results show
that the complexity of solving the game class up to ǫ accuracy is at least of order 1 as ǫ tends to zero. This result is
Gγ γ2ǫ2
established by deriving an asymptotic expansion for the Kullback-Leibler (KL) distance between a non-Gaussian probability
distribution function(PDF) and its shifted version, under some mild assumptionson the non-GaussianPDF. More precisely, it
is shown that the KL distance between a PDF and its shifted version can be written, up an error term, as a monomial which
is quadratic in the shift parameter and linear in the Fisher information of the corresponding PDF with respect to the shift
parameter.
This paper is organized as follows. Section II discusses our modeling assumptions and problem formulation. Section III
discussesourmainresultsalongwith theirinterpretations.Alltheproofsarerelegatedto SectionIVto improvethe readability
of the paper. Section V concludes the paper.
II. SYSTEMMODEL
A. Game-theoretic Set-up
Consider a non-cooperativegame with N players indexed over = 1, ,N . Let xi (i ), and x= x1, ,xN ⊤
N { ··· } ∈N ···
denote the action of the ith player and the collection of all players’ actions, respectively.The utility function of the ith player
(cid:2) (cid:3)
is denotedby u xi,x i where x i is the vector of those otherplayers’ actionsthataffectthe ith player’sutility. The utility
i − −
function of the ith player quantifies the desirability of any point in the action space for the ith player. The actions of players
are limited by L(cid:0)convex(cid:1)constraints denoted by g(x) 0 where g() = [g1(), ,gL()]⊤ is a mapping from RN to RL.
The set of constraint functions is indexed over = p≤ N:1 p · L . Let· ·d·e·note th·e action space of players, i.e.,
L { ∈ ≤ ≤ } S
= x RNs.t.g(x) 0 .
S ∈ ≤
We assume that is a compact and convex subse(cid:8)t of RN. (cid:9)
S
3
T⋆ ( , )=inf T N: s.t. sup infPr x X ,Yˆ ,Zˆ ǫ δ . (1)
ǫ,δ G O ( ∈ ∃A U() i NEi,U(·)−AT+1 1:T 1:T 1:T ≥ ≤ )
· ∈F (cid:16)(cid:13) (cid:16) (cid:17)(cid:13) (cid:17)
(cid:13) (cid:13)
(cid:13) (cid:13)
In non-cooperative games, each player is interested in maximizing its own utility function, irrespective of other players.
Since the maximizersof utility functionsof playersdo notnecessarily coincide with each other, a trade-offis required.In this
paper, the Nash equilibrium is considered as the canonical solution concept of the non-cooperative game among players. Let
x be the NE of the game among players. Then, at the NE, no player has incentive to unilaterally deviate its action
NE
∈ S
from its NE strategy, i.e.,
xiNE =argxi m(xa−xi ) ui xi,x−NEi ,∀i∈N,
∈S NE,C
(cid:0) (cid:1)
where x i is the collection of NE strategies of players which are coupled with the ith player through constraints, and
−NE,C
x i is the set of possible actions of the ith player given x i . The vector of all utility functions is denoted by
S −NE,C −NE,C
U(cid:16)(x)=[u(cid:17)1(x), ,uN(x)]⊤.
Let denote th··e·class of functions from RN to RN such that any N-player non-cooperative game with the constraint set
F
and utility function vector in admits at least one NE. By the class of non-cooperative games = , , , we mean
S F G hN S Fi
the set of all games with N players, the action space , and the utility function vector in , i.e., U() .
S F · ∈F
B. Communication Model
In this paper, we consider a distributed Nash seeking set-up wherein, at each time-step, players communicate with a set of
utilitysystemnodes(USNs)andconstraintsystemnodes(CSNs)toobtaintherequiredutility/constraintrelatedinformationfor
updating their actions. A USN computesutility-related informationfor a set of players, e.g., the utility functionsof players or
their partialderivatives.A CSN evaluatesa subsetofconstraintsbasedon the receivedactionsof players.Eachutility function
or constraintis evaluated at only one USN or CSN, respectively.The number of USNs and CSNs are denoted by K and K ,
u c
respectively, with K N and K L. We use USN (l 1,...,K ) and CSN (n 1,...,K ) to refer to the lth
u c l u n c
≤ ≤ ∈ { } ∈ { }
USN and nth CSN, respectively.
At each time-step, player i transmits its action to USN if its action affects at least a utility function evaluated by USN .
l l
The set of players which transmit their actions to USN is denoted by . We use the mapping π(), from 1, ,N to
l Nusnl · { ··· }
1,...,K , to indicate the USN which computes the utility-related information for a given player, i.e., π(i) = l if USN
u l
{ }
computes the utility-related information for the ith player. Thus, the utility-related information for the ith player is computed
by USN .
π(i)
Similarly,ateachtime-step,playeritransmitsitsactiontoCSN ifitsactionaffectsatleastoneconstraintfunctionevaluated
n
by CSN . The set of players which transmit their actions to CSN is represented by . We use the mapping φ(), from
n n Ncsnn ·
1, ,L to 1,...,K , to indicate the CSN which evaluatesa givenconstraintfunction,i.e., φ(p)=n if CSN evaluates
c n
{ ··· } { }
the pth constraint function. Hence, the pth constraint function is evaluated by CSN . The set of constraint functions which
φ(p)
are affected by the ith player’s action are denoted by .
i
L
The communication topology between players and system nodes is given by a bipartite digraph in which the players and
the system nodes form two disjoint sets of vertices. There exists a directed edge, in the communication graph, from the ith
player to USN if i . Also, there exists a directed edge from USN to the ith player for all i . Furthermore,
l ∈ Nusnl π(i) ∈ N
there exist a directed edge from the ith player to CSN , and a directed edge from CSN to the ith player if i . We
n n ∈ Ncsnn
refer to communication channels from players to system nodes as uplink channels and the communicationschannels between
systemnodesandplayersasdownlinkchannels.Fig.1showsapictorialrepresentationofthecommunicationtopologybetween
system nodes and players.
Playerscommunicatewith system nodesusing frequencydivisionmultiplexing(FDM) ortime divisionmultiplexing(TDM)
schemes, i.e., each player broadcasts its action to its neighboringsystem nodes in the communicationgraph using a dedicated
time or frequency band. Similarly, system nodes communicate with players via FDM or TDM communication schemes. The
communicationbetweenplayersandsystem nodesis performedovernoisycommunicationchannels,i.e.,playersreceivenoisy
information from system nodes, and system nodes receive noisy versions of players’ actions. This will be made more explicit
in the next subsection.
C. Nash Seeking Algorithms
1) TheUpdateRule: Inthispaper,weconsiderageneralstructurefortheNashseekingalgorithmswhichallowseachplayer’s
action to be updated using the past actions of that player as well as the past received utility/constraint related information by
4
that player. Let be such a Nash seeking algorithm. Then, under , the ith player’s action at time k, i.e., xi, is updated
A A k
according to the update rule
xi = i Xi ,Yˆi ,Zˆi ,
k Ak 1:k−1 1:k−1 1:k−1
(cid:16) (cid:17)
where Xi is the history of the ith player’s actions from time 1 to k 1, Yˆi denotes the sequence of received utility-
related in1f:ko−rm1ation by the ith player from time 1 to k 1, and Zˆi −denot1e:sk−th1e sequence of received constraint-related
tihnafotrXmaition,bYyˆithe itahnpdlaZˆyier fromcantimbeew1rtiottekn−as1. Here−, Aik(·,·,·)1i:sk−a1mapping from Rk−1×Rk−1×R(k−1)|Li| to R. Note
1:k 1 1:k 1 1:k 1
− − −
Xi = xi k−1,
1:k 1 t t=1
−
Yˆi =(cid:8)yˆi(cid:9)k−1,
1:k 1 t t=1
−
k 1
Zˆi =(cid:8)zˆi(cid:9),p,p − ,
1:k−1 t ∈Li t=1
n o
, respectively, where xi is the action of the ith player at time t, yˆi denotes the received utility-related information by the ith
t t
player at time t and zˆi,p denotes the received information regarding the pth constraint by the ith player at time t.
t
The kth step of the algorithm is denoted by
A
X ,Yˆ ,Zˆ = i Xi ,Yˆi ,Zˆi ,
Ak 1:k−1 1:k−1 1:k−1 Ak 1:k−1 1:k−1 1:k−1 i
(cid:16) (cid:17) n (cid:16) (cid:17)o
where
X1:k−1 = xit :i∈N tk=−11,
Yˆ1:k−1 =(cid:8)yˆti :i∈N(cid:9)tk=−11,
k 1
Zˆ =(cid:8)zˆi,p :i (cid:9),p − .
1:k−1 t ∈N ∈Li t=1
n o
We refer to
= X ,Yˆ ,Zˆ ,
k 1:k 1 1:k 1 1:k 1
A A − − − k
n (cid:16) (cid:17)o
as the Nash seeking algorithm .
2) Communication and ComAputation At USNs: The received action of the ith player by USN at time k, i.e., xˆi , can
l k,usnl
be written as
xˆi =xi +Wi ,
k,usnl k k,usnl
whereWi isthenoiseintheuplinkchannelfromtheithplayertoUSN .LetXˆusnl = xˆi :i k denotethe
k,usnl l 1:k t,usnl ∈Nusnl t=1
history of actions received by USN from time 1 to time k. At time k, USN computes yi, i.e., the utility-related information
l l k(cid:8) (cid:9)
for player i at time k, for all i such that π(i)=l.
