Game-theoretic Understanding of Price Dynamics in Mobile Communication Services
aa r X i v : . [ c s . G T ] A ug Game-theoretic Understanding of Price Dynamicsin Mobile Communication Services
Seung Min Yu and Seong-Lyun Kim
Abstract —In the mobile communication services, users wishto subscribe to high quality service with a low price level,which leads to competition between mobile network operators(MNOs). The MNOs compete with each other by service pricesafter deciding the extent of investment to improve qualityof service (QoS). Unfortunately, the theoretic backgroundsof price dynamics are not known to us, and as a result,effective network planning and regulative actions are hard tomake in the competitive market. To explain this competitionmore detail, we formulate and solve an optimization problemapplying the two-stage Cournot and Bertrand competitionmodel. Consequently, we derive a price dynamics that theMNOs increase and decrease their service prices periodically,which completely explains the subsidy dynamics in the realworld. Moving forward, to avoid this instability and ineffi-ciency, we suggest a simple regulation rule which leads toa Pareto-optimal equilibrium point. Moreover, we suggestregulator’s optimal actions corresponding to user welfare andthe regulator’s revenue.
Index Terms —Network economics, game theory, competi-tion, price dynamics, regulation, mobile communications.
I. I
NTRODUCTION
A. Conflict of Interests among Mobile Network Operators,Users, and the Regulator
In mobile communication services, there is interactionamong mobile network operators (MNOs) , users , and theregulator (Figure 1). Each MNO makes an investment inits network to improve the quality of service (QoS) andsets a service price to maximize its profit. The users decidewhich MNO is more appropriate to subscribe to the networkservice considering the service price and the QoS. Finally,the regulator aims to maximize the welfare of all users.Therefore, there should be some equilibrium points for theservice price and network investment (QoS) between MNOsand users. Theoretically, finding such equilibrium points isnot easy. The situation becomes even more complicatedwhen there are multiple MNOs competing with each other.Maximizing profit is the primary concern of MNOs,which might be achieved by having a high price level andlow investment on the network. On the other hand, userswish to maximize their utility by consuming high QoSwith a low service price. The QoS is directly related tothe network investment from MNOs. Therefore, there isa conflict of interests among these players and the roleof the regulator is very important. From the regulatory Seung Min Yu and Seong-Lyun Kim are with the School of Electricaland Electronic Engineering, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 120-749, Korea. E-mail: { smyu, slkim } @ramo.yonsei.ac.kr.All correspondence should be addressed to Prof. Seong-Lyun Kim. Fig. 1. The interaction among MNOs, users, and the regulator. perspective, solely maximizing profit by MNOs shouldbe avoided if it is at the cost of sacrificing user welfaresignificantly.For making efficient regulations, we firstly investigatecharacteristics of the competitive mobile communicationservices. An important question for MNOs is how much ofthe network capacity should be provisioned and how highthe service price should be. Price competition between twooperators was previously studied by Walrand [1], wherethe network capacity was assumed to be given. Here, weanalyze how each MNO determines the optimal investmenton the network and the service price as a response tothe strategy of its competitor. For this purpose, we applyCournot and Bertrand competition models [2]-[5].In the Cournot model, MNOs compete with each otherdeciding the extent of investment on their networks. On theother hand, in the Bertrand model, MNOs engage in pricecompetition to attract more subscribers for a given networkcapacity. We combine the Cournot and Bertrand modelsso that the network capacity is determined in the Cournotphase and afterwards the service price is determined inthe Bertrand phase. The Cournot and Bertrand models areinterlinked and we achieve joint optimization of the networkcapacity and service price. Our main viewpoint of this jointoptimization is in investigating the dynamics of competitionbetween MNOs and also in finding an optimal role of theregulator. B. Price Competition and Subsidization
The dynamics of price competition among networkoperators was studied in some previous works [6]-[10].Particularly, in [6], we showed that there would be a pricedynamics that network operators increase and decreasetheir service prices periodically. In the real world, however,
Fig. 2. Quarterly marketing expenses as percentage of sales of the majorMNOs (SKT, KT and LGU+) in South Korea. network operators’ billing systems are very similar in thesame country or state and the price dynamics does not seemto occur.For discussing the reality of the price dynamics, letus consider the monthly charging structure in the mobilecommunication service. In most countries, many MNOsgive a subsidy to attract new subscribers (potential users ortheir competitors’ subscribers) [11]. The subsidy is offeredas part of a contract that includes a stipulated time period.Therefore, we should consider the subsidy amount forexamining the price competition among MNOs.To show that there can be a kind of price dynamics bysubsidization in the real world, we plot quarterly marketingexpenses as percentage of sales of the major MNOs (SKT,KT and LGU+) in South Korea as an example (Figure 2).Note that the investigated marketing expenses are mostlyused for subsidization. In the figure, the MNOs increaseand decrease their marketing expenses repeatedly, andthis can be interpreted as the service price dynamics bysubsidization. Then, why do the MNOs use subsidizationas an indirect method for increasing or decreasing theirservice prices? The MNOs cannot increase the serviceprices easily due to regulations. On the other hand, there arefew regulations on subsidization and people are relativelygenerous about change of the subsidy amount because itis believed that subsidization is a means of lowering thecost of new subscriber’s entry to the mobile communi-cation services [12]. Therefore, the MNOs compete witheach other by adjusting their subsidy amounts, making theprice dynamics in the real world. Unfortunately, theoreticbackgrounds of price dynamics are not known to us, and asa result, effective network planning and regulative actionsare hard to make in the competitive market.In this paper, we analyze the price dynamics betweenMNOs using the two stage competition model, where theMNOs increase and decrease their service prices periodi-cally without an equilibrium point. This kind of price dy-namics is not desirable to any player due to the instability.For example, it is unfair that users’ payments for the mobilecommunication service are different in different start timesof the subscription even though they are served by the sameMNO. Based on our analysis, to avoid such instability,we suggest a simple regulation rule that guarantees an equilibrium point of price levels, which is Pareto-optimal.
C. Related Work
In [13], the author shows that service price and QoSare inter-related in communications networks, and suggestsParis metro pricing (PMP) for Internet. PMP is a kind ofprice discrimination over different QoS levels; the higherQoS, the higher price. In that paper, the author finds that theservice price and QoS will converge to an equilibrium pointafter a number of interactions. PMP is further extendedby Walrand [1], who formulates an Internet pricing modelunder price and QoS constraints. In that work, the authorinvestigates how much PMP improves the operator’s profitcompared to a single optimal service price. The authoralso analyzes price competition between two homogeneousnetwork operators, the network capacities of which arefixed. In [6], we show the dynamics of price competition( price war ) using the Walrand model [1], and suggest aregulation for price level convergence.The price war in communication service is observed in[6]-[10]. Particularly, in [7] and [8], if one operator lowersits price to increase revenue or to monopolize the entiremarket, then the other operators will also lower their priceto match the price leader. The price down competition willoccur repeatedly among all operators, eventually damagingevery operator with a revenue decrease.Competition among network operators occurs not onlyby price differentiation. The capacity of the network isanother important variable. This is because users will selecta network operator based on decision criteria includingnot only service price but also QoS level, and the QoSis directly related to the network capacity. Therefore, eachoperator jointly optimizes the service price and networkcapacity. All of the previous work mentioned above focusesonly on price competition, assuming the network capacityis given as an external value. In [14], the authors considercompetition among multiple network operators with single-or two-service classes. In that work, service prices are fixedand price competition does not occur. To attract more users,the operators decide only the network capacity.Suppliers of a homogeneous good/service compete witheach other by deciding their amount of output. This is calledCournot competition (quantity competition) [4]. Generally,the market price decreases as the total amount of outputincreases. On the other hand, Bertrand competition refersto price competition where the suppliers compete with eachother by controlling the product price [5]. In the Bertrandcompetition model, consumers buy all of a particular prod-uct from the supplier with the lowest price.We analyze mobile communications markets usingCournot and Bertrand competition models [15]-[18]. In[15]-[17], we suggest spectrum policies and subsidizationschemes for improving user welfare in mobile communi-cations. In [18], we investigate the effect of allocation ofasymmetric-valued spectrum blocks on mobile communi-cations markets. However, our previous works focus onspectrum allocation and have not dealt with price dynamicsin mobile communications.
D. Main Contribution of This Paper
Using the two stage model [19], we will show that MNOssequentially decrease their service prices (i.e., increase sub-sidies) as in [7] and [8], but one MNO suddenly increasesits price when the competitor’s price is lower than a certainthreshold. Therefore, the price levels increase and decreaseperiodically without an equilibrium point. We call this pricewar with long jumps , which is not desirable to any player.The main contributions and results of this paper aresummarized below. • Description of price dynamics : In the real world,MNOs tend to compete with each other changingtheir service prices by subsidization (see Figure 2).Using a two-stage Cournot and Bertrand competitionmodel with network congestion, we mathematicallyanalyze the competition between MNOs. Based ona game-theoretic approach, we show that there existsprice (subsidy) dynamics in the mobile communicationservice, which well explains the subsidy dynamics inFigure 2. • Regulation for price convergence : To avoid instabilityand inefficiency, we propose a simple regulation thatlimits the number of price level changes. We show thatthe regulation guarantees an equilibrium point of pricelevels that is Pareto-optimal. We also introduce morerealistic regulations that bring the same effect of theprice regulation. • Regulation under a two-stage Cournot and Bertandmodel : Using the two-stage model, we calculate anequilibrium point of the network capacity and theservice price. From the result, we suggest a regulator’soptimal action (exacting taxes) corresponding to userwelfare or the regulator’s revenue. This is an extensionof our previous work [6] to the two-stage model.The rest of this paper is organized as follows. In thenext section, we describe our system model. Section IIIpresents our optimization problem in two-stage duopolycompetition. In Section IV, we derive a solution to theoptimization problem in the Bertrand stage and explain howtwo operators’ prices vary. We suggest a simple regulationthat drives the price levels to converge on an equilibriumpoint. In Section V, we combine the Bertrand model withthe Cournot model. Using a backward induction method, wesolve the optimization problem in the Cournot stage andfind an equilibrium point. From the results, we describecharacteristics of the communications service market andintroduce the role of the regulator. Finally, Section VIconcludes the paper.II. S
YSTEM M ODEL
Consider a service area covered by two competitiveMNOs. There are M users for the mobile communicationservice. Nonnegative values k and k , respectively, denotethe first and the second MNO’s capacity, which determinethe quality of service (QoS) of the networks. MNOs deter-mine the optimal k and k values in the Cournot stage. Inthe Bertrand stage, MNOs compete by controlling p and p , the first and the second MNO’s price for the service.These service prices include the subsidy amounts (i.e., theinitial service price minus the subsidy amount). Therefore,the MNOs can control the service prices by adjusting thesubsidy amounts even if there are some regulations thatprohibit the initial price level changes. Without loss ofgenerality, we assume p and p are normalized valuesover the interval [0 , . Each MNO can provide only oneprice to all users at a given time. The QoS of a networkdepends on the congestion level of the network. We denotethe QoS of each MNO’s network by q and q , respectively.Without loss of generality, the values of q and q arealso normalized over the interval [0 , . A value closer to 0denotes a higher congestion level (lower QoS).Each user decides whether to subscribe to the communi-cation service or not by selecting its serving MNO. Someusers prefer high QoS (low congestion) even though theyhave to pay more. On the other hand, some other userswill accept a low QoS if the service price is low, as wasalso noted by Paris metro pricing [13]. Willingness-to-payand the QoS required by users are positively correlated. Tomodel user behavior, we define the user type as in [1] and[14]. The user type α is a variable over [0 , that quantifiesthe user’s willingness-to-pay. At the same time, it quantifiesthe QoS level required by the user. The value α is closeto when the user is willing to pay a high price for highQoS. At the other extreme ( α −→ ), the user prefers lowQoS with a low price. Since it is difficult to figure out theuser type value of each user, we assume that it is a randomvariable (e.g., uniform distribution in [0 , ).Consider a user with user type value α . For the user tosubscribe to the communication service offered by MNO i ,both the price and QoS levels should be satisfied. In otherwords, α ≥ p i and α ≤ q i , where we regard the first andsecond inequalities as the price condition (PC) and QoScondition (QC), respectively. The PC is commonly usedin microeconomics [20], but it is not sufficient to modelcommunication service, where some users whose PCs aresatisfied may not use the service because the QoS is lowerthan expected due to congestion. If multiple MNOs satisfyboth conditions, then the user will select the MNO offeringthe lowest price.We use a linearly decreasing QoS model as in [1] and[14], which mirrors the perception of service quality [21]: q = 1 − d k M , q = 1 − d k M , (1) The user type α has a dual role as willingness-to-pay and QoS require-ment. This seems to be open to dispute because those two criteria cannotmerged into one-dimensional parameter space. In this paper, however, weassume that each user’s willingness-to-pay and QoS requirement are highlycorrelated and can be modeled as a one-dimensional parameter for themathematical tractability. The characteristics of users’ MNO selection are based on the assump-tion that each user wants a specific service (or application) requiring sometarget QoS level. Then, each user’s utility function becomes a step functionwith a step at QoS level. In other words, if a QoS level is higher than α ,then the utility from the service is equal to α . Otherwise, the utility is zero.Therefore, a user with user type α subscribes to MNO i ’s communicationservice only if both α ≥ p i and α ≤ q i are satisfied. Fig. 3. User type intervals of the users subscribing each MNO’s service. where d and d denote the number of users accessingthe first and the second MNO, respectively. We define thereference capacity, which makes the QoS level equal tozero when all users access one of the MNOs. k and k are normalized values by the reference capacity. In otherwords, if k = 1 ( k = 1 ) and d = M ( d = M ), thenthe QoS level is q = 0 ( q = 0 ). The user demand isexpressed as follows: d = M Z α max1 α min1 f ( α ) dα, d = M Z α max2 α min2 f ( α ) dα, (2)where α min1 , α max1 , α min2 and α max2 denote the minimumand the maximum values of α among the users accessingthe first and the second MNO, respectively. f ( α ) denotes aprobability density function of α . Equation (2) is a demandfunction derived from integration of the willingness-to-paydistribution.Figure 3 illustrates a perfectly segmented market andnon-segmented market. Assume p > p . In the perfectlysegmented market ( p ≥ q ), the values of α min1 and α min2 are determined by PC, and α max1 and α max2 are determinedby QC. On the other hand, in the non-segmented duopolymarket ( p < q ), α min1 is determined by α max2 . This isbecause if there are users whose PC and QC are satisfiedby both MNOs , then the users whose user types are within [ p , q ] will access the second MNO with the lower price, p .III. M OBILE N ETWORK O PERATOR ’ S O PTIMIZATION P ROBLEM
Figure 4 explains our two-stage model. In the Cournotstage, MNOs decide their capacity considering the invest-ment cost. Each MNO cannot change its network capacityin the short-term after observing a competitor’s networkinvestment. Thus, this capacity competition can be mod-eled as a simultaneous game. Hereafter, let i denote thedecision maker index and j denote the competitor’s. Then,we formulate the optimization problem of MNO i in theCournot stage as follows:max k i ≥ f Ri ( k i , k j ) − f Ci ( k i ) , (3)where f Ri ( · ) and f Ci ( · ) denote revenue and cost functionsof MNO i . Note that the MNO’s revenue depends not For the users in [ p , q ], both PC and QC are satisfied by both MNOs. Fig. 4. Two-stage Cournot and Bertrand competition between two MNOs. only on its own capacity, but also on its competitor’s. Therevenue function is determined by the result of the Bertrandstage.In the Bertrand stage, MNOs compete with each otherby controlling their prices for the given capacity determinedin the Cournot stage. Here the price includes the subsidy.Thus high price implies low subsidy, and low price implieshigh subsidy. Each MNO can change its price repeatedlyafter observing a competitor’s. Thus, this price competitioncan be modeled as an infinite sequential game. We excludethe case of pricing equal to that of the competitor. Then,we formulate the optimization problem of MNO i in theBertrand stage, which is divided into two cases: Usingeither a higher or lower price than the competitor’s. • Using a lower price ( p i ≤ p j ) :max ≤ p i ≤ p i d i s.t. d i = M Z − dikiM p i f ( α ) dα. In this case, the number of users for MNO i is independentof the competitor’s price, which is just like the monopolyled by the low pricing MNO. Thus, the upper and lowerlimits of the integral in the constraint are replaced by α max i = q i = 1 − d i /k i M and α min i = p i as in Figure3. • Using a higher price ( p i > p j ) :max ≤ p i ≤ p i d i s.t. d i = M Z − dikiM max n p i , − djkjM o f ( α ) dα. In this case, the competitor affects the number of users ofMNO i . If the competitor guarantees PC and QC of a user,then the user will access the competitor network. Notingthat α max i is determined by the QC, the upper limit of theintegral is given by α max i = q i = 1 − d i /k i M . On the otherhand, α min i is determined by max { p i , q j } as explained inFigure 3. In [14], the authors assume that all MNOs’ service prices are thesame due to MNO competition. Thus, price dynamics does not occur.Each MNO in our model, on the other hand, has no reason to matchits competitor’s service price because it can profit more by lowering itsservice price a little, which will eventually lead to a price dynamics.
IV. B
ERTRAND S TAGE : P
RICE (S UBSIDY )C OMPETITION
A common method for analyzing a multi-stage game isthe backward induction method. This method is used to findthe equilibrium that represents a Nash equilibrium in everystage (or subgame) of the original game. We start with theBertrand stage. The capacities k i and k j are assumed to begiven, and will be optimized in the next section describingthe Cournot stage. Hereafter, we assume that the user type α is uniformly distributed. This assumption was also usedin [1] and [14]. We will show how the price dynamicschanges with more general distributions of user type α inthe last of this section. A. Price War with Long Jumps
We derive the optimal price of MNO i , which is sum-marized in the following lemmas: Lemma 1 : In the case that the MNO’s price p i is lowerthan its competitor’s price p j , the optimal solution p Li is: p Li = (cid:26) if p j > p j − ε if p j ≤ , (4) where ε is a minimum unit of price level changes and verysmall positive value. Proof : Under the assumption that α is uniformly dis-tributed, we get the following equation from the constraintof the optimization problem for the lower price case: d i = M Z − dikiM p i f ( α ) dα = M (cid:18) − d i k i M − p i (cid:19) . (5)We calculate d i and the objective function p i d i fromEquation (5) as follows: d i = k i (1 − p i ) k i + 1 M, p i d i = k i p i (1 − p i ) k i + 1 M. (6)The objective function is a quadratic function whose max-imum is at p i = 1 / . Therefore, if p j > / , then theoptimal solution will be p Li = 1 / . Otherwise, the optimalsolution will be p Li = p j − ε . (cid:4) Lemma 2 : In the case that the MNO’s price p i is higherthan its competitor’s price p j , the optimal solution p Hi is: p Hi = ( k j + p j k j +1 if p j ≥ − k j if p j < − k j . (7) Proof : Under the assumption that α is uniformly dis-tributed, we get the following equation from the constraintof the optimization problem for the higher price case: d i = M Z − dikiM max n p i , − djkjM o f ( α ) dα = M (cid:18) − d i k i M − max (cid:26) p i , − d j k j M (cid:27)(cid:19) . (8) We calculate d i from Equation (8) as follows: d i = min (cid:26) k i (1 − p i ) k i + 1 M, k i d j ( k i + 1) k j (cid:27) . (9)Then, the objective function is: p i d i =min (cid:26) k i p i (1 − p i ) k i + 1 M, k i p i d j ( k i + 1) k j (cid:27) =min (cid:26) k i p i (1 − p i ) k i + 1 M, k i p i (1 − p j )( k i + 1) ( k j + 1) M (cid:27) . (10)The second equality in Equation (10) holds by Equation (6).In the minimum operator of the objective function, the leftside is a quadratic function whose maximum is at p i = 1 / ,and the right side is a linear function whose slope is k i (1 − p j ) M/ (( k i + 1)( k j + 1)) . Therefore, if p j < (1 − k j ) / ,then the optimal solution is p Hi = 1 / , which is the apexof the quadratic function. Otherwise, the optimal solutionis p Hi = ( k j + p j ) / ( k j + 1) , which is the intersection ofthe quadratic and linear functions. (cid:4) From Lemmas 1 and 2, we derive the MNO’s bestresponse function (optimal strategy) in the duopoly com-petition.
Lemma 3 : Given the competitor’s price p j , the MNO’s bestresponse function p ∗ i is • Case 1 ( k j < : p ∗ i = if p j > or ≤ p j < − k j p j − ε if k j +2 < p j ≤ k j + p j k j +1 if − k j ≤ p j ≤ k j +2 . (11) • Case 2 ( k j ≥ : p ∗ i = if p j > p j − ε if k j +2 < p j ≤ k j + p j k j +1 if ≤ p j ≤ k j +2 . (12) Proof : Using the results of Lemmas 1 and 2, we calculatethe optimal values for the higher and lower price cases asfollows: p Li d Li = ( k i k i +1) M if p j > k i ( p j − ε )(1 − p j + ε ) k i +1 M ≈ k i p j (1 − p j ) k i +1 M if p j ≤ ,p Hi d Hi = ( k i k i +1) M if p j ≥ − k j k i ( k j + p j )(1 − p j )( k i +1)( k j +1) M if p j < − k j . Above all, we consider Case 1 ( k j < ). In this case, wecalculate the best response function of MNO i as follows: • If p j > / : To compare the optimal values ofthe higher and lower price cases, we calculate thefollowing: p Li d Li − p Hi d Hi = k i M k i + 1) − k i ( k j + p j ) (1 − p j ) M ( k i + 1) ( k j + 1) = k i ( k i + 2 p j − M k i + 1) ( k j + 1) . This value is positive because we consider the duopolymarket (i.e., k i and k j are positive) and assume p j > / . Therefore, the best response function is p ∗ i = p Li = 1 / . • If (1 − k j ) / ≤ p j ≤ / : To compare the optimalvalues of the higher and lower price cases, we calculatethe following: p Li d Li − p Hi d Hi = k i p j (1 − p j ) Mk i + 1 − k i ( k j + p j ) (1 − p j ) M ( k i + 1) ( k j + 1) = k i k j ( k j + 2) (1 − p j ) M ( k i + 1) ( k j + 1) (cid:18) p j − k j + 2 (cid:19) . All values except p j − / ( k j + 2) are positive becausewe consider the duopoly market (i.e., k i and k j arepositive) and assume (1 − k j ) / ≤ p j ≤ / .Therefore, if p j > / ( k j + 2) , then the best responsefunction is p ∗ i = p Li = p j − ε . Otherwise, the bestresponse function is p ∗ i = p Hi = ( k j + p j ) / ( k j + 1) . • If ≤ p j < (1 − k j ) / : To compare the optimal valuesof the higher and lower price cases, we calculate thefollowing: p Hi d Hi − p Li d Li = k i M k i + 1) − k i p j (1 − p j ) Mk i + 1= k i Mk i + 1 (cid:18) − p j (1 − p j ) (cid:19) . This value is positive under the assumption ≤ p j < (1 − k j ) / . Therefore, the best response function is p ∗ i = p Hi = 1 / .Noting that the intervals [0 , (1 − k j ) / and [(1 − k j ) / , / ( k j + 2)] in Case 1 are merged into an interval [0 , / ( k j + 2)] in Case 2, we can derive the MNO’s bestresponse function in Case 2 similarly. (cid:4) Using Lemma 3, we describe two MNOs dynamicsbelow.
Proposition 1 : In the duopoly competition of two MNOs,there is no pure Nash equilibrium, and price levels increaseand decrease periodically. Proof : Suppose for a contradiction that a pure Nash equi-librium point ( p NEi , p
NEj ) exists. We first consider Case 1( k j < ) as follows: • If p NEj > / , then p NEi should be p NEi = 1 / . Then, p NEj should be p NEj = 1 / − ε , which contradicts theassumption that p NEj > / . • If / ( k j + 2) < p NEj ≤ / , then p NEi should be p NEi = p NEj − ε < / . Then, p NEj should satisfy at Even though there is no Nash equilibrium in pure strategies, there canbe a Nash equilibrium in mixed strategies. This means that the MNOs canrandomize their service prices. However, in the sequential game that eachMNO can change its price after observing a competitor’s, the best responsefunction-based strategy will be used rather than the random strategy. Thisis the reason that we focus on the price dynamics. least one of the following equations: p NEj = p NEi − ε = p NEj − ε, (13) p NEj = k i + p NEi k i + 1 = k i + p NEj − εk i + 1 . (14)Obviously, Equation (13) has a contradiction. FromEquation (14), we find that p NEj is p NEj = 1 − ε/k i ≈ , which contradicts the assumption that p NEj ≤ / . • If (1 − k j ) / ≤ p NEj ≤ / ( k j + 2) , then p NEi shouldbe p NEi = ( k j + p NEj ) / ( k j + 1) . Then, p NEj shouldsatisfy at least one of the following equations: p NEj = 12 , (15) p NEj = p NEi − ε = k j + p NEj k j + 1 − ε, (16) p NEj = k i + p NEi k i + 1 = k i + k j + p NEj k j +1 k i + 1 . (17)Based on the assumption that p NEj ≤ / ( k j + 2) ,Equation (15) means that k j = 0 and p NEi = p NEj =1 / . This has a contradiction because p NEi = 1 / and p NEj = 1 / are not the best response to eachother. Using Equation (16), we calculate that p NEj is p NEj = 1 − ε − ε/k j ≈ , which contradicts theassumption that p NEj ≤ / ( k j + 2) . Using Equation(17), we calculate that p NEj is p NEj = 1 , whichcontradicts the assumption that p NEj ≤ / ( k j + 2) . • If ≤ p NEj < (1 − k j ) / , then p NEi should be p NEi =1 / . This means that p NEj should be p NEj = 1 / − ε ≈ / , which contradicts the assumption that p NEj < (1 − k j ) / .Therefore, we conclude that there is no pure Nash equilib-rium point in Case 1. The only difference between Case 1and Case 2 ( k j ≥ ) is that the intervals [0 , (1 − k j ) / and [(1 − k j ) / , / ( k j + 2)] in Case 1 are merged in theinterval [0 , / ( k j + 2)] in Case 2. Therefore, we can usesimilar proof for Case 2 and conclude that there is no pureNash equilibrium point in Case 2. (cid:4) Figure 5 shows the MNOs’ best response functions.In the figure, we assume the symmetric capacity ( k = k = k ). We see how each MNO sequentially decreasesits price to a little lower level than its competitor whenthe competitor’s price is within (1 / ( k + 2) , / . This is a price war [7], [8]. After that, if the competitor’s price isless than or equal to / ( k +2) , then one MNO will increaseits price to ( k + 1) / ( k + 2) . This is a long jump . Then, thecompetitor sets its price to / . This situation is repeatedperiodically and there is no equilibrium point. We call this price war with long jumps . We plot Figure 6 to illustratethe process of the price war with long jumps. In the realworld, this kind of price dynamics tends to occur by changeof the subsidy amount (see Figure 2). B. Regulation for Convergence
The price war with long jumps is not desirable because ofinstability and inefficiency. We suggest a simple regulation
Fig. 5. Best response functions of both MNOs. We assume that the MNOs’ capacities are equal ( k = k = k ).Fig. 6. Price war with long jumps ( k = k = k ). that leads to an equilibrium point of price levels, which isPareto-optimal. More details are contained in Lemma 4 andProposition 2. Lemma 4 : A regulation that limits the number of price levelchanges makes the price levels converge to an equilibriumpoint ( p Ei , p Ej ) as follows: ( p Ei , p Ej ) = (cid:16) k i +2 , k i +1 k i +2 (cid:17) if k i < k j (cid:16) k i +2 , k i +1 k i +2 (cid:17) or (cid:16) k j +12( k j +1) , (cid:17) if k i = 2 k j (cid:16) k j +12( k j +1) , (cid:17) if k i > k j , where the last opportunity for a price level change is givento MNO j . Proof : When we use the regulation, two MNOs dynamicsis modeled as a finite sequential game and we can calculatethe equilibrium point using the backward induction method.We first consider Case 1 ( k j < ). Let t denote the last stageof the price level change. Then, MNO i will choose the beststrategy at the ( t − stage (i.e., its last choice stage) inorder to maximize its revenue. To calculate the best strategyfor MNO i , we divide the strategy set of MNO i into fourdisjoint subsets. Let p ∗ i ( t − and p ∗ j ( t ) denote the optimalstrategy of MNO i at the t − stage and the optimal strategyof MNO j at the last stage in each subset. Using Lemma 3,we calculate p ∗ i ( t − in each subset. Then, from Equations(6) and (10), we calculate the revenue of MNO i r ∗ i ( t ) asfollows: Strategy 1 ( p i ( t − > / : p ∗ j ( t ) = 12 −→ p ∗ i ( t −
1) = k j + k j + 1 = 2 k j + 12( k j + 1) ,r ∗ i ( t ) = k i p ∗ i ( t ) (1 − p ∗ i ( t )) k i + 1 M = k i (2 k j + 1)4 ( k i + 1) ( k j + 1) M. Strategy 2 (1 / ( k i + 2) < p i ( t − ≤ / : p ∗ j ( t ) = p i ( t − − ε −→ p ∗ i ( t −
1) = 12 ,r ∗ i ( t ) = k i p ∗ i ( t − (cid:0) − p ∗ j ( t ) (cid:1) ( k i + 1) ( k j + 1) M = k i (1 + 2 ε )4 ( k i + 1) ( k j + 1) M. Strategy 3 ((1 − k i ) / ≤ p i ( t − ≤ / ( k i + 2)) : p ∗ j ( t ) = k i + p i ( t − k i + 1 −→ p ∗ i ( t −
1) = 1 k i + 2 ,r ∗ i ( t ) = k i p ∗ i ( t ) (1 − p ∗ i ( t )) k i + 1 M = k i ( k i + 2) M. Strategy 4 (0 ≤ p i ( t − < (1 − k i ) / : p ∗ j ( t ) = 12 −→ p ∗ i ( t −
1) = 1 − k i − ε,r ∗ i ( t ) = k i p ∗ i ( t ) (1 − p ∗ i ( t )) k i + 1 M = k i (cid:16) − ( k i + ε ) (cid:17) k i + 1) M. From these equations, we prove that some strategies arestrictly dominated as follows: • Strategy 2 is strictly dominated by Strategy 1: k i (2 k j + 1)4 ( k i + 1) ( k j + 1) M − k i (1 + 2 ε )4 ( k i + 1) ( k j + 1) M = k i k i + 1) ( k j + 1) (cid:18) k j k j + 1 − ε (cid:19) M > . • Strategy 4 is strictly dominated by Strategy 3: k i ( k i + 2) M − k i (cid:16) − ( k + ε ) (cid:17) k i + 1) M> k i ( k i + 2) M − k i (cid:0) − k i (cid:1) k i + 1) M = k i ( k i + 3)4 ( k i + 2) M > . Therefore, the MNO i never chooses Strategies 2 and 4. Tofinish this proof, we compare Strategies 1 and 3 as follows: k i (2 k j + 1)4 ( k i + 1) ( k j + 1) M − k i ( k i + 2) M = k i ( k i + 2 ( k i + 1) k j ) ( k i − k j )4 ( k i + 1) ( k i + 2) ( k j + 1) M. If k i < k j , then MNO i will set its price to / ( k i + 2) andMNO j will set its price to ( k i + 1) / ( k i + 2) . Therefore,in this case, the equilibrium point is ( p Ei , p Ej ) = (1 / ( k i +2) , ( k i + 1) / ( k i + 2)) . Likewise, if k i > k j , then the equi-librium point is ( p Ei , p Ej ) = ((2 k j + 1) / (2( k j + 1)) , / .Note that, if k i = 2 k j , then both equilibriums are possible.The only difference between Case 1 and Case 2 is that theintervals [0 , (1 − k j ) / and [(1 − k j ) / , / ( k j + 2)] inCase 1 are merged into the interval [0 , / ( k j + 2)] in Case2. Therefore, we can use a similar proof for Case 2 andconclude that the equilibrium in Case 2 is same to that inCase 1. (cid:4) To verify the efficiency of the regulation, we need tocheck the Pareto-optimality of the equilibrium point. Asufficient condition for Pareto-optimality is given in thefollowing lemma:
Lemma 5 : Let subscripts l and h denote MNOs whoseprices are lower and higher than that of their competitor,respectively. If p l ≤ / , p h ≥ / and p h ≥ ( k l + p l ) / ( k l +1) , then ( p l , p h ) and ( p h , p l ) are Pareto-optimal. Proof : If p l < / , then MNO l can increase its revenueby increasing its price toward / . However, it alwaysmakes MNO h ’s revenue decrease. If p l = 1 / , thenMNO l cannot increase its revenue. Therefore, MNO l cannot increase its revenue without decreasing the revenueof MNO h . Likewise, MNO h cannot increase its revenuewithout decreasing the revenue of MNO l because the onlymethod to increase the revenue of MNO h in the perfectlysegmented condition ( p h ≥ ( k l + p l ) / ( k l +1) ) is to decrease p l and p h simultaneously. Therefore, the point that satisfiesthe conditions in Lemma 5 is Pareto-optimal. (cid:4) Number of price level changes P r i c e p i p j regulation Fig. 7. Price level changes of both MNOs (uniform user type case).
Lemma 5 indicates that the MNOs’ prices should besufficiently separated (i.e., perfectly segmented market)for Pareto-optimality. The equilibrium point under oursuggested regulation (Lemma 4) satisfies the conditionsin Lemma 5. Therefore, the equilibrium point is Pareto-optimal. This finding is summarized below.
Proposition 2 : A regulation that limits the number ofprice level changes makes the price levels converge to anequilibrium point that is Pareto-optimal.
The price war with long jumps occurs due to the MNOs’short-sighted way of thinking. However, the MNOs cannotuse a myopic strategy under Proposition 2.We can find similar regulations of Proposition 2. InSouth Korea, the market-dominating enterprise (SKT) can-not change its service price without government permission.This is similar to giving the other MNOs (KT and LGU+)the last opportunity for a price level change. The priceregulation also can be implemented by restricting subsi-dization. In other words, notifying MNOs that subsidiza-tion will be banned after a certain time is equal to theregulation that limits the number of price level changes.For instance, Korea Communications Commission (KCC)actually prohibited subsidization by MNOs from 2003 to2008. This kind of subsidy regulation also can be observedin Finland [22]. Another way that brings the same effect ofsuch price regulation is to regulate the time interval betweenprice level changes. If MNOs cannot change their serviceprice or subsidy levels for a long time, then they will usefar-sighted strategies, which leads to the equilibrium pointdescribed in Lemma 4.To verify our analysis, we conduct simulations, whichshow the price level changes of both MNOs. In the sim-ulations, the minimum unit of price level changes andeach MNO’s network capacity are set to ε = 0 . and k i = k j = 1 . Also, we set each MNO’s initial priceto . and apply the price regulation that limits the Number of price level changes P r i c e (a) p i p j regulation Number of price level changes P r i c e (b) p i p j regulation Number of price level changes P r i c e (c) p i p j regulation Fig. 8. Price level changes of both MNOs in non-uniform user type cases: (a) f ( α ) . (b) f ( α ) . (c) f ( α ) . number of price level changes to . Figure 7 illustratesthe results, where both MNOs initially decrease their pricesrepeatedly but one MNO suddenly increases its price whenthe competitor’s price is lower than some threshold (i.e.,price war with long jumps). After the price regulation, theprices converge on an equilibrium point, which coincideswith our analysis. C. Non-uniform User Type Case
So far, we assume that user type α of (2) is uniformlydistributed. We will now see how the price dynamicschanges with more general distributions of α by means ofsimulations. For this, we adopt three additional distributionsof α in [23] as follows: f ( α ) = 2 − α,f ( α ) = 2 α,f ( α ) = (cid:26) α, if 0 ≤ α ≤ , − α, if < α ≤ . Using these distributions, we can reflect network scenariosconsisting of high population of users having low, high andmiddle user type, respectively.Figure 8 shows the results, where price war with longjumps occurs like the uniform user type case. Moreover,in the non-uniform user type cases, the prices alwaysconverge on an equilibrium point after the price regulation.The equilibrium price tends to be biased towards the highdensity of user type.V. C
OURNOT S TAGE : C
APACITY C OMPETITION
In this section, we combine the result of the Bertrandstage with that of the Cournot stage. That is, using theresults of Section IV, we rewrite the optimization problem(Equation (3)) of the Cournot stage and solve it. The mainmotivation of this section is to completely understand thecompetitive actions of each MNO, and thus to derive theoptimal response of the regulator.
A. Characteristics of Communications Service DuopolyMarket
Without loss of generality, we assume that the last oppor-tunity for a price level change is given to MNO j . Using the revenue equations (Equations (6) and (10)), we calculate therevenue function f Ri ( k i , k j ) at the equilibrium price of theBertrand stage (Lemma 4). We use the linearly increasingcost function as in [20]. That is, f Ci ( k i ) = γM k i , where γ is the unit cost per capacity. Then, we can rewrite theoptimization problem (Equation (3)) of the Cournot stage,which is divided into two cases. • Case 1 ( k i ≤ k j ) : max k i ≥ M k i ( k i + 2) − γM k i , (18) max k j ≥ M ( k i + 1)( k i + 2) k j k j + 1 − γM k j . (19) • Case 2 ( k i ≥ k j ) : max k i ≥ M (2 k j + 1)4 ( k j + 1) k i k i + 1 − γM k i , (20) max k j ≥ M k j k j + 1 − γM k j . (21)To solve these optimization problems, we need somemathematical knowledge given in the lemmas below. Lemma 6 : Consider the following optimization problem: max x ≥ bxx + a − cx, (22) where a , b and c are positive. Then, the optimal solution x ∗ is x ∗ = max ( , r abc − a ) . (23) Proof : We calculate the first and second order derivativesof the objective function as follows: (cid:18) bxx + a − cx (cid:19) ′ = ab ( x + a ) − c, (24) (cid:18) bxx + a − cx (cid:19) ′′ = − ab ( x + a )( x + a ) . (25)The problem is a convex optimization problem because thesecond order derivative is always negative in the feasible set. Therefore, using the first order condition, we calculatethe optimal solution as follows: x ∗ = max ( , r abc − a ) . (26) (cid:4) Lemma 7 : Consider the following optimization problem: max x ≥ bx ( x + a ) − cx, (27) where a , b and c are positive. Then, the optimal solution x ∗ is x ∗ = max (cid:26) , − a + s abc + r a b c + b c + s abc − r a b c + b c (cid:27) . (28) Proof : We calculate the first and second order derivativesof the objective function as follows: bx ( x + a ) − cx ! ′ = − bx + ab ( x + a ) − c, (29) bx ( x + a ) − cx ! ′′ = 2 bx − ab ( x + a ) . (30)The objective function is partially concave because thesecond order derivative is positive when x > a . However,from the first order derivative, we know that the objectivefunction decreases as x increases when x > a . Therefore,we only consider ≤ x ≤ a as the feasible set, andthe optimization problem becomes a convex optimizationproblem in the set. Then, using the first order condition andthe root formula of the third order equation, we calculatethe optimal solution x ∗ . (cid:4) Using Lemmas 6 and 7, we calculate the MNOs’ optimalsolutions k ∗ i and k ∗ j as follows: • Case 1 ( k i ≤ k j ) : k ∗ i = max (cid:26) , − s γ + r γ + 127 γ + s γ − r γ + 127 γ (cid:27) , (31) k ∗ j = max ( , s k i + 1( k i + 2) γ − ) . (32) • Case 2 ( k i ≥ k j ) : k ∗ i = max ( , s k j + 14 ( k j + 1) γ − ) , (33) k ∗ j = max (cid:26) , r γ − . (cid:27) (34) From this result, we find the equilibrium of the originaltwo-stage game as follows: Proposition 3 : Under the regulation that limits the pricelevel changes by MNOs, if the unit cost per capacity γ satisfies the following inequality, F ( γ ) = s γ + r γ + 127 γ + s γ − r γ + 127 γ > , (35) then there is an equilibrium point ( k Ei , k Ej , p Ei , p Ej ) : (cid:0) k Ei , k Ej , p Ei , p Ej (cid:1) = k Ei , s k Ei + 1 (cid:0) k Ei + 2 (cid:1) γ − , k Ei + 2 , k Ei + 1 k Ei + 2 ! = F ( γ ) − , s F ( γ ) − F ( γ ) γ − , F ( γ ) , − F ( γ ) ! , (36) which is Pareto-optimal. Proof : In Case 2 ( k ∗ i ≥ k ∗ j ) , to be an equilibrium point,the optimal solution should satisfy the following equation: k ∗ i = max , vuut k ∗ j + 14 (cid:0) k ∗ j + 1 (cid:1) γ − ≤ max , vuut k ∗ j + 2 k ∗ j + 14 (cid:0) k ∗ j + 1 (cid:1) γ − = max (cid:26) , r γ − (cid:27) = k ∗ j , (37)which contradicts the assumption k ∗ i ≥ k ∗ j because k ∗ i and k ∗ j are non-zero. Therefore, there is no equilibrium pointin Case 2.In Case 1 ( k ∗ i ≤ k ∗ j ) , to be an equilibrium point, theoptimal solution should satisfy the following equation: k ∗ j − k ∗ i = 2 s k ∗ i + 1( k ∗ i + 2) γ − ( k ∗ i + 2) ≥ ⇒ s k ∗ i + 1( k ∗ i + 2) γ ≥ k ∗ i + 2 . (38)The left and right hand side equations of the last inequalityare positive. Thus, we compare the squares of them asfollows: s k ∗ i + 1( k ∗ i + 2) γ ! − ( k ∗ i + 2) = 4 k ∗ i + 4 − ( k ∗ i + 2) γ ( k ∗ i + 2) γ = 4 k ∗ i + 4 − ( k ∗ i + 2) ( − k ∗ i + 2)( k ∗ i + 2) γ = k ∗ i ( k ∗ i + 4)( k ∗ i + 2) γ ≥ . γ F ( γ ) (a) γ =0.25, F( γ )=2 γ C apa c i t y (b) k iE k jE 0 0.05 0.1 0.15 0.2 0.2500.10.20.30.40.50.60.70.80.91 γ P r i c e (c) p iE p jE Fig. 9. (a) Feasibility function F ( γ ) . (b) Equilibrium capacities k Ei and k Ej . (c) Equilibrium prices p Ei and p Ej . Note that the second equality holds because we use anequation ( k ∗ i +2) γ = − k ∗ i +2 from the first order conditionof the optimization problem. From the above calculations,we conclude that if k ∗ i > (i.e., F ( γ ) > ), then therewould be an equilibrium. Using these results and Lemma4, we can calculate the equilibrium as in Proposition 3. (cid:4) We call F ( γ ) of (35) the feasibility function because wecan discriminate between feasible and infeasible marketsusing this function. If the unit cost γ is expensive and doesnot satisfy Equation (35), then the market will be infeasible(i.e., market failure) and, k Ei and k Ej are negative values.Note the equilibrium point is a function of k Ei in Equation(36). This means MNO i has market power even though theregulator gives the last opportunity for changing the pricelevel to MNO j .Figure 9-(a) shows the feasibility function, where the unitcost γ should be less than . to avoid market failure.We plot the equilibrium point in Proposition 3 varying γ ∈ [0 , . . Figures 9-(b) and 9-(c) show the result. Weobserve that p Ej is always higher than p Ei , and MNO j always invests more than MNO i . This means the userswith high user type (i.e., high QoS requirement and highwillingness-to-pay) are targeted by MNO j . On the otherhand, MNO i can make a profit by making a relatively smallinvestment because its target users have low user types. Aninteresting observation is that the price gap between bothMNOs decreases as the unit cost per capacity increasesin Figure 9-(c). This is because the high cost makes bothMNOs reduce their investment levels and concentrate onlucrative targets (i.e., the users whose user types are near . ). B. Role of the Regulator
The regulator’s key concern is to improve user welfare[15], [16]. User welfare means the sum of all users’ utilities.If a user with user type θ purchases MNO i ’s networkservice, its net utility will be θ − p i . On the other hand, if theuser consumes neither of MNOs’ network services, then itsutility will be zero. The regulator can achieve its purposeby exacting taxes from the MNOs or giving subsidies to By a feasible market, we mean there is at least one operator wishingto exist for the market. −0.05 0 0.05 0.1 0.1500.010.020.030.040.050.060.07 γ t U s e r w e l f a r e Fig. 10. User welfare as a function of γ t ( γ c = 0 . ). The value isdivided by M . them. So far, we assume that the unit cost per capacity γ is a given parameter. However, we can divide γ into γ c and γ t (i.e., γ = γ c + γ t ). The value γ c denotes the fixedcost, and γ t denotes the tax ( γ t > ) or subsidy ( γ t < ).We plot user welfare as a function of γ t in Figure 10. Inthe figure, we set γ c = 0 . . The figure shows that userwelfare decreases as γ t increases. Even though this resultis predictable, the regulator can use it to forecast results ofexacting taxes or giving subsidies.From the regulatory perspective, another important thingis to secure finances. We plot the regulator’s revenue asa function of γ t in Figure 11. Intuitively, the regulatorruns a deficit when it gives subsidies (i.e., γ t < ) toimprove user welfare. Therefore, the regulator should strikea balance between securing finances and improving userwelfare. Figure 11 also shows that very high taxes lead tothe revenue loss. This is because the burden of high taxesmakes MNOs cut their investments. If the regulator’s goalis to maximize its revenue, then γ t should be set to . .VI. C ONCLUSIONS
MNOs tend to compete with each other changing theirservice prices by subsidization in the real world (see −0.05 0 0.05 0.1 0.15−0.1−0.08−0.06−0.04−0.0200.020.040.06 γ t R egu l a t o r r e v enue Fig. 11. The regulator’s revenue as a function of γ t ( γ c = 0 . ). Thevalue is divided by M . Figure 2). In this paper, to theoretically explain the pricedynamics in the mobile communication service, we useda two-stage Cournot and Bertrand competition model thatis well understood in microeconomics. The Cournot andBertrand models are interlinked and we perform a jointoptimization of network capacity and service price. Basedon a game-theoretic approach, we show that there is a pricewar with long jumps. This price dynamics explains thesubsidy dynamics in the real world. To avoid the instabilityand inefficiency, we propose a regulation that ensures anequilibrium point of price levels, which is Pareto-optimal.Based on our results in the Cournot stage, we describecharacteristics of the duopoly market and suggest theregulator’s optimal actions (exacting taxes) correspondingto user welfare and the regulator’s revenue. Although ouranalytic results are derived under some assumptions formathematical tractability, it will provide good intuition forunderstanding the price dynamics and imposing regulationsin the mobile communication service.R
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