Network: A Lower Bound on the Price of Anarchy. Ninoslav Marina and Are Hjørungnes. UNIK - University Graduate Center. University of Oslo. Instituttveien 25 ...
Power Allocation Game in a Four Node Relay Network: A Lower Bound on the Price of Anarchy Ninoslav Marina and Are Hjørungnes UNIK - University Graduate Center University of Oslo Instituttveien 25, P. O. Box 70 2027 Kjeller, Norway Abstract—In this paper, we propose an idea on how game and In [15], we have shown that if users do not coperate the information theoretic results can be combined to analyze the Nash equilibrium is achieved by a selfish transmission without performance of wireless cooperative networks. More precisely, helping the other user. A possible measure of how good is the we consider a four node wireless network, where the transmit nodes help each other acting as relays during the periods in performance of the system when users do not cooperate could which they do not transmit their own information. In order to be equilibrium efficiency, i.e., the ratio of users’ sum rate when help the other node, each node has to use a part of its available they are in equilibrium and the best possible cooperative sum power to relay the signal of the other transmitter. The network rate. We derived an upper bound on the worst case equilibrium is modeled as a non-cooperative game in which each player efficiency, in game theory, known as price of anarchy. Here, (node) maximizes its own utility function (information rate). The goal of the game designer (network provider) is to maximize we extend this result by deriving a lower bound and comparing the objective function (in this case the sum rate) in order to it to the upper bound derived in [15]. get better network efficiency. Here, we analyze the so called The remainder of this paper is organized as follows: Secequilibrium efficiency, as the ratio between the objective function tion II describes the system model. The main results about at the worst Nash equilibrium and the optimal objective function. the lower bound are presented in Section III, while the Using game theoretical language, it is the price of anarchy of the proposed game. In this scenario, the Nash equilibrium is achieved comparisons of the bounds are done in Section IV. Section by selfish (non-cooperative) behavior between the players. In V concludes the paper. other words, in order to maximize its own utility function each II. S YSTEM M ODEL node chooses a strategy to use its available power only for itself, We use the same network model as in [15] shown in and not helping the other node. Earlier, we derived an upper bound for the worst case equilibrium efficiency and in this paper Figure 1. In order to make the paper easy to read, we we present a lower bound. From the comparisons, we conclude repeat the preliminary steps from [15]. Nodes 1 and 2, that for path loss coefficients that are of practical importance the proposed bounds are tight. Our results show that the worst case equilibrium efficiency for the proposed simple network is very small (below 10%). Hence, there is a large possibility for improvements if the network nodes are encouraged to cooperate by designing certain mechanisms.
I. I NTRODUCTION In present day wireless communication networks there is an increased need for higher communication rates. Wireless ad-hoc networks consist of various mobile terminals, communicating with each other on a peer-to-peer basis, without the assistance of a central entity. It has been proved [1]–[6] that, in such systems, it is possible to improve the performance by relaying the transmitted data through one or several terminals or nodes. A limitation to wireless ad hoc networks is that for large networks the capacity per node goes to zero as inverse of the square root of the total number of nodes [7]. A promising method to increase the network spectral efficiency is the node cooperation [8]–[14] by relaying. The idea is that some nodes, while not active can offer their resources to other nodes, most often in their vicinity, and, therefore, helping them to achieve better performance, i.e., better transmission rate. In many situations, it is not straightforward to “motivate” the nodes to cooperate. In this paper, we compare the overall system performance when users cooperate versus when they do not. The goal of the system designer is to “encourage” nodes to cooperate in order to increase the total system efficiency.
3 4
1 Fig. 1.
2
A network with two source nodes and two destination nodes.