In this paper, we study the complexity of solving non-cooperativegames under two computation models at USNs. We first
considerageneralcomputationmodelinwhichyi isallowedtobeanyarbitraryfunctionofu ()andtheinformationavailable
k i ·
at USN from time 1 to k, i.e.,
l
yi = Xˆusnl,u () i:π(i)=l. (2)
k Ok,i 1:k i · ∀
(cid:16) (cid:17)
where (, ) is a functional. This formulation allows us to capture the complexity of solving the game class under a
k,i
O · · G
general class of computation models at USNs in Theorem 1. We refer to = Xˆusnπ(i),u () as the general
O Ok,i 1:k i · k,i
computation model at USNs. n (cid:16) (cid:17)o
We also study the complexity of solving non-cooperative games under the partial-derivative computation model in which
USN at time k evaluates the partial derivative of the utility function of the ith player with respect to its action, i.e., yi =
l k
∂ u xi,x i for all i with π(i)=l. We refer to the partial-derivative computational model for USNs as
∂(xi) i − Xˆusnl
(cid:12) k
(cid:0) (cid:1)(cid:12)(cid:12) 1 = 1 Xˆusnπ(i),u () (3)
O Ok,i k i · k,i
n (cid:16) (cid:17)o
where Xˆusnl = xˆi :i denotes the set of actions received by USN at time k and
k k,usnl ∈Nusnl l
n o
∂
1 Xˆusnπ(i),u () = u xi,x i .
Ok,i k i · ∂(xi) i − (cid:12)x=Xˆusnπ(i)
(cid:16) (cid:17) (cid:0) (cid:1)(cid:12) k
(cid:12)
(cid:12)
5
Then, USN transmits yi to the ith player for all i with π(i)=l. The received utility-related information by the ith player
l k
at time k can be written as
yˆi =yi +Vi,
k k k
where Vi is the noise in the downlink channel from the USN to the ith player.
k π(i)
3) Communication and Computation At CSNs: The received action of the ith player by CSN at time k, i.e, xˆi , can
n k,csnn
be written as
xˆi =xi +Wi ,
k,csnn k k,csnn
where Wi is the noise in the uplinkchannelfrom the ith player to CSN . The collection of receivedactions at time k by
k,csnn n
the CSN is denoted by Xˆcsnn = xˆi :i . At time k, CSN evaluates its associated constraint functions using
n k k,csnn ∈Ncsnn n
the received actions at time k, i.e.,n o
zp =g Xˆcsnn , p:φ(p)=n
k p k ∀
Finally, CSN broadcasts zp to the players which th(cid:16)eir acti(cid:17)ons affect g (). If the action of the ith player affects the pth
n k p ·
constraint, the ith player will receive
zˆi,p =zp+Vi,p,
k k k
at time k where Vi,p is the noise in the downlink channel from CSN to the player i.
k φ(p)
Remark 1: Although, we assume that the CSN at time k transmits g Xˆcsnn to the ith player (if p ), our results
n p k ∈ Li
continue to hold when other computation models are implemented at the(cid:16)CSNs,(cid:17)e.g., when the CSNn at time k transmits
∂ g (x) to the ith player.
∂xi p x=Xˆcsnn
k
(cid:12)
(cid:12) TABLEI
TABLEOFTHEMAINVARIABLES
Variable Description
The set of players
N
The set of constraints
L
The set of constraints which are affected by the ith player action
i
L
USN The USN which computes utility-related information for the ith player
π(i)
CSN The CSN which evaluates the pth constraint
φ(p)
The set of players which transmit their actions to USN
Nusnl l
The set of players which transmit their actions to CSN
Ncsnn n
xi Action of the ith player at time k
k
Xi Actions of the ith player from time 1 to k
1:k
X Actions of all the players from time 1 to k
1:k
xˆi The received action of the ith player by USN at time k
k,usnl l
Xˆusnl The collection of received actions by USN at time k
k l
Xˆusnl The collection of received actions by USN from time 1 to k
1:k l
yi The utility-related information computed by USN for the ith player
k π(i)
yˆi The received utility-related information by the ith player
k
Yˆi The history of received utility-related information by the ith players from time 1 to k
1:k
Yˆ The history of received utility-related information by all the players from time 1 to k
1:k
xˆi The received action of the ith player by CSN at time k
k,csnn n
Xˆcsnn The collection of received actions by CSN at time k
k n
zp The value of the pth constraint at time k evaluated by CSN
k φ(p)
zˆi,p The received value of the pth constraint at time k by the ith player
k
Zˆi The history of received constraint-related information by the ith players from time 1 to k
1:k
Zˆ The history of received constraint-related information by all the players from time 1 to k
1:k
Wi The additive noise in the uplink channel from the ith player to USN at time k (i )
k,usnl l ∈Nusnl
Wi The additive noise in the uplink channel from the ith player to CSN at time k (i )
k,csnn n ∈Ncsnn
Vi The additive noise in the downlink channel from USN to the ith player at time k
k π(i)
Vi,p The additive noise in the downlink channel which transmits zp to the ith player at time k (p )
k k ∈Li
6
D. The Complexity Criterion
Consider the class of games and and the computationmodel . Then,the (ǫ,δ)-complexityof solving the class of games
G O
with the computation model , denoted by T⋆ ( , ), is defined in (1) where x is a NE of the non-cooperative
gGamewith the utilityfunctionveOctorgivenbyU(ǫ·,δ) G O. Accordingto (1), the (ǫ,δ)-cNoEmi,pUl(e·x)ityofsolvingthe class ofgames
∈F
with the computation model is defined as the smallest positive integer T for which there exists an algorithm such
G O A
that, for any game in , the probability of ǫ deviation of the algorithm’s output at time T +1 from at least a NE of the game
G
is less than δ. Note that (1) assigns a positive integer to any class of games. For a given pair of ( , ), a small value of
G O
T⋆ ( , ) indicates that the class of games with the computation model can be solved faster compared to a large value
ǫ,δ G O G O
of T⋆ ( , ). The complexityof solving the game class under the computation model 1 can be defined in a similar way.
ǫ,δ G O G O
Remark 2: The ǫ-Nash equilibrium (ǫ-NE) is a closely related solution concept to the NE which is defined as the point
such that no play can gain more than ǫ by unilaterally deviating its strategy from its ǫ-NE strategy. However, an ǫ-NE is not
always close to a NE since game-theoretic problems are not necessarily convex problems and a NE is not necessarily the
maximizer/minimizer of utility functions of all players [12]. Hence, we do not consider ǫ-NE as a solution concept in this
paper.
E. Modeling Assumptions
Inthispaper,weimposethefollowingassumptionsontheNashseekingalgorithmsandthenoisetermsintheuplink/downlink
communication channels:
1) X is specified by the algorithm , and the algorithm uses the same value of X for solving any game.
1 1
A A
2) Wi ,i isacollectionofzeromean,independentandidenticallydistributed(i.i.d.)randomvariableswith
k,usnl ∈Nusnl k
vnariance σu2snl >0 foor all 1≤l≤Ku.
3) Wi ,i isa collectionofzeromean,i.i.d.randomvariableswithvarianceσ2 >0forall1 n K .
k,csnn ∈Ncsnn k csnn ≤ ≤ c
4) nVi,Vi,p,p ois a collection of i.i.d. random variables with zero mean and variance σ2 >0 for all i .
k k ∈Li k i ∈N
5) nAll the uplink/dowonlink noise terms are jointly independent.
F. Organization of The Paper and Notations
The rest of this paper is organized as follows. Section III states our main results on the complexity of solving two classes
of non-cooperativegames. Section IV presents the derivation of our results, and Section V concludes the paper.
In the rest of this paper, we use the following notations from asymptotic analysis literature. For two positive functions
f(x) and g(x), we say f(x) = Ω(g(x)) if liminf f(x) > 0. We also say f(x) = Θ(g(x)) if liminf f(x) > 0 and
x↓0 g(x) x↓0 g(x)
limsup f(x) < . Our main notations are summarized in Table I.
x↓0 g(x) ∞
III. RESULTSAND DISCUSSIONS
Inthissection,weestablishvariouslowerboundsonthecomplexityofsolvingtwogameclassesunderdifferentassumptions
on the distribution of uplink/downlinknoise terms and different computation models at USNs. In Subsection III-A, we derive
two lower bounds on the complexity of solving the game class under the general computation model at USNs without
G
assuming any particular distribution for the uplink/downlink noise terms. In Subsection III-B, we establish a lower bound on
the complexityof solving a subclass of , denoted by , under Gaussian uplink/downlinkchannels and the partial-derivative
γ
G G
computation model. Subsection III-B presents a lower bound on the complexity of solving the game class under noiseless
γ
G
uplink channels, non-Gaussian downlink channels, and the partial-derivative computation model. Subsection III-D discusses
the complexity of solving the game class under the partial-derivative computation model when both uplink and downlink
γ
G
channels are non-Gaussian distributed.
A. General Computation model at USNs and General Uplink/Downlink Channels
In this subsection, we study the computational complexity of solving the game class without imposing any particular
G
structure on the computation model at the USNs, or imposing any specific probability distribution on the noise in the
uplink/downlink channels. To this end, we first give the definition of the total capacity of downlink channels, the notion
of 2ǫ-distinguishable subsets of , and the Kolmogorov capacity of .
S S
The total capacity of downlink channels from USNs to players is defined as
C = max I y1, ,yN;yˆ1, ,yˆN
down
pY(y),E[kYk2]≤α ··· ···
(cid:2) (cid:3)
where yi and yˆi are the input and the output of the downlink channel from USN to the ith player, respectively, Y =
π(i)
y1, ,yN ⊤, pY (y) is the joint distribution of Y, and α is the total average power constraint of the downlink channels
···
between USNs and players.
(cid:2) (cid:3)
7
Definition 1: A subset of is 2ǫ-distinguishable if the distance between any two of its points is more than 2ǫ [13].
S
Definition 2: Let ( ) denote the cardinality of maximal size 2ǫ distinguishable subsets of . Then, the Kolmogorov
2ǫ
M S S
capacity of is defined as log ( ) [13].
2ǫ
S M S
The next theorem establishes a lower bound on T⋆ ( , ).
ǫ,δ G O
Theorem 1: Let T⋆ ( , ) denote the complexity of the class of N-player non-cooperative games with the continuous
ǫ,δ G O G
action space . Then, we have
S
(1 δ)log ( ) 1
T⋆ ( , ) − M2ǫ S − (4)
ǫ,δ G O ≥ C
down
where C is the total capacity downlink channels from USNs to players, and log ( ) is the Kolmogorov 2ǫ-capacity
down 2ǫ
M S
of the action space .
S
Proof: See Subsection IV-A.
Theorem1establishesanalgorithm-independentlowerboundontheorderofcomplexityofsolvingthegameclass .According
G
to this theorem, T⋆ ( , ) is lower bounded by the ratio of the Kolmogorov 2ǫ-capacity of the action space to the total
ǫ,δ G O S
Shannon capacity of the downlink channels. Note that the Kolmogorov 2ǫ-capacity of can be interpreted as a measure of
S
players’ ambiguity about their NE strategies. Thus, as log ( ) becomes large, T⋆ ( , ) is expected to increase since
M2ǫ S ǫ,δ G O
players have to search in a bigger space to find their NE strategies. Based on Theorem 1, C has a reverse impact on
down
T⋆ ( , ). Note that C is an indication of the informationtransmission quality from USNs to players. That is, as C
ǫ,δ G O down down
decreases, players will receive noisier information regarding their utility functions compared with a large value of C .
down
Theorem 1 depends on the 2ǫ-capacity of the constraint set which is usually hard to compute unless the action space of
S
players is restricted to special geometries. As ( ) is just the maximum number of ǫ-balls that can be packed into , it is
2ǫ
asymptotically equalto VVooll((BSǫ)) = VαoNl(ǫSN) as ǫMtends tSo zero, where Bǫ is the N-ball of radiusǫ, and αN is the N-dimenSsional
sphericalconstantunderthe assumednorm.Thus,the complexityis atleast of orderlog1 asǫ becomessmall. The nextresult
ǫ
establishes a non-asymptotic lower bound of the same order, by lower bounding ( ) using a result from lattice theory.
2ǫ
M S
Corollary 1: Thecomplexityofsolvingthe class ofN-playernon-cooperativegames with continuousactionspace can
G S
be bower bounded as
(1 δ)Nlog 1 +log(Vol( ) ǫP( )) 1
T⋆ ( , ) − 2ǫ S − S − (5)
ǫ,δ G O ≥ C
(cid:0) down (cid:1)
where Vol( ) and P( ) are the volume and the surface area of the action space of players, respectively.
S S
Proof: See Subsection IV-B.
Based on Corollary 1, the lower boundon T⋆ ( , ) increasesat least linearly with the numberof players. This is due to the
ǫ,δ G O
fact thatthe amountof uncertaintyaboutthe NE increasesas the numberof playersbecomeslarge.Recall that log ( ) is
2ǫ
M S
a quantitativeindicator of ambiguityabout the NE. Furthermore,ǫ has a logarithmiceffect on T⋆ ( , ), i.e., the complexity
ǫ,δ G O
of solving the class of games increases according to Ω log1 as ǫ becomes small. Hence, based on Corollary 1, the game
G ǫ
class cannotbesolvedfasterthanΘ log1 time-stepsregardlessofuplink/downlinknoisedistributions,andthecomputation
G ǫ (cid:0) (cid:1)
model at the USNs.
(cid:0) (cid:1)
According to Corollary 1, the lower bound on the complexity of solving the game class increases as the volume of the
G
action space of players becomes large. Also, for a given surface area of action space of players, i.e., P( ), the volume of
S
action space of players can be upper bounded using the isoperimetric inequality for convex bodies [14] as follows:
Vol(B) N
Vol( ) P( )N−1 (6)
S ≤ (P(B))NN−1 S
where B is the closed unit ball in N-dimensional Euclidean space RN. Note that the equality in (6) is achieved if and only
if is a ball in RN[14]. Thus, for a given surface area of action space of players P( ), the lower bound on the complexity
ofSsolving games in the class increases as the action space of players becomes clSoser to a ball in RN with the volume
Vol(B) P( )NN−1. G
(P(B))NN−1 S
B. Partial-derivative Computation Model at USNs and Gaussian Uplink/Downlink Channels
In this section, we establish a lower bound on the complexity of solving a subclass of , denoted by , under the partial-
γ
G G
derivativecomputationmodel(seeequation(3))whenthecommunicationnoiseintheuplinkanddownlinkchannelsisGaussian
distributed.We also comparethe complexityof solvingthe gameclass with the complexityof solving the class of strongly
γ
G
convexoptimizationproblems.Tospecifythegameclass ,wefirstdefinethenotionofpseudo-gradientforautilityfunction
γ
G
vector as follows.
Definition 3: The pseudo-gradient of the utility function vector U(x) = u1 x1,x−1 , ,uN xN,x−N ⊤ is defined
···
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
8
as
∇˜U(x)= ∂(∂x1)u1 x1,x−1 ,··· ,∂(x∂N)uN xN,x−N ⊤
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
We use J (x) to denote the Jacobian matrix of the vector valued function ˜U(x), i.e.,
˜U ∇
∇
∂2
[J (x)] = u xi,x i , 1 i,j N
˜U ij ∂(xj)(xi) i − ≤ ≤
∇
(cid:0) (cid:1)
We next specify a set of utility vector functions, denoted by , which is used to define the game class .
γ γ
Definition 4: The set of utility vector functions is defiFned as the set of all vector valued functionsGU(x) from RN to
γ
RN such that F
1) The N-player non-cooperativegame with utility vector function given by U(x) and the constraint set admits at least
S
a Nash equilibrium (NE).
2) The matrix J (x) exists for all x in .
˜U S
∇
3) The matrix J (x) satisfies J γ >0 for all x where J (x) denotes the matrix norm of J (x).
∇˜U ∇˜U(x) ≥ ∈S k ∇˜U k ∇˜U
The next definition specifies the gam(cid:13)e class(cid:13) .
(cid:13) (cid:13)Gγ
Definition 5: The class of games(cid:13) = (cid:13), , is defined as the set of all non-cooperative games with N players, the
γ γ
G hN S F i
constraint set , and the utility function vector U() in .
γ
S · F
Note that the game class reduces to the class when γ is equal to zero. The complexity of solving the game class
γ γ
G G G
heavily depends on J (x) as shown in Theorem 2.
˜U
Remark 3: We note∇thatboth ˜U(x)andJ (x)playimportantrolesinthegametheoryandsystemtheoryliterature.To
∇ ˜U
clarify this point, consider an unconstrained N∇-player game with the utility vector function U(x)= u xi,x i such that
i − i
eachu xi,x i isconcavein xi.Then,anysolutionof ˜ (x)=0 willbea NEofthisgame.Also, considerthedynamical
i − ∇U (cid:2) (cid:0) (cid:1)(cid:3)
system x˙ = ˜U(x). Then, any NE of the aforementioned game will be an equilibrium of this dynamical system and the
(cid:0) ∇ (cid:1)
eigenvalues of the matrix J (x) determine the local stability of this dynamical system around its equilibria. Moreover, the
˜U
matrix J (x) can be used∇to study the uniqueness of the NE in non-cooperative games [15].
˜U
The ne∇xt theorem studies the complexity of solving the game class under the partial-derivative computation model and
γ
G
Gaussian distributed uplink/downlinkchannels.In the derivationof Theorem2, it is assumed that the constraintset contains
S
a 2-ball with radius √2ǫ, i.e., the set of all points in a 2-dimensional plane with the distance √2ǫ from a point in .
S
Theorem 2: Let T⋆ , 1 denote the complexity of solving the game class under the partial-derivativecomputation
ǫ,δ Gγ O Gγ
model at USNs. Then, for Gaussian distributed uplink/downlink channels and δ 0.5, we have
(cid:0) (cid:1) ≤
(2(1 δ) 1)min σ2
T⋆ , 1 − − i i , (7)
ǫ,δ Gγ O ≥ 4γ2ǫ2
(cid:0) (cid:1)
where σ2 is the variance of noise at the player i’s receiver.
i
Proof: See Subsection IV-C.
Theorem 2 establishes an algorithm-independent lower bound on the complexity of solving the game class . According
γ
G
to this result, the game class cannot be solved faster than Θ 1 under the partial-derivative computation model and
Gγ γ2ǫ2
Gaussian noise model for uplink and downlink channels. (cid:16) (cid:17)
It is helpful to compare the complexity of solving the game class with that of solving a black-box convex optimization
γ
G
problem. The complexity of solving black-box optimization problems is studied under an oracle-based setting in which
optimization algorithms rely on an oracle for function evaluation. In this setting, the oracle receives noise-free queries from
the optimization algorithm, and, the algorithm receives a noisy version of oracle’s response [6]. The next corollary studies
the complexity of solving the game class in a similar setting, i.e., under noiseless uplink channels and Gaussian downlink
γ
G
channels.
Corollary 2: LetT⋆ , 1 denotethecomplexityofsolvingtheclass usingthepartial-derivativecomputationmodel.
ǫ,δ Gγ O Gγ
Then, for noiseless uplink channels, Gaussian distributed downlink channels and δ 0.5, we have
(cid:0) (cid:1) ≤
(2(1 δ) 1)min σ2
T⋆ , 1 − − i i .
ǫ,δ Gγ O ≥ 4γ2ǫ2
(cid:0) (cid:1)
Proof: The proof is similar to that of Theorem 2 and is skipped.
Next, we use Corollary 2 to compare the complexity of solving the class with the complexity of solving a class of convex
γ
G
optimization problems using the oracle-based setting. To this end, consider the following optimization problem
min f(x),
x
∈S
9
where is a convex set, and f(x) belongs to the class of continuous and strongly convex functions . The complexity of
sc
S F
solving the class of convex optimization problems with the objective function in is defined as [6]
sc
F
inf T N: s.t. sup Pr(f(x ) f⋆ ǫ) δ
T+1
( ∈ ∃A f(·)∈Fsc − ≥ ≤ )
where x is the outputof the algorithm after T queries,andf⋆ =inf f(x). Itis shown in [6] that the complexityof
T+1 x
A ∈S
solvingthe class ofstrictly convexoptimizationproblemsunderthesubgradientcomputationmodelandGaussian noisemodel
isgivenbyΩ 1 .AccordingtotheCorollary2,thegameclass ishardertosolvecomparedwiththeclassofstrictlyconvex
ǫ Gγ
optimization problems since the games are non-convex problems, and NE is more sophisticated solution concept compared
(cid:0) (cid:1)
with the minimizer of a convex function (see Remark 2 for more details).
C. Partial-derivative Computation Model at USNs, Non-Gaussian Downlink Channels and Noiseless Uplink Channels
In this subsection, we study the complexity of solving the game class under the partial-derivative computation model
γ
G
when the downlink channels are not necessarily Gaussian and the uplink channels are noiseless. To this end, let p (x)
Vi
denote the common probability distribution function (PDF) of the collection of random variables Vi , i.e., the collection
k k
of noise terms in the downlink channel from USN to player i. To investigate the complexity of the game class in the
π(i) (cid:8) (cid:9) Gγ
non-Gaussian setting, we assume that p (x) satisfies the following mild assumptions for all 1 i N
Vi
≤ ≤
1) The PDF p (x) is non-zero everywhere on R.
Vi
2) The PDF p (x) is at least 3 times continuously differentiable, i.e., p (x) C3.
Vi Vi
∈
3) There exist positive constants β ,β ,β >0 such that
1 2 3
d3
logp (x) β +β xβ3 x R
dx3 Vi ≤ 1 2| | ∀ ∈
(cid:12) (cid:12)
(cid:12) (cid:12)
4) The tail of the random variable Vki(cid:12)(cid:12) decays faster th(cid:12)(cid:12)an x−(β3+1), i.e., we have
(cid:12) (cid:12) lim x(β3+1+r)Pr Vi x =0 k
(cid:12) (cid:12) x k ≥ ∀
→∞
for some r >0. (cid:8)(cid:12) (cid:12) (cid:9)
(cid:12) (cid:12)
The next theorem derives a lower bound on the complexity of solving the game class in the non-Gaussian setting.
γ
G
Theorem 3: Let T⋆ , 1 denote the complexity of solving the N-player non-cooperativegames in the class using
ǫ,δ Gγ O Gγ
the partial-derivativecomputationmodel at USNs. Assume that the PDFs of the downlink noise terms satisfy the assumptions
(cid:0) (cid:1)
1-4 above and the uplink channels are noiseless. Then, for δ 0.5 we have
≤
2(1 δ) 1
T⋆ , 1 − − .
ǫ,δ Gγ O ≥ 4Nǫ2γ2max +O(ǫ3)
i i
I
(cid:0) (cid:1)
where is the Fisher information of the PDF p (x) with respect to a shift parameter.
i Vi
I
Proof: See Subsection IV-D.
Theorem 3 establishes a lower bound on the complexity of solving the game class under noiseless uplink channels and
γ
G
non-Gaussian downlink channels. In Corollary 2, we showed that the complexity of solving the game class under the
γ
G
partial-derivativecomputation model is at least of the order 1 , as ǫ becomes small, when the uplink channels are noiseless
γ2ǫ2
andthedownlinkchannelsareGaussiandistributed.AccordingtoTheorem3,thelowerboundonthecomplexityofsolvingthe
game class is also of the order 1 when the uplink channels are noiseless and the downlink channels are not necessarily
Gγ γ2ǫ2
Gaussian distributed.
Theorem 3 is established by deriving an asymptotic expansion for the Kullback-Leibler (KL) distance between the PDF
p (x) and its shifted version. More precisely, we show that
Vi
1
D[p (x) p (x)]= (A τ)2+O τ 3
Vi k Aiτ+Vi 2Ii i k k
(cid:16) (cid:17)
where A is a 1-by-N row vector, τ is an N-by-1 column vector, p (x) is the PDF of A τ +Vi, and is the Fisher
i Aiτ+Vi i Ii
informationofp (x)withrespecttoashiftparameter.SincetheTaylorseriesofarealfunctionisnotnecessarilyconvergent,
Vi
Theorem 3 is proved using Taylor expansion Theorem. The assumptions 1-4 above are used to bound the remainder integral
which appears in the Taylor expansion (see Lemma 6 in Subsection IV-D and its proof for more details).
D. Partial-derivative Computation Model at USNs With Arbitrarily Distributed Uplink and Downlink Channels
The next theorem establishes a lower bound on the complexity of solving the game class under the partial-derivative
γ
G
computation model when the uplink and downlink channels are not necessarily Gaussian distributed.
10
Theorem 4: The complexity of solving the game class using the partial-derivative computation model at USNs is lower
γ
G
bounded by
(1 δ)log 1
T⋆ , 1 sup − |S2ǫ|− (8)
ǫ,δ(cid:0)Gγ O (cid:1)≥A:kAk≥γ,S2ǫ∈S I x⋆M;−Ax⋆M +WˆA
h i
where is a 2ǫ-distinguishable subset of , A=[a ] is an N-by-N symmetric, negative definite matrix, x⋆ is a random
S2ǫ S ij N M
vector taking value in with uniform distribution, the random vector Wˆ = Wˆi is defined as
S2ǫ A 1 i=1
h i
Wˆi = a Wj +Vi, 1 i N
1  ij 1,usnπ(i) 1 ≤ ≤
j∈NXusnπ(i)
 
and I x⋆ ; Ax⋆ +Wˆ is the mutual information between x⋆ and Ax⋆ +Wˆ .
M − M A M − M A
Phroof: See Subsectioin 4.
Theorem 8 derives a lower bound on the complexity of solving the game class which depends on the constraint set ,
γ
G S
the constant γ and the noise distribution in the uplink and downlink channels. The optimization in (8) is over the set of all
symmetric, negative definite matrices with norm greater than or equal to γ, and the set of all 2ǫ-distinguishable subsets of
. The matrix A in (8) stems from the construction of quadratic utility functions in the proof of Theorem 4, the set and
2ǫ
tShe matrix A jointly represent a finite subset of the function class , and Wˆi represents the combined impact of uplSink and
Fγ 1
downlink channels at player i’s receiver under the constructed quadratic utility functions (see the proof of this theorem for
more details).
Theorem 4 can be used to numerically obtain a lower bound on the complexity of solving the game class up to ǫ
γ
G
accuracy when the uplink/downlink channels are not Gaussian distributed. Note that according to (8), T⋆ , 1 can be
ǫ,δ Gγ O
lower bounded as
(cid:0) (cid:1)
(1 δ)log 1
T⋆ , 1 − |S2ǫ|− (9)
ǫ,δ Gγ O ≥ I x⋆ ; Ax⋆ +Wˆ
(cid:0) (cid:1) M − M A
h i
where A is a symmetric, negative definite matrix with A γ and is a 2ǫ-distinguishable subset of . Thus, by
2ǫ
k k ≥ S S
numerically evaluating the mutual information term in (9), one can obtain a lower bound on T⋆ , 1 .
ǫ,δ Gγ O
The lower bound in Theorem 4 has the following information-theoreticinterpretation. Consider an auxiliary multiple-input-
single-output(MISO) broadcastchannelwith x⋆ as input and the Ax⋆ +Wˆ as output.Here,(cid:0)the chan(cid:1)nelinput, i.e., x⋆ ,
M − M A M
takes value from the finite set of input alphabets with uniform distribution. The symmetric, positive definite matrix A
2ǫ
acts on the input, and the received signal by playSer i is given by A x⋆ +Wˆi where A is the ith row of A. Note −that
− i M 1 i
log can be intuitively interpreted as the transmitter’s bit-rate and I x⋆ ; Ax⋆ +Wˆ can be intuitively deemed as the
|S2ǫ| M − M A
amount of common information between the transmitted signal and thehset of received signials by players. Therefore,
I x⋆ ; Ax⋆ +Wˆ
M − M A
R(A, )=
S2ǫ h(1 δ)log 1i
2ǫ
− |S |−
can be viewed as the relative common information between the transmitted signal and the set of received signals by players
for a particular choice of the set and the matrix A. Note that I x⋆ ; Ax⋆ +Wˆ H[x⋆ ] = log as x⋆ is
S2ǫ M − M A ≤ M |S2ǫ| M
uniformly distributed over 2ǫ. Thus, according to (8), the complexityhof solving the gamie class γ is limited by the choice
S G
of and A such that the transmitted signal and the set of received signals by players have the smallest amount of relative
2ǫ
S
common information.
IV. DERIVATIONS OF RESULTS
A. Proof of Theorem 1
The proof of Theorem 1 is based on the following four steps:
1) Firstly, we construct a finite subset of , denoted by (see subsection IV-A1 for more details).
′
F F
2) Secondly, for the function class , the Nash seeking problem is reduced to a hypothesis test problem (see subsection
′
F
IV-A2 for more details).
3) Thirdly, the generalized Fano inequality is used to obtain a lower bound on the error probability of the hypothesis test
problem (see subsection IV-A2 for more details).
4) Finally, information-theoretic inequalities are used to obtain an upper bound on the mutual information term which
appears in the generalized Fano inequality.