Stable Nash equilibria of medium access games under symmetric, socially altruistic behavior
aa r X i v : . [ c s . G T ] A ug Stable Nash equilibria of medium access gamesunder symmetric, socially altruistic behavior
G. Kesidis Y. Jin A.P. Azad and E. AltmanCS&E and EE Depts Sogang University INRIA Sophia AntipolisPenn State University Seoul, Korea University of Avignon, [email protected] [email protected] { amar.azad,eitan.altman } @sophia.inria.fr Abstract
We consider the effects of altruistic behavior on random medium access control (slotted ALOHA)for local area communication networks. For an idealized, synchronously iterative, two-player gamewith asymmetric player demands, we find a Lyapunov function governing the “better-response” Jacobidynamics under purely altruistic behavior. Though the positions of the interior Nash equilibrium pointsdo not change in the presence of altruistic behavior, the nature of their local asymptotic stability does.There is a region of partially altruistic behavior for which both interior Nash equilibrium points arelocally asymptotically stable. Variations of these altruistic game frameworks are discussed consideringpower (instead of throughput) based costs and linear utility functions. Also, for a power control gamewith a single Nash equilibrium, we show how its stability changes as a function of the altruism parameter.
I. I
NTRODUCTION
Game theoretic models for telecommunication systems have recently been surveyed in [4].They are motivated by the need to model the very significant effects of end-user behavior. Inthe Internet, TCP congestion control, a protocol presuming cooperative end-user behavior, hasbeen exploited by end-users and their client applications acting in selfish ways. As common insome wireless settings, e.g. , tactical mobile ad hoc networks (MANETs), network nodes may
The work was supported by NSF CISE grants 0524202 and 0915928 and by a Cisco Systems URP gift.
October 3, 2018 DRAFT engage in cooperative/coalitional or simply altruistic behavior with respect to some of theirpeers. Altruistic action can be for the purpose of routing [7], medium access, etc . In [10], thebehavior of permanent seeder peers in on-line BitTorrent swarms (characterized as altruistic) isasymmetric; clearly if one peer is completely selfish and the other completely altruistic, thenthe greedy peer stands to benefit. We consider a symmetric situation where the players havesimilar communication priorities and degrees of altruism. This said, the demands of the playersare, however, generally assumed asymmetric in the following.In distributed systems, altruistic actions can easily be shown to be counter-productive in thepresence of • limited information and/or observation errors, • excessive communication overhead to (securely) convey more accurate network state infor-mation, and/or • deliberate injection of false observation information by enemy actors.In these cases, selfish behavior alone will give better performance. In coalitional games, thecost of cooperation is typically weighed against its benefits, e.g. , [5]. Extensive prior work hasconsidered the effects of imperfect learning of the utilities of other players and the parametersof the environment in which they all interact, e.g. , [11], [26], [25].In this paper, we assume perfectly correct and complete information about the player who isthe target of altruistic behavior, and this at no cost. Despite this assumption, we show that it ispossible that negative performance effects are possible under altruistic behavior. In particular,we will consider iterative games where utilities are a function of average throughput. The gamesare “quasi stationary” in that the (mixed or pure) strategies of players were based on observedplayers’ actions over time [23].That Nash equilibria of iterative games are not necessarily asymptotically stable is well known, e.g. , [22], [1], [27]. In slotted-ALOHA medium access under congestion, each user (player) i attempts to transmit with probability p i ( i.e. , his/her control variable or strategy), so that theprobability of successful transmission is γ i = p i Q j = i (1 − p j ) . In [12], [13], using a Lyapunovfunction for arbitrary N ≥ players, noncooperative two-player ALOHA was shown to havetwo different interior Nash equilibria though only one was locally asymptotically stable, see i.e. , not including the stable boundary deadlock equilibrium. October 3, 2018 DRAFT also [15]. These local stability properties will change in the presence of altruistic behaviorstudied herein. In some partially altruistic cases, both interior Nash equilibria will be locallyasymptotically stable. Local asymptotic stability properties are generally important for robustperformance in the presence of modeling, estimation and quantization errors, e.g. , [6]. Wesimilarly explore simple distributed power control medium access dynamics having a singlefeasible equilibrium.We focus on two-player games because: (a) the communication overhead required to accuratelyinform altruistic behavior will not scale to many users, (b) the players are assumed to havedifferent utilities so one of them could be a model for an aggregation of users, and (c) thepurely altruistic dynamics have a vector field (and Lyapunov function) expressible in closedform.We also assume synchronous player action; the effects of asynchronous player action can beconsidered using, e.g. , the methods of [9], [8]. Even in these more general scenarios, poten-tially unstable Nash equilibria for synchronous two-person sub-game would remain a significantchallenge for altruistic behavior.This paper is organized as follows. In Section II, we give an overview of our game-theoreticframework. In Section III, we give the main results on the stability properties of the Nashequilibria, as a function of the “degree of altruism”, of a two-player, synchronously iterativeALOHA medium access game. In Section IV, we consider variations of our game framework inwhich the utility is linear (instead of strictly concave) and costs are based on energy expenditure(rather than throughput). In Section V, we study medium access by power control involving asingle feasible Nash equilibrium point. Finally, we conclude with a brief summary.II. A
LTRUISTIC GAMES
By using his/her control action (strategy) q i , the i th player seeks to maximize the compositeof the net utilities V j ( γ j ) := U j ( γ j ) − γ j M , i.e. , X j α ij ( U j ( γ j ) − γ j M ) , (1)where i and j index the players, X k α ik = 1 and α ij ≥ ∀ i, j, (2) October 3, 2018 DRAFT γM is the usage-based charge for service γ , and all utilities U i are strictly concave, i.e. , U ′′ i ( γ i ) < ∀ γ i ∈ R + . (3)The player actions q are related to the service γ through the network’s dynamics. For the exampleof slotted ALOHA used below, γ i = q i Y j = i (1 − q j ) , where the (re)transmission probabilities ≤ q i ≤ for all i . So, the users need to be aware ofeach others’ actions and utilities . For all collective-action vectors q , we will typically assume ∂γ i ∂q i ( q ) > ∀ q, i, and ∂γ i ∂q j ( q ) < ∀ q, i = j. (4) A. Purely selfish/non-cooperative games
Regarding player i ’s net utility, ∂V i ∂q i ( q ) = ( U ′ i ( γ i ( q )) − M ) ∂γ i ∂q i ( q ) , where we have now explicitly written the γ i as functions of q . So, at a Nash equilibrium point(NEP) ˆ q of a non -altruistic game, i.e. , where α ii = 1 for all i , γ i (ˆ q ) = ( U ′ i ) − ( M ) =: y i ∀ i. (5)Note that by the concavity assumption of the U i , at a NEP ˆ q is ∂ V i ∂q i (ˆ q ) = U ′′ i ( γ i (ˆ q )) (cid:18) ∂γ i ∂q i (ˆ q ) (cid:19) < . That is, (5) and (3) are the conditions for q to be an NEP. In the purely selfish/non-cooperative games ( α ii = 1 ∀ i ) of [14], actions were taken based on observations in quasi steady-state thus not requiring any coordination between presumed selfish/non-cooperative peers. October 3, 2018 DRAFT
B. Altruistic invariance of NEPsClaim 1:
Under (3), the NEPs of the purely selfish game ( α ii = 1 for all i ) are also NEPsunder the altruistic objective (1) for all [ α ij ] satisfying (2). Proof:
Note that for objectives (1), the first order condition for player i is X j α ij ( U ′ j ( γ j ( q )) − M ) ∂γ j ∂q i ( q ) = 0 . So, clearly this condition is satisfied when q = ˆ q , i.e. , under (5). Moreover, the second ordercondition at such a ˆ q (again, under (5)) reduces to X j α ij U ′′ j ( γ j (ˆ q )) (cid:18) ∂γ j ∂q j (ˆ q ) (cid:19) < . When the γ j are all linear functions of q , a qualified converse of the previous claim, i.e. , thatthere are no additional interior NEPs under altruistic behavior, is given in [18]. C. Asymptotic stability of NEPs under symmetric altruism
Again, it is possible that NEPs are not asymptotically stable. Though the positions of theNEPs may not change under altruistic behavior, the nature of their stability may change withthe ( α ij ) parameters (as in the bifurcation theory of dynamical systems). Exploring this issuefor specific examples is the main goal of the balance of this paper.In the following, we restrict our attention to “symmetric” altruism where for some fixed α ∈ [0 , , α ii = α and α ij = (1 − α ) / ( N − for all N ≥ players i , and all j = i . Symmetricaltruism is consistent with players whose communication is of equal priority, though they mayvalue their own communication differently through different utility functions.III. T WO - PLAYER
ALOHA
GAME WITH SYMMETRIC ALTRUISM
In the following, we simplify to the symmetric case by assuming the α ii are the same ( α ) forall players i . For the selfish, non-cooperative ALOHA game ( i.e. , α = 1 ), the (re)transmissionprobabilities q i obey the following continuous-time Jacobi approximation ( e.g. , (10) of [13]): ˙ q ( t ) = F ( q ( t )) − q ( t ) , (6) October 3, 2018 DRAFT where for players i ∈ { , } : F i ( q ) = y i − q − i . The deadlock equilibrium q i = 1 ∀ i is stable in the purely noncooperative case, and opt-outequilibrium q i = 0 ∀ i is stable in the purely altruistic case; at either equilibrium point, allplayers have zero throughput. To address this problem, we can restrict q i ∈ [ q min , q max ] for all i ,so that F i ( q ) = min { q max , max { q min , y i − q − i }} , where large q max < is set to prevent a deadlock (or “capture” by just one player) and small q min > is set to prevent opt-out. Our focus in this section will be on the stability of interiorNash equilibria for concave utilities.In a distributed system, the Jacobi approximation is further justified when players take smallsteps toward their currently optimal play ( i.e. , only “better response” based on their currentknowledge of the state of the game including the actions of the other players in particular, i.e. , fictitious play [23]). One reason for this is that players act simultaneously so the optimalplays may change significantly. Thus, small steps may avoid large oscillations in the networkdynamics, i.e. , oscillations that will be harmful to performance ( e.g. , the balance between tentativeand aggressive action taken by TCP in its distributed congestion control strategy). Also, smallsteps may ensure convergence to stable interior equilibria, even under asynchronous updates,and avoid deadlock or opt-out boundary equilibria.A Lyapunov function governing the Jacobi iteration (6) is a function Λ such that for all times t , the inner product h∇ Λ( q ( t )) , ˙ q ( t ) i ≤ . For the non-cooperative N -player game [12], [13], Λ( q ) = − N Y i =1 y i − q i + N X i =1 (cid:18) q i − q i + log(1 − q i ) (cid:19) Y j = i y j . In the following, we will assume N = 2 players. A. Purely altruistic game ( α = 0 ) The throughput of player − i is (1 − q i ) q − i , so that the purely altruistic choice for player i is q i = 1 − y − i /q − i . Thus, player i ∈ { , } updates q i = 1 − y − i q − i =: G i ( q ) . (7) October 3, 2018 DRAFT
That is, replace F with G in (6).By Claim 1, the fixed points for this purely altruistic game ( α = 0 ) include those of the purelyselfish game ( α = 1 ), i.e. , both given by the solution of y i = q i (1 − q − i ) for i ∈ { , } . Claim 2:
The Lyapunov function of the purely altruistic game is Λ ∗ ( q ) = − Y i =1 (1 − y i q i ) + X i =1 y i log q i . (8) Proof: d Λ ∗ d t ( q ) = h∇ Λ ∗ ( q ) , ˙ q i = − X i =1 ( G i − q i ) y i q i ≤ . B. Stability of purely altruistic or purely cooperative behavior
Let H ∗ , respectively H , be the Hessian corresponding to Λ ∗ , respectively Λ , i.e. , H ∗ ij = ∂ Λ ∗ /∂q i ∂q j for i, j ∈ { , } . Claim 3: (a) The NEP q is locally stable under purely altruistic ( α = 0 ) behavior ⇔ σ ∗ := y y q q < . (9)(b) The NEP q is locally stable under purely selfish ( α = 1 ) behavior ⇔ σ := y y (1 − q ) (1 − q ) < . (10) Proof: H ∗ is positive definite ( i.e. , with two positive, real eigenvalues) at the NEP q ( i.e. , where G ( q ) = q ) if and only if (9). Similarly, H is positive definite at the NEP q ( i.e. , where F ( q ) = q ) if and only if (10).The example selfish two-player ALOHA game of [12], [13] with ( y , y ) = (8 / , / hadNEPs ( q , q ) = (2 / , / and (4 / , / :NEP q σ σ ∗ (2 / , /
5) 1 / / , /
3) 2 1 / So, for purely altruistic (cooperative α = 0 ) actions, (4/5,1/3) is the locally asymptoticallystable NEP but (2/3,1/5) is unstable. On the other hand, for purely selfish (non-cooperative α = 1 ) actions, (2/3,1/5) is the locally stable NEP but (4/5,1/3) is unstable. Clearly, σ < and σ ∗ < is not possible if q + q < , but note that q + q > for the NEP (4 / , / . October 3, 2018 DRAFT
C. Numerical example for partial altruism
Given the current play q − i of player − i , let Q i ( α, q − i ) be the play of the i th player thatmaximizes (1), i.e. , Q is used instead of F in (6). In particular, Q i ( α, q − i ) = F i if α = 1 G i if α = 0 with no such closed-form expression available for Q i when < α < .For utilities of the form U i ( γ ) = M (1 + y i ) arctan( γ ) , (11)we numerically evaluated the local asymptotic stability of the two NEPs. That is, we choseutilities normalized by the price M so that we can simply take M = 1 . Stability can beascertained by the Hartman-Grobman theorem (linearizing the dynamics at the equilibrium pointand checking the eigenvalues of the Jacobian), i.e. , even if a Lyapunov function is not available.Note from the following table that the stability of the NEP (4/5,1/3) changes at α ≈ . andsimilarly the stability of the NEP (2/3,1/5) changes at α ≈ . .NEP \ α ∈ [0 , .
42) [0 . , .
58] (0 . , / , / unstable stable stable (4 / , / stable stable unstableSo, in the range . ≤ α ≤ . both interior NEPs are locally asymptotically stable. Thecontours of the Lyapunov function Λ are given in Figure 1(a) and of Λ ∗ in Figure 1(c), whereone clearly sees the saddle contour in the latter. Given the large interval about α = 0 . whereboth NEP are stable, by continuity we expect that the this condition will hold for all sufficientlysmall asymmetries in altruism between the players, i.e. , α = α . D. Discussion: Alternative altruistic strategies
We now give a Lyapunov function as a function of α for an alternative altruistic play, i.e. , onethat does not necessarily maximize (1).Generally, Q i ( α, q − i ) will be a non-linear function of α . We can consider an alternative playthat is a linear combination: Q oi := αF i + (1 − α ) G i . October 3, 2018 DRAFT (a) Stable α = 1 (b) Stable α = 0 . (c) Unstable (saddle) α = 0 Fig. 1. The ALOHA NEP (2/3,1/5)
Moreover, instead of the purely altruistic G i , consider ˜ G i = (cid:18) − y − i q − i (cid:19) y i ( q − i − , corresponding to the α -linear action ˜ Q i = αF i + (1 − α ) ˜ G i . At a fixed point q for a purely altruistic game using ˜ G i , q i can be easily shown to be an increasingfunction of G i y i . Claim 4:
The Jacobi iteration (6) using ˜ Q instead of F has the following Lyapunov function: Λ α ( q ) = − α Y i =1 y i − q i − (1 − α ) Y i =1 h i ( q i )+ X i =1 (cid:18) q i − q i + log(1 − q i ) (cid:19) y − i October 3, 2018 DRAFT0 where h i ( q i ) := y i − y i /q i . Note: We have modified the Lyapunov function (7) of [12], [13] by adding the α factors andthe second term. Proof: d Λ α d t ( q ) = < ∇ Λ α ( q ) , ˙ q > = − X i =1 ( ˙ q i ) y − i (1 − q i ) ≤ . We can similarly modify the purely altruistic iteration and find a Lyapunov function modifiedfrom Λ ∗ . Note that for these dynamics, the positions of interior NEPs may change as a functionof α . IV. D ISCUSSION : V
ARIATIONS OF THE GAME FRAMEWORK
In this section we discuss other extensions and variations of the synchronous, two-player gameframework of the previous section.
A. Linear utilities If U i ( γ ) ≡ u i γ for a scalar u i > M , then the net utility of player i is simply γ i ( u i − M ) .So, in the purely selfish ( α = 1 ) case, player i will simply maximize γ i . That is, the selfishstrategy is simply q i = 1 if q j < ∀ j = i . Thus, any play q such that at least one player i usesthe “pure” strategy q i = 1 is a NEP. These include the deadlocked NEPs where two or moreplayers choose q = 1 so that γ = 0 for all players.In the purely altruistic, two-player case, each player i will simply maximize γ − i = q − i (1 − q i ) (of the other player) over q i , i.e. , choose q i = 0 if q − i > . So, any play q such that at leastone player uses the pure (opt-out) strategy q = 0 is an NEP.For partially altruistic behavior, (1) with < α < remains a linear form. For the two-playergame at equilibrium, q i = 1 if α (1 − q − i )( u i − M ) − (1 − α ) q − i ( u − i − M ) > ⇔ q − i < α ( u i − M ) α ( u i − M ) + (1 − α )( u − i − M ) =: φ − i ( α ) . So, there are two stable NEPs (0,1) and (1,0) and the unstable saddle NEP φ ( α ) between them.More precisely, the following table indicates convergence trend based on the starting point q . October 3, 2018 DRAFT1 staring q ∈ NEP q [0 , φ ( α )) × ( φ ( α ) , (0,1) ( φ ( α ) , × [0 , φ ( α )) (1,0) [0 , φ ( α )) × [0 , φ ( α )) φ ( α )( φ ( α ) , × ( φ ( α ) , φ ( α ) Note that both (0,1) and (1,0) are NEPs for α = 0 and α = 1 as well. B. Power based costs
Now instead of a cost of the form
M γ ( i.e. , what the network charges for actual throughput),consider a cost of the form M q ( i.e. , the average cost for power experienced by the user). Power-based costs are important for the context communications relying on limited energy supply inorder to account for energy expenditure for failed communication due to interference (as here)and/or noise.Under power-based costs, the NEPs may change position as a function of the degree ofaltruistic behavior, α , i.e. , Claim 1 does not necessarily hold, cf. , Section V-E.
1) Strictly concave utilities:
Here, the purely selfish update rule is F i ( q − i ) = 1 Q j = i − q i ( U ′ i ) − M Q j = i − q i ! . (12)For scalar parameters β i , u i > , consider utilities of the form U i ( γ ) = M u i β i arctan( β i γ ) ⇒ ( U ′ i ) − ( z ) = 1 β i r M u i z − . For two players, (12) becomes F i ( q − i ) = 1 β i (1 − q − i ) p u i (1 − q − i ) − ≈ y i √ − q − i , where here y i := √ u i /β i and the approximation holds when u i (1 − q − i ) ≫ . (13)Under (13), the two-player Lyapunov function for purely selfish behavior is Λ ( q ) = − Y i y i √ − q i + X i ( p − q i + 1 √ − q i ) Y j = i y i . October 3, 2018 DRAFT2
Under purely altruistic behavior, we see that ∂∂q i ( U − i ( γ − i ) − M q − i ) = − q − i U ′ − i ( γ − i ) ≤ . Thus, the only NEPs q are such that q i = 0 for at least one player i , i.e. , the opt-out action.Suppose each utility U i is strictly concave only for an interval [0 , ˆ γ i ] and U ′ i ( γ i ( q )) = 0 when γ i ( q ) > ˆ γ i , ∀ i ∈ { , } , (14) i.e. , the utility “saturates” after ˆ γ i < . Clearly here there will be additional NEPs under purelyaltruistic behavior: the set of plays q which jointly satisfy (14), assuming this set has non-emptyintersection with the feasibility region [0 , for ( q , q ).
2) Linear utilities:
Here again take U i ( γ ) = u i γ for a scalar u i . For the partially altruistictwo-player case, q i = 1 if α [(1 − q − i ) u i − M ] − (1 − α ) q − i u − i ) > ⇔ q − i < α ( u i − M ) αu i + (1 − α ) u − i ) =: ψ M3 − i ( α ) . The situation here is as in Section IV-A, except that for α = 1 there is an additional unstableinterior NEP, ψ M (1) .For α > and u i = u j =: u (identical players), we can derive a price ˆ M ( α ) for purely selfishbehavior that mirrors α -partially altruistic behavior: ψ M ( α ) = ψ ˆM( α ) (1) ⇒ ˆ M ( α ) = u − α ( u − M ) . V. D
ISTRIBUTED POWER CONTROL GAMES
Game-theoretic models for power control have been extensively studied, e.g. , [21], [20], [2],[24], [16], including consideration of issues of robust convergence to equilibria, e.g. , [3]. In thefollowing, we consider a game played by unidirectional flows between pairs of nodes, includingthe mesh networking case where all one-hop flows share a single (gateway) node, but that nodedoes not act as a central authority (base station or, possibly, mesh access point), rather the systemis distributed in its decision making regarding transmission powers.We further assume that a transmission attempt occurs in every time-slot by every player.The signal to interference and noise ratio (SINR):
SINR i ( q ) := q i h ii N + P j = i h ji q j , October 3, 2018 DRAFT3 where N is the ambient noise power, the power q i ≥ pertains to the transmitter of flow i , and h ji are the path gains between the transmitter of flow j and the receiver of flow i . Indeed, SINR i is the SINR at the receiver of flow i . See Figure 2 illustrating two flow with transmitters T k andreceivers R k , k ∈ { , } . T R T R h h h h Fig. 2. Two interfering flows
The demands of each player i y i := ( U ′ i ) − ( M ) , as above.Note that bidirectional links are not considered above due to self-interference at the trans-mitters/receivers. Typically, each way communication of a bidirectional link will be separatedby TDMA, FDMA or CDMA/spread-spectrum means. In the TDMA setting for a distributedmultihop wireless network, a spatial scheduling problem ensues, e.g. , [19]. A. From SINR to QoS, γ Shannon’s expression for capacity log(1 +
SINR ) is often used to map SINR to service quality.For different modulation frameworks, we can idealize γ i ( q ) := Γ( SINR i ( q )) for correspondingly different increasing functions Γ , so that (4) holds [17]. If the number of bitsper frame n is large and p e an exponential function of SINR (as, e.g. , under DBPSK or GFSKmodulation), then Γ( SINR ) ≈ exp( − n exp( − SINR )) . We used this expression in our numerical studies below. cf. , the Appendix. October 3, 2018 DRAFT4
B. The selfish game
In a quasi-stationary selfish game, the users observe their interference and user i sets q i = Υ i ( N + X j = i q j h ji ) =: F oi , where Υ i := Γ − ( y i ) /h ii . (15)Note that this system is simply affine in q and a unique NEP q such that q = F o ( q ) can bedetermined if the matrix I − Ψ is nonsingular where Ψ ji := h ji Υ i ∀ j = i and Ψ ii := 0 ∀ i (again,here we are not considering constraints on power). That is, the NEP would be q T = N Υ T ( I − Ψ) − . (16)For a two-player game, the Lyapunov function of the continuous-time Jacobi iteration ˙ q = F o ( q ) − q is the quadratic form Λ o ( q ) = X i h i, − i Υ − i ( 12 q i − N Υ i q i ) − Y i q i h i, − i Υ i . The system is (globally) asymptotically stable with unique “interior” NEP if I − Ψ has alleigenvalues with modulus < . Equivalently, we can specify stability in terms of the Hessiansof the quadratic Lyapunov function, Λ o . The result is that the NEP is stable if Y i h i, − i Υ i < . (17) C. The purely altruistic game
For an altruistic two-player game with information sharing as above, user i sets q i = 1 h − i,i ( q − i Υ − i − N ) =: G oi . Note that G oi is also a simple affine function. Again, we can show that the NEP (16) holds heretoo and similarly study its stability properties as in the non-cooperative case.The Lyapunov function for the altruistic ( ˙ q = G o − q ) two-player game is Λ + ( q ) = X i h i, − i Υ i ( Nh − i,i q i + 12 q i ) − Y i q i h i, − i Υ i . October 3, 2018 DRAFT5
The NEP stability condition for altruistic dynamics is Y i h i, − i Υ i > , (18)which obviously cannot be true if (17) holds.Recalling (15), we see a potentially beneficial role for symmetric altruism in the case where Q i Γ − ( y i ) > Q i h ii /h i, − i so that (17) does not hold, and therefore (18) does. D. Numerical example
Consider an example where the frame sizes n = 1024 bits (128 bytes), the desired meancorrect frame transmission probabilities ( y , y ) = ( . , . , the noise power N = 1 . (so thetransmission powers ( q , q ) are normalized with respect to noise), the path gains h i,i = 0 . and h i, − i = 0 . for all i , and the utilities are of the arctan form (11).The unique feasible NEP is ( q , q ) = (224 , with corresponding SINRs of 10.4 and 10.8,respectively, and comparable noise and interference magnitudes. This NEP does not changeposition as the altruism parameter α changes, i.e. , Claim 1. The asymptotic stability conditionfor altruistic dynamics (18) does not hold for this example, giving a saddle contour for Λ + asin Figure 1(c). E. Game framework variation: power-based costs
With the cost
M q instead of
M γ , as α ↓ (to purely altruistic behavior) the equilibrium pointconverged to an opt-out equilibrium, i.e. , ( q , q ) = (0 , , consistent with Section IV-B.VI. S UMMARY
We considered synchronously iterative two-player medium access games under simple ALOHAor power-control dynamics. Assuming symmetric altruistic behavior (but not necessarily sym-metric demand), we showed how the local asymptotic stability properties of the Nash equilibriachanged as a function of the degree of altruism. Even assuming the necessary information is freeof cost and perfect, such altruistic behavior may not have net beneficial effects in this regard. The coding strategy will, in many cases, additionally reduce the interference factor h i, − i beyond propagation attenuation, i.e. , by a “processing gain” factor of the code, here assumed to be dB = 20 = h i,i /h i, − i . October 3, 2018 DRAFT6
In particular, partial altruistic behavior for an example ALOHA game can cause both interior,feasible Nash equilibria to be stable. A beneficial stabilizing effect is possible for some casesof a power control game when the NEP is unstable in the non-cooperative setting.
Acknowledgement:
John F. Doherty for helpful discussions on power control.R
EFERENCES [1] A. Al-Nowaihi and P.L. Levine. The stability of the Cournot oligopoly model: A reassessment.
Journal of Economic Theory , , p. 307-321, 1985.[2] T. Alpcan, T. Basar, R. Srikant and E. Altman. CDMA uplink power control as a noncooperative game. Wireless Networks,Vol. 8, Nov. 2002.[3] T. Alpcan, T. Basar and S. Dey. A Power Control Game Based on Outage Probabilities for Multicell Wireless Data Networks.In Proc. IEEE ACC , 2004.[4] E. Altman and T. Boulogne, R. El-Azouzi, T. Jim´enez and L. Wynter. A survey on networking games in telecommunications.
Comput. Oper. Res. (2): pp. 286–311, 2006.[5] A. Aram, C. Singh, S. Sarkar and A. Kumar. Cooperative profit sharing in coalition based resource allocation in wirelessnetworks. In Proc. IEEE INFOCOM , Rio de Janeiro, Brazil, April 2009.[6] G. Arslan and J.S. Shamma. Distributed convergence to Nash equilibria with local utility measurements. In
Proc. IEEECDC , Atlantis, Paradise Island, Bahamas, Dec. 2004.[7] A. P. Azad, E. Altman and R. Elazouzi. From Altruism to Non-Cooperation in Routing Games. Networking and ElectronicCommerce Research Conference (NAEC), Lake Garda, Italy, Oct. 2009.[8] T. Basar and G. J. Olsder.
Dynamic noncooperative game theory , 2nd Ed. Academic Press, 1995.[9] D.P. Bertserkas and J. N. Tsitsiklis. Convergence rate and termination of asynchronous iterative algorithms. In
Proc. 3rdInternational Conference on Supercomputing , 1989.[10] J. Bieber, M. Kenney, N. Torre and L.P. Cox, An empirical study of seeders in BitTorrent. Duke University, ComputerScience Dept, Technical Report No. CS-2006-08, 2006.[11] D. Fudenberg and D.K. Levine.
The Theory of Learning in Games.
MIT Press, Cambridge, MA, 1998.[12] Y. Jin and G. Kesidis. A pricing strategy for an ALOHA network of heterogeneous users with inelastic bandwidthrequirements. In
Proc. CISS, Princeton , March 2002.[13] Y. Jin and G. Kesidis. Equilibria of a noncooperative game for heterogeneous users of an ALOHA network.
IEEECommunications Letters : (7), pp. 282-284, 2002.[14] Y. Jin and G. Kesidis. Dynamics of usage-priced communication networks: the case of a single bottleneck resource. IEEE/ACM Trans. Networking , Oct. 2005.[15] I. Menache and N. Shimkin. Fixed-rate equilibrium in wireless collision channels. In
Proc. Network Control andOptimization (NET-COOP) , Avignon, France, June 2007.[16] F. Meshkati, M. Chiang, H.V. Poor, and S.C. Schwartz. A Game-Theoretic Approach to Energy-Efficient Power Controlin Multicarrier CDMA Systems.
IEEE JSAC , (6), June 2006.[17] J. G. Proakis, Communication Systems Engineering , Prentice Hall International Editions, 1994.[18] J.B. Rosen. Existence and uniqueness of equilibrium points for concave N -person games. Econometrica , (3), pp. 520-534, 1965. October 3, 2018 DRAFT7 [19] S. Roy and J. Zhu. A 802.11 Based Slotted Dual-Channel Reservation MAC Protocol for In-Building Multi-hop Networks.
Mobile Networks and Applications , (5), Oct. 2005.[20] C.U. Saraydar, N.B. Mandayam and D.J. Goodman. Power control in a multicell CDMA data system using pricing. In Proc. IEEE VTC , pp. 484-491, 2000.[21] C.U. Saraydar, N.B. Mandayam and D.J. Goodman. Pareto efficiency of pricing based control in wireless data networks.In
Proc. IEEE WCNC , pp. 231-234, 1999.[22] J. Seade. The stability of Cournot revisited.
Journal of Economic Theory , , p. 15-27, 1980.[23] J.S. Shamma and G. Arslan. Dynamic fictituous play, dynamic gradient play, and distributed convergence to Nash equilibria. IEEE Trans. Auto. Contr. , (3):312-327, 2005.[24] M. Xiao and N.B. Shroff, and E.K.P. Chong. A utility-based power-control scheme in wireless cellular systems. IEEE/ACMTrans. Networking , Vol. 11, April 2003.[25] Y. Xing and R. Chandramouli. Stochastic learning solution for distributed discrete power control game in wireless datanetworks.
IEEE/ACM Trans. Netw. (4), 2008.[26] H.P. Young. Strategic Learning and Its Limits . Oxford University Press, New York, 2004.[27] A. Zhang and Y. Zhang. Stability of Nash equilibrium: The multiproduct case.
Journal of Mathematical Economics , (4),p. 441-462, 1996. A PPENDIX : F
ROM