called Sources, try to communicate with Nodes 3 and 4, called destinations, respectively; see [15]. We assume that the sources use a time-sharing protocol to communicate with their destinations. Although the use of the time sharing protocol is suboptimal, it allows to get simpler expressions in the derivation of the proosed bounds. Using a model in which both nodes can transmit simultaneously would be more appropriate, however, in that case, deriving closed form expressions could be extremely challenging. In a decentralized non-cooperative network, the time sharing method is not the best model, however, we thought it might be a good starting point. At the end, the proposed bounds are tight for the path-loss exponents of practical interst. Certain intervals of discrete time periods are devoted to one of the sources, say Source i ∈ {1, 2}, in which Source i tries to communicate with its destination while
the other source acts as a relay. We will denote the portion of the time devoted to Source 1 by τ 1 ∈ (0, 1), which implies, the portion of the time devoted to Source 2 is τ 2 1 − τ1 . The average power per channel use is limited to P i > 0 for the Source i. Each Source i uses some portion k i ∈ [0, 1] of its limited power to communicate with its own destination and uses the remaining portion k¯i 1 − ki to act as a relay to help the other source communicate with its destination. Therefore, the average power used by the Source i for its own transmission equals k i Pi /τi , whereas the average power used by the Source i to help the other source equals k¯i Pi /¯ τi where τ¯i = 1 − τi . In each period, we have a single relay channel whose various properties are discussed in [3], [5], [16]. The relay channel that arises when Source 2 helps Source 1, called the first relay channel, is described by Y3 [n] =
h31 X1 [n] + h32 X2 [n] + Z3 [n],
Y2 [n] =
h21 X1 [n] + Z2 [n],
{Ui }i∈{1,2} are properly aligned with U (k 1 , k2 ). In fact, the one of the goals of the network designer (which is beyond the scope of this paper) is to carefully design the individual utility functions in order to optimize some network-level performance measure such as (1). A profile of solutions k ∗ = (k1∗ , k2∗ ) constitutes a Nash equilibrium or simply equilibrium if ∗ ∗ ) = max Ui (ki , k−i ), ∀i ∈ {1, 2}. Ui (ki∗ , k−i ki ∈[0,1]
In other words, no source has an incentive to unilaterally deviate from k i∗ , i.e., it is person-by-person optimal. Given the individual utilities {Ui }i∈{1,2} , an equilibrium may or may not exist, and when it exists, it may or may not be unique. In our context, we expect that the sources will find their way towards an equilibrium using a learning algorithm [18]. An equilibrium will typically be inefficient with respect to the network designers objective U , or in other words, it will typically not optimize U . A useful measure of efficiency of an equilibrium k ∗ with respect to U can be given as
whereas the relay channel that arises when Source 1 helps Source 2, called the second relay channel, is described by Y4 [n] = Y1 [n] =
h42 X2 [n] + h41 X1 [n] + Z4 [n], h21 X2 [n] + Z1 [n],
where Xi [n] is the symbol transmitted by the Source i at time n, Yi [n] is the symbol received by Node i at time n, Z i [n] are additive white Gaussian noise with unit variance, and h ij are the channel coefficients between the receiving Node i and transmitting Node j. We will assume that the magnitude of the channel coefficient satisfies |h ij |2 = d−β ij , where dij is the distance between the Nodes i and j, and β is the path loss coefficient [17]. We will denote the capacity of the i−th τ−i , relay channel per channel use by C i ki Pi /τi , k¯−i P−i /¯ where we use the notation −i to refer to the source helping the Source i. As in [15], we adopt a non-cooperative game theoretic framework in which each source is viewed as an autonomous decision maker by deciding the allocation k i in order to optimize its own utility function U i (ki , k−i ). Note that, although the game is called non-cooperative, we assume that the sources have to agree on the time division strategy. Hence, in the sequel, by non-cooperation we mean individual decision on the allocation ki for each source. Each Source i is non-cooperative in the sense that it is interested in selfishly optimizing its own utility function U i . However, a network designer would typically be interested in the optimization of a networklevel performance metric U (k 1 , k2 ), for instance, the average network rate per channel use U (k1 , k2 ) = (1) τi Ci ki Pi /τi , k¯−i P−i /¯ τ−i . i∈{1,2}
Although chaotic behavior can emerge out of non-cooperative interactions, it is also possible that a network-level performance metric U (k 1 , k2 ) can be optimized through noncooperative players provided the individual utility functions
η(k∗ )
U (k∗ ) max(k1 ,k2 )∈[0,1]2 U (k1 , k2 )
.
Now, an important consideration in designing the individual utilities {Ui }i∈{1,2} is to make a resulting equilibrium k ∗ efficient, i.e., η(k∗ ) = 1. Ideally, we would like all the equilibria to be fully efficient; however, this may not always be an easy task unless one obtains an optimal solution and retrofits the individual utilities into the optimal solution. Hence, we investigate the efficiency of the unique equilibrium ˜k = (k˜1 , k˜2 ) = (1, 1) of the game characterized by the utilities τ−i , Ui (ki , k−i ) = Ci ki Pi /τi , k¯−i P−i /¯ with respect to U given by (1). Note that, in this case, we have U = τ1 U1 + τ2 U2 , and, therefore, the average network rate would be optimized when each source optimizes its own average rate. In [15], it has been shown that the equilibria ˜k = (1, 1) could be quite inefficient. Here we try to derive a lower bound on the worst-case efficiency of the equilibrium ˜k = (1, 1) defined as [15] ηmin inf G
˜ U (k) max(k1 ,k2 )∈[0,1]2 U (k1 , k2 )
,
(2)
where G is the network geometry defined by G = {(d ij ) : d21 > 0, d31 > 0, d42 > 0, |d21 − d31 | ≤ d32 ≤ d21 + d31 , |d21 − d42 | ≤ d41 ≤ d21 + d42 }, and U is given by (1). For continuity, we repeat the main result from [15] about the upper bound of the worst case equilibrium efficiency. Theorem 1 [15] For any fixed P 2 ≥ P1 > 0, τ1 ∈ (0, 1), and β > 0, the worst-case equilibrium efficiency η min defined in (2) is upper bounded as ηmin ≤ (ρ∗ )β < 2−β , ∗
where ρ ∈ (0, 1) is the unique solution of P1 ρ−β = P1 + P2 (1 − ρ)−β .
(3)
III. M AIN R ESULT In the following text all logarithms are base e. Before presenting the main result, we introduce two lemmas that will be necessary to derive our proposed lower bound. Lemma 2 For Bi > 1 and ai > 0, for all i = 1, 2, . . . , N N a log(1 + xi ) = max1 B . (4) inf Ni=1 i j xi ≥0 a log(1 + B x ) i i i i=1 j Proof: Note that in (2), the function within the inf is not defined at (x1 , x2 , . . . , xN ) = (0, 0, . . . , 0). Then, for any permutation π = [π1 , π2 , . . . , πN ] of the set {1, . . . , N }, it is true that N ai log(1 + xi ) 1 . (5) lim lim . . . lim Ni=1 = xπ1 →0 xπ2 →0 xπN →0 Bπ1 i=1 ai log(1 + Bi xi ) From the generalized Bernoulli inequality [19, Ch. III], for every integer real number r ≥ 1, and x > −1 (1 + x)r ≥ 1 + rx.
(6)
The strict inequality is satisfied for x > 0 and r > 1 [19, Ch. III, Eq. (1.2)]. Here, for x i > 0 > −1 and Bi > 1 for all i = 1, . . . , N , taking logarithm on both sides of (6) we get log(1 + Bi xi ) < Bi log(1 + xi ).
Proof: Note that the function c(ρ) is continuous in ρ, i.e., ρ) = c2 (˜ ρ), where c1 (˜ −1 1/β ρ˜ = 1 + (p2 /p1 ) . (7) The function c 1 (ρ) is monotonically decreasing for 0 < ρ < 1. The function c 2 (ρ) is continuous and differentiable on 0 < ρ ≤ ρ˜, with a maximum at ρ ∗ , with ρ∗ < ρ˜. The value of ρ ∗ could not be analytically determined since we have to solve the following transcendental equation ∂c 2 (ρ)/∂ρ = 0. In order to upper bound c(ρ), note that p1 p2 p2 +2 c2 (ρ) < log 1 + 1+ d p1 (1 − ρ)β p1 (1 − ρ)β 2 p1 p2 = log 1 + , (8) 1+ d p1 (1 − ρ)β p2 where the inequality follows from the fact that p1 (1−ρ) β > 0 and ρ−β − 1 < ρ−β + 1. Then, we have 2 p p 1 2 c(ρ) ≤ c2 (ρ∗ ) < log 1 + 1+ d p1 (1 − ρ∗ )β 2 p1 p2 < log 1 + , (9) 1+ d p1 |1 − ρ˜|β
where the last inequality is true since (8) is increasing in ρ Hence, we have 2 and ρ∗ < ρ˜. Replacing (7) in (9) concludes the proof. N N Since the expression for the capacity is not known, we use ai log(1 + xi ) i=1 ai log(1 + xi ) > Ni=1 the following expression for its upper bound [4, Eqns (10) and N i=1 ai log(1 + Bi xi ) i=1 ai Bi log(1 + xi ) (12)] N a log(1 + x ) 1 i i i=1 ≥ , = dβ31 P1 N 1 UB max Bj , (10a) C (P1 , P2 ) = log 1 + 1 + β maxj Bj i=1 ai log(1 + xi ) j 2 d21 dβ31 where in the weak inequality, equality holds if B 1 = B2 = when P1 |d31 − d21 |β ≤ P2 dβ21 , and ⎛ · · · = BN . From (5), we see that the latest coincide with the β d31 P limit for any permutation in which π 1 = arg max Bj . This 2 β − 1 1 P1 d21 ⎜ j β C UB (P1 , P2 ) = log ⎝1 + β + concludes the proof. 2 d 2 d31 |d31 − d21 |β 31 β + 1 d21 Lemma 3 Let p1 , p2 , d > 0 and ⎞ β β β P d31 /d21 d P1 P2 p1 1 + d31 − |d31 −d 2 |d231 −d β β β 21 | 21 | ⎟ dβ c1 (ρ) log 1 + 1 + ρ−β , 21 31 ⎟ (10b) d + β β ⎠ 1 + d31 /d21 p1 p2 (ρ−β − 1) c2 (ρ) log 1 + 1+ d p1 (1 − ρ)β (1 + ρ−β ) when P1 |d31 − d21 |β ≥ P2 dβ21 . ⎞⎞ p2 ρ−β p2 −β 2 p1 (1−ρ)β 1 + ρ − p1 (1−ρ)β ⎟⎟ These bounds are direct consequence of the results in [3]. ⎟⎟ , + ⎠⎠ 1 + ρ−β Letting ρ = d21 /d31 , for d21 < d31 the expression in (10a) is c1 (ρ)/2, and the expression in (10b) is c 2 (ρ)/2 for P1 =
p1 , P2 = p2 and dβ31 = d. Hence, using Lemma 3 we can β c1 (ρ), if p1 (1/ρ − 1) ≤ p2 , c(ρ) upper bound the capacity of the relay channel by c2 (ρ), if p1 (1/ρ − 1)β ≥ p2 . C ≤ C UB (P1 , P2 ) Then for any 0 < ρ < 1 ⎛ ⎛ ⎞2 ⎞ ⎛ β1 β2 ⎛ ⎞2 ⎞ β/2 1/β P2 1 P1 ⎜ ⎠ ⎟ p2 p1 ⎝ < log ⎝1 + β ⎝1 + 1 + ⎜ ⎠ . (11) ⎠ ⎟ c(ρ) < log ⎝1 + 1+ 1+ 2 P ⎠. 1 d31 d p 1
Theorem 4 For any fixed P 1 , P2 , τ ∈ (0, 1) and β ≥ 1, the worst case equilibrium efficiency η min is lower bounded by ⎞ ⎛ 1/β β/2 −2 P2 P1 ⎠ . , (12) ηmin > ⎝1 + 1 + max P1 P2 Proof: From (2) consider the following sequence of inequalities and equalities ηmin = inf G
U (1, 1) max U (k1 , k2 ) k1 ,k2
(a)
≥ inf G
U (1, 1) ¯ ¯ k k P max C UB 1τ 1 , 2τP2 + C UB k2τ¯P2 , k1τ¯P1 k1 ,k2
(b)
= inf G1
max C UB
k1 ,k2 (c)
≥ inf G1
max k1 ,k2
C UB
U (1, 1) ¯ + C UB k2τ¯P2 , k1τ¯P1
¯2 P2 k1 P1 k τ , τ
U (1, 1) ¯ + max C UB k2τ¯P2 , k1τ¯P1
¯2 P2 k1 P1 k τ , τ
k1 ,k2
U (1, 1) = inf UB G1 C (P1 /τ, P2 /τ ) + C UB (P2 /¯ τ , P1 /¯ τ) U (1, 1) (e) = inf UB G2 C (P1 /τ, P2 /τ ) + C UB (P2 /¯ τ , P1 /¯ τ) P1 τ τ¯ + 2 log 1 + (f ) 2 log 1 + τ dβ 31 > inf P1 τ¯ d31 ,d42 >0 τ + log 1 + B log 1+ 1 β 2 2 τd (d)
31
P2 τ¯dβ 42
P2 B2 τ¯dβ 42
(g)
= min{1/B1 , 1/B2 } β/2 −2 (h) 1/β = 1 + 1 + max {P2 /P1 , P1 /P2 } , where (a) follows from the upper bound on the capacity given by (10a) and (10b), and G = {d 21 > 0, d31 > 0, d42 > 0, |d31 − d21 | ≤ d32 ≤ d31 + d21 , |d42 − d21 | ≤ d41 ≤ d42 + d21 }, (b) since the denominator is maximized if the nodes are in line [15], and, therefore, we maximize over G1 = {d21 > 0, d31 > 0, d42 > 0, d32 = |d31 − d21 |, d41 = |d42 − d21 |}. The inequality in (c) comes from the fact that maxA (C1 (A)+C2 (A)) ≤ maxA C1 (A)+maxA C2 (A), while (d) since C UB (k1 P1 /τ, k¯2 P2 /τ ) is maximized for (k 1 , k2 ) = τ , k¯1 P1 /¯ τ ) for (k1 , k2 ) = (0, 1). Equal(1, 0) and C UB (k2 P2 /¯ ity (e) is true since the denominator is maximized if the relay nodes are in between the source and the destination nodes, i.e., d21 < min{d31 , d42 }, or in other words, ρ < 1. That means we maximize over G 2 = {min{d31 , d42 } > d21 > 0}. The strict inequality (f ) follows from (11) since ρ < 1 with β/2 2 1/β B1 = 1 + 1 + (P2 /P1 ) , β/2 2 B2 = 1 + 1 + (P1 /P2 )1/β . Using the result of Lemma 2 for N = 2, and replacing a 1 = −β τ /2, a2 = (1 − τ )/2, x1 = P1 d−β τ , since 31 /τ , x2 = P2 d42 /¯
B1 > 1 and B2 > 1 we get (g). Finally, (h) is satisfied since ⎞−2 ⎛ β1 β2 1 1 P P 2 1 ⎠ min , , = ⎝1 + 1 + max B1 B2 P1 P2 and this concludes the proof. 2 In the following text, without loss of generality, we assume P2 ≥ P1 . IV. C OMPARISON OF THE B OUNDS In this section, we compare the lower bound of η min given by Theorem 4 to its upper bounds of Theorem 1, from [15]. So far we have been able to prove that ⎛ ⎞ 1/β β/2 −2 P 2 ⎝1 + 1 + ⎠ < ηmin ≤ (ρ∗ )β < 2−β , P1 where ρ∗ ∈ (0, 1) is the unique solution to P 1 ρ−β = P1 + P2 (1 − ρ)−β . In Fig. 2, we observe the tightness of the bounds for different ratios P2 /P1 . Note that as P2 /P1 increases, the solution ρ∗ to P1 ρ−β = P1 + P2 (1 − ρ)−β on (0, 1) decreases, and consequently the upper bound (ρ ∗ )β on the worst-case equilibrium efficiency η min becomes significantly tighter than 2−β . The loose upper bound 2 −β is proposed since it is easy to remember as a function of β. As P 2 /P1 increases the proposed lower bound is getting closer to the stronger upper bound (ρ∗ )β , therefore, offering a good estimate . The results for some path loss coefficents of practical interest for the worst case when P 2 = P1 are summarized in Table IV. There we show the lower bound (LB) given by the RHS of (12) and both upper bounds (UB) given by (3). TABLE I B OUNDS ON ηmin IN % FOR P2 = P1 FOR VARIOUS β. β 2.0 2.5 3.0 3.5 4.0 4.5 5.0
LB 11.11 8.76 6.82 5.25 4.00 3.02 2.26
UB (ρ∗ )β 22.00 16.16 11.74 8.45 6.06 4.32 3.08
UB (2−β ) 25.00 17.68 12.50 8.84 6.25 4.42 3.13
In [15], we also noted that the upper bound can be extended to general relay networks with n sources and n destinations. In that case, the upper bound becomes looser. The result for the lower bound cannot be straightforwardly extended for larger networks. V. C ONCLUSION In this paper, we derived an analytical lower bound for a simple four node cooperative network when it is modeled as a non-cooperative game. We confirmed the results obtained in [15] that the equilibrium emerging from such a noncooperative setting may lead to a significantly inferior sum rate compared to the optimal sum rate. Our main message is that, in a general relay network in which it is not feasible to require the sources to jointly maximize the sum rate, the sources should not (be allowed to) maximize their own individual rates, but
P =P 2
on the upper bound is more fundamental since it states that using this model, by a non-cooperative behavior, we cannot gain more than (ρ ∗ )β of the optimal solution. Moreover, the upper bound is also true for a network of n communicationg pairs [15]. The lower bounds offers a better estimate of the price of anarchy in a simple network of four nodes. ACKNOWLEDGMENT Authors would like to thank for the support from the Research Council of Norway through the project 176773/S10 entitled “Optimized Heterogeneous Multiuser MIMO Networks OptiMO”, as well as from the Swiss National Science Fondation under the Grant PA 002–117385. R EFERENCES
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Fig. 2. Upper and lower bounds for different ratios P2 /P1 : the loose upper bound 2−β (plain line), the tight upper bound (ρ∗ )β (dashed line), and the proposed lower bound (LB) (dash-dot line).
rather they should optimize some carefully designed “utility functions” leading to a reasonably high sum rate. Designing such utility functions is a challenging future research topic since it will encourage network nodes to cooperate and, hence, improve the performance in terms of the total sum rate. Our results demonstrate that there is a big room for improvement since for path loss exponents of practical interest (3 ≤ β ≤ 6) the price of anarchy is bounded by less than 10 %. The result
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