Edge Computing Resource Management and Pricingfor Mobile Blockchain

Zehui Xiong1, Shaohan Feng1, Dusit Niyato1, Ping Wang1, and Zhu Han2

1School of Computer Engineering, Nanyang Technological University (NTU), Singapore2Department of Electrical and Computer Engineering, University of Houston, Texas, USA

Abstract—The mining process in blockchain requires solving aproof-of-work puzzle, which is resource expensive to implementin mobile devices due to the high computing power and energyneeded. In this paper, we, for the first time, consider edgecomputing as an enabler for mobile blockchain. In particular,we study edge computing resource management and pricingto support mobile blockchain applications in which the miningprocess of miners can be offloaded to an edge computing serviceprovider. We formulate a two-stage Stackelberg game to jointlymaximize the profit of the edge computing service provider andthe individual utilities of the miners. In the first stage, the serviceprovider sets the price of edge computing nodes. In the secondstage, the miners decide on the service demand to purchase basedon the observed prices. We apply the backward induction toanalyze the sub-game perfect equilibrium in each stage for bothuniform and discriminatory pricing schemes. For the uniformpricing where the same price is applied to all miners, the existenceand uniqueness of Stackelberg equilibrium are validated byidentifying the best response strategies of the miners. For thediscriminatory pricing where the different prices are appliedto different miners, the Stackelberg equilibrium is proved toexist and be unique by capitalizing on the Variational Inequalitytheory. Further, the real experimental results are employed tojustify our proposed model.

Index Terms—edge computing, offloading, mobile blockchain,proof-of-work puzzle, pricing, mining, game theory, VariationalInequality.

I. INTRODUCTION

Electronic trading with digital transactions is becomingpopular than ever in e-commerce society, where the consen-sus is reached through trusted centralized authorities. Theintroduction of centralized authorities incurs additional cost,i.e., nominal fees which become more excessive when thenumber of digital transactions becomes large. In 2008, a newpeer-to-peer electronic payment system called “Bitcoin” wasintroduced that avoids this additional cost caused by digitaltransactions [1]. As one popular digital cryptocurrency, Bitcoincan record and store all digital transactions in a decentralizedappend-only public ledger called “blockchain”. The Bitcoin isthe first application of blockchain technologies. Subsequently,the blockchain technologies have generated remarkable publicinterests via a distributed network with independence fromcentral authorities. With blockchain, a transaction can takeplace in a decentralized fashