Non-Cooperative Operators in a Game-Theoretic Framework Mehdi Bennis
Juan Lara
Antti Tolli
Centre for Wireless Communications (CWC) University of Oulu, Finland
[email protected]
Telefonica Barcelona, Spain
[email protected]
Centre for Wireless Communications (CWC) University of Oulu, Finland
[email protected]
Abstract—In this paper, we address the problem of noncooperative operators trying to maximize their profits by offering extra spectral resources to other operators starving for spectrum. An oligopoly1 market is used to model this game-theoretical setting where several operators seek to maximize their profit while customers try to sustain or increase their quality-of-service (QoS). In addition, a Bertrand game model is used to maximize the payoff of operators where the Nash Equilibrium is computed. Simulations validate the proposed game model.
I. I NTRODUCTION Recently, there has been a huge interest in flexible and dynamic spectrum allocation [1-2], [6]-[8] where efficient radio resource allocation is the goal to be attained. Clearly, future radio systems need to have high spectrum efficiency and be able to share and coexist in an efficient manner. This coexistence issue has brought up several new challenges such as the issue of selfishness and cooperation. These issues need to be tackled in order to yield efficient systems where all resources are efficiently utilized between different radio access networks (RANs). In this work, radio resources are traded between several operators that can be either primary or secondary. Primary systems refer to system that have priority in accessing the spectrum whereas secondary systems can utilize the spectrum when the primary system is idle. Particularly, if primary systems are not using their resources during a certain period of time, these resources can be sold to other operators. In this way, resources are not wasted but efficiently managed by satisfying both sellers (primary) and buyers (secondary). This problem is straightforward in the case of one operator. However, it becomes more challenging in the presence of several operators trying to gain more customers by offering significant prices. This brings up the notion of pricing as to how operators decide about their pricing strategies in order to satisfy their customers. Clearly, the issue of pricing is a very critical aspect and need to be tackled accordingly so that both players are happy with the outcome of the game. In this paper, we model the spectrum trade as a game from an economical perspective. An oligopoly market is considered where operators compete with each other for the offered price to get the maximum profit. 1 Oligopoly markets comprise a few sellers and many competitive buyers. In the special case of two sellers, this market is known as a duopoly. Moreover, in an oligopoly, the buyers cannot influence the price or quantities offered.
978-1-4244-2644-7/08/$25.00 © 2008 IEEE
Fig. 1.
Several operators negotiating a price for spectrum.
Game theory [5], [9] is a useful tool to analyze resource allocation and mathematical models of conflict and cooperation among intelligent rational decision makers. It aims to represent and understand the actions of these decisions makers as well as the outcome of their interactions. In 1950, John Nash [5] introduced the concept of a ”Nash Equilibrium” (NE), which became the organizing concept under Game Theory – even though the concept actually stretches as far back as Cournot (1838). The spectrum sharing problem is well suited for analysis using game theoretic models where radios interact to share the available spectrum. The understanding of the outcome can aid in determining whether the interactions will converge as well as whether the outcome is desirable. In this work, we focus on non-cooperative games which has been used to analyze packet slotted Aloha [8] to mention a few of its applications. Recently, in [15] some work has been done aiming at defining spectrum management architectures supporting spectrum trading and sharing among different actors in the spectrum landscape, following different modalities and time-scales (definitions were mainly done at the system level). However, there is still work to be done to understand and define the procedures and behavior of actors (e.g. operators). In this context, the game theory models developed in this paper give insight on how operators implicitly evolve their pricing towards reaching an equilibrium, if any. And, thus foresee preventive policies to be implemented by the system capable of inter-operator
spectrum sharing, aiming at both fairness and optimality. This paper is organized as follows: Section II considers the system model used in this work where the spectrum utility function is formulated then a game is designed. Numerical results are provided in Section III as well as the impact of costs on the profit of operators. Finally, conclusions are drawn in Section IV. II. SYSTEM M ODEL First, we start by defining our utility function for the operator starving for spectrum (i.e. secondary) according to the commonly used utility function in [3]: 1 U (q1 , q2 , I) = α1 q1 +α2 q2 − (β1 q12 +β2 q22 +2γq1 q2 )+I (1) 2 The rationale behind using this quadratic utility function is due to its concavity which makes it easy for further analysis and which are of particular interest in game theory because they constitute second-order approximation to other types of nonlinear cost functions. For simplicity and without loss of generality, let us assume that β1 = β2 = 1. Thus, the utility function is quadratic in the consumption of the goods (spectrum) q and linear in the consumption of other goods I. The parameters γ ∈ [−1, 1] measures the substitutability between the products. If γ = 0, each firm has monopolistic market power, while if γ = 1, the products are perfect substitutes. A negative γ implies that the goods are complementary. Finally, αi measures quality (for instance channel quality or spectral efficiency). Thus, an increase in αi increases the marginal utility of consuming good qi . The spectrum utility function used in (1) can easily be extended to allow n firms producing one product variety each. U (q, I) =
n
n 1 2 qi αi − ( qi + 2γ qi qj ) + I 2 i=1 i=1
(2)
i=j
pi q i + I ≤ m
(3)
i=1
where parameter m denotes the operator’s income. We have therefore the following optimization problem to solve using the Lagrangian method. For this, we take the first order derivative: n n ∂ 1 2 [ qi αi − ( qi + 2γ qi qj ) − ∂qi i=1 2 i=1 i=j
n pi qi + I − m)] = 0 λ1 × ( i=1
i=j
n pi qi + I − m)) = 0 λ1 × (
(5)
i=1
From Equations (4) and (5), we find out that λ∗1 = −1 gives the maximum spectrum utility function expressed in Equation (6). Umax n(q) n n = i=1 qi αi − 12 ( i=1 qi2 + 2γ i=j qi qj ) − i=1 pi qi Next, to derive the optimum spectrum demand, we use the first-order condition determining the optimal consumption of goods qk ∂U (q) = αk − qk − γ qj − pk = 0, (6) ∂qk j=k
The following inverse demand function for firm k producing substitute goods is used in all Cournot treatments. ⇒ pk = αk − qk − γ qj (7) j=k
A. Cournot Competition In the Cournot model, the selling price is the same for all operators. Cournot oligopoly [3] situation occurs when there are n firms with identical products, which operate in a market in which the market demand function is known. Every spectrum seller (primary operator) rationally assumes that the production of other operators will not be modified as a response to their moves and vice-versa (best-response). The profit of every primary operator (given that costs are normalized to zero) is given by: Π(qk , q−k ) = pk ×
n
qk = (αk − qk − γ
k=1
Where q¯ = {q1 , q2 , ..., qn } is the spectrum set. Operators (i.e consumers) maximize their utility subject to their budget constraints (herein, p refers to price): n
n n 1 2 ∂ ( qi αi − ( qi + 2γ qi qj ) − ∂I i=1 2 i=1
qj ) × qk , (8)
j=k
where q−k denotes the traded spectrum of operator other than operator k. Moreover, all firms know the total number of firms in the market and take the output of the others as given in order to maximize their respective profits. This is precisely the essence of one-shot games. We take the first-order derivative of the profit with respect to the quantity qk and get: ∂Π(qk , q−k ) =0 ∂qk αk − γ i=j qj ⇒ qk (q−k ) = 2 Summing over all operators and noting that: qi = qk + qj i
(4) i
(9) (10)
i=j
αi = αk +
i=j
αj
(11)
as follows: Rev(i) = c1 Mi Wi − q i 2 Cost(qi ) = c2 Mi (BWireq − αi ) Mi
Fig. 2. In case (A) the primary operator sells his free spectal band to the secondary operator, whereas in case (B) the primary operator has to share the band, hence some costs are incured.
we find the Cournot equilibrium for demand and price by solving the equations in (10): αk [γ(n − 2) + 2] − γ j=k αj c c (12) q k = pk = (2 − γ)[γ(n − 1) + 2] Subscript C indicates Cournot equilibrium and n is the number of primary operators. B. Bertrand Competition In a more realistic scenario, the Bertrand model is a model of price-setting oligopolists. In this case, prices are differentiated and the secondary operators seek the cheapest price. Summing over all operators, Equation (6) can be written: αi − qi − γ(n − 1) qi − pi = 0 (13) i
i
i
i
After some mathematical derivation, the spectrum demand function can be expressed as follows: (αk − pk )[γ(n − 2) + 2] − γ j=k (αj − pj ) qk (pk , p−k ) = (1 − γ)[γ(n − 1) + 1] (14) C. Revenue and Cost Function For The Offering Operator In this section, we consider the case where a primary operator is sharing spectrum with a secondary system. Sharing his frequency band incurs some costs that will be modeled. In order to develop a cost function, the Quality of Service (QoS) of the primary system needs to be considered. Therefore, we consider two cases. In the first case, a spectrum portion band is shared with the secondary system whereas in the second case the portion band is dedicated to the secondary system. This is illustrated in Figure 2 and accounted for into the cost model through the weight c2 . The revenue2 and costs for primary operator i are calculated 2 It is a partial revenue as the primary operator also obtains revenue due to q i Pi
(15) (16)
where c1 and c2 denote weights for the revenue and cost functions. BWireq denotes the throughput for a primary operator and αi is the spectral efficiency for primary operator i. Mi is the number of users of the primary connections and Wi is the spectrum size of operator i. Based on the aforementioned model, we formulate our game where the players are the primary operators offering spectrum whose strategy is the price per unit of spectrum pi and finally the payoff for every operator is the profit (revenue-costs) after the spectrum transaction. The solution of this game is called Nash equilibrium. First, the profit for every operator is calculated as: P rofi (p) = qi pi + Revi − Costi
(17)
where pi denotes the set of prices offered by all players in the game. p¯ = {p1 , ..., pN } is the set of prices. By definition, the Nash Equilibrium of a game is a strategy profile where no player can increase his payoff by choosing a different action, given other players’s actions. In this case, the Nash equilibrium is obtained using the best response function which is the best strategy of one player given others’s strategies. Hence, the best response function of operator i, given a set of prices offered by other primary operators pi is: Bi (p−i ) = argmaxpi (P rofi (pi , p−i ))
(18)
Mathematically, in order to compute the N.E we calculate the i (p) first order of the profit: ∂P rof = 0 for all operators i. One ∂pi has to notice that the spectrum demand is taken into account in order to compute the primary operator’s profit. Wi − qi (p) 2 ) Mi (19) i (p) ∗ = 0, we get the corresponding p which is Using ∂P rof i ∂pi the Nash Equilibrium of the game.
P rofi (p) = qi (p)pi +c1 Mi −c2 Mi (BWireq −αi
D. Special case with 2 primary operators and 1 secondary operator For the sake of illustration, we consider 2 primary operators trying to sell spectrum to another secondary operator. As a result, deriving Equation (17) with respect to the price pi gives: ∂qi (p) ∂P rofi (p) = 0 = qi (p) + pi ∂pi ∂pi Wi − q i αi ∂qi (p) req −2c2 Mi (BWi − αi )(− ) Mi Mi ∂pi
(20)
Similarly, by using Equation (18), we find the corresponding prices p∗1 and p∗2 . It is worth noting that in the case of non-shared portion band, c2 = 0 and therefore Equation (19) is further simplified.
3.5
23 22.5
3
Best Response
22
2.5
γ = 0.4
p2=3.5
21
2 p2
profit operator 1
21.5
20.5
1.5
20 p2=0.05
19.5
1
Nash Equilibrium
19 0.5
γ = 0.1
18.5 18
0
0
0.2
0.4
0.6
0.8
1 p1
1.2
1.4
1.6
1.8
2
Fig. 3. Profit of primary operator 1 as a function of the offered price to secondary operator 3. This is plotted for several values of the offered price from primary operator p2 .
III. N UMERICAL R ESULTS In order to validate our model in the previous Section, we consider 3 operators from which 2 are primary. The bandwidth allocated to operators is Wi = 20 MHz. We further assume Mi = 10 primary users using the bands per operator. SNR = 9 dB and the target BERt for the secondary operator is BERt = 10−4 . Based upon that we calculate [14] the spectral efficiency. Furthermore, constant spectral efficiency is assumed for every operator, both primary or secondary. Moreover, the bandwidth requirement is set to be 2 Mbps and c1 = c2 = 2 (for example). Finally, the substitutability factor γ varies between 0.1 to 0.6. The variation on the γ accounts for the non total substitutability of the spectrum. Due to technological or business reasons, a secondary operator might have a reference for buying spectrum from a particular primary operator. Figure 3 depicts the profit of primary operator 1 as a function of the offered price. Clearly, when the offered price increases, so does the profit since more revenue is generated due to higher price. Nonetheless, for a certain value p1 , the revenue starts to dwindle due to the fact that the secondary operator decreases its spectrum demand due to the expensive prices charged by the primary operator. Furthermore, we can see that the price which results in the highest profit for the operator is the best response: given the prices of operator 2, operator 1 maximizes its profit accordingly. Besides, in Figure 3 the best response for operator 1 increases with increasing p2 . This is intuitive since the primary operator has to increase its prices as well. On the other hand, Figure 4 gives some insights about the Nash Equilibrium with respect to p1 and p2 . The best response functions for both primary operators are shown, in function of the substitutability γ values (γ = 0.1 and γ = 0.4), where the N.E point gives less prices for both operators meaning that the secondary operator has the freedom of changing and picking up the best primary operator. In this
0
0.5
1
1.5
2
2.5
3
3.5
p
1
Fig. 4. Best response function and Nash equilibrium as a function of the substitutability parameter γ.
way, both primary operators have no choice but to decrease their respective prices, otherwise the secondary system will not seek any further spectrum. This suggests that a third-party system responsible for mediating the spectrum trading among operators should enforce secondary operators to be agnostic with respect to primary operators, when offering their bids. This would contribute to make spectrum a more substitutable commodity. A. Impact of costs on operators profits We are interested in seeing the impact on the costs of sharing a spectrum band with other primary operators. Figure 5 depicts the profit of 2 primary operators in function of each other. This is shown for several values of the cost parameter c2 as depicted in Equation (16). It can be seen that sharing the band with another operator decreases the profit for both operators. Nevertheless, when no costs are incured (c2 = 0), a maximum profit is achieved. B. Discussion Given the assumption of a non-cooperative scenario, the Nash equilibrium might yield non-efficient outcome (price) for operators. Therefore, the concept of collusion can be envisaged where there is some kind of cooperation between operators or pre-assigned agreement. This is to be avoided. Further, the model assumes each operator being able to know the profit of each other. However this is arguable since in a real-life noncooperative setting operators are not willing to disclose their profits. On the other hand, the spectrum management architectures blueprinted in [15] a centralized entity conveys and enforces policies (spectrum etiquette) in the spectrum sharing scenario. Due to this spectrum etiquette, it is therefore feasible to obtain other operators profits as long as this knowledge leads to a more profitable equilibrium among operators.
22 c2 = 4 c2 = 0
20
c =2 2
profit operator 2
18
16
14
12
10
8 20.05
20.1
20.15
20.2 20.25 20.3 profit operator 1
20.35
20.4
20.45
Fig. 5. Comparison between profits of both primary operators as a function of the incured costs, for c2 = 0, 2 and 4.
IV. C ONCLUSION AND F UTURE W ORK With the upsurge in cognitive radio studies for next wireless generations, pricing has received a lot of interest recently. In this paper, we have presented a game-theoretic model to obtain the optimal pricing for dynamic spectrum sharing, where several operators compete with each other to offer spectrum to secondary systems. In this context, both primary and secondary systems are satisfied with the outcome of the game. Bertrand game was used to model the game where primary and secondary operators are trading available resources in the network. The optimal Nash Equilibrium (N.E) was computed. Moreover, parameters influencing the outcome of the game have been evaluated providing some insights for practical systems. Future work will extend the current work to account for partial knowledge in addition to punishment-based mechanisms will be investigated when primary systems tend to deviate from their strategy. V. ACKNOWLEDGEMENT This research was supported by the Finnish Funding Agency for Technology and Innovation (Tekes), in the framework of the WINNER+ European project. R EFERENCES [1] J. Mitola, ”Cognitive radio for flexible multimedia communications,” in proc. MoMuC 99, pp. 3-10, 1999. [2] Q. Zhao and B. Sadler, ”A survey of dynamic spectrum access: signal processing, networking and regulatory policy,” IEEE Signal Processing Mag., vol. 24, no. 3, pp. 79-89, May 2007. [3] N. Singh and X. Vives, ”Price and Quantity Competition in a Differentiated Duopoly”, RAND Journal of Economics, 546-554, 1984. [4] S. Haykin, ”Cognitive Radio: Brain empowered Wireless Communication” IEEE Journal on Selected Areas in Communication, Vol. 23, No. 2, Feb. 2005. [5] M. Osborne, An introduction to Game Theory, Oxford University Press, 2003. [6] Federal Communications Commission, Cognitive Radio Technologies Proceeding (CRTP), http://www.fcc.gov/oet/cognitiveradio/.
[7] J. Mitola, ”Cognitive radio: An integrated agent architecture for software defined radio,” Doctor of Technology, Royal Institute of Technology (KTH), Stockholm, Sweden 2000. [8] M. Magnus, Y. Joseph, L. Li, and V. Mirrokni, ”On spectrum sharing games,” in proc. ACM symposium on Principles of Distributed Computing, pp. 107-114, 2004. [9] J. Nash, ”Non-cooperative games,” Annals of mathematics, 54 (1951), pp. 286-295. [10] X. Vives, ”On the efficiency of Bertrand and Cournot equilibria with product differentiation,” Journal of Economic Theory 36, 1985, 166175. [11] G. Symeonidis, ”Comparing Cournot and Bertrand equilibria in a differentiated duopoly with product R&D,” International Journal of Industrial Organization 21 (2003) 39-55. [12] E. Altman, N. Bonneau and M. Debbah, ”Correlated Equilibrium in Access Control for Wireless Communications,” Networking 2006, Coimbra, Portugal ,May 2006. [13] E. Altman, K. Avrachenkov, N. Bonneau, M. Debbah, R. Azouzi, D. Menasche, ”Constrained Stochastic Games in Wireless Networks,” GLOBECOM 2007, Washington, USA, Nov. 2007. [14] A. Goldsmith and S. Chua, Variable rate variable power MQAM for fading channels, IEEE Trans. Commun., vol. 45, no. 10, pp. 1218-1230, Oct. 1997. [15] M. Bennis, J. Kermoal, P. Ojanen, J. Lara, S. Abedi, R. Pintenet, S. Thilakawardana and R. Tafazolli, ”Advanced Spectrum Functionalities for Future Radio Networks” Wireless Personal Communication Journal, Springer, November 2007.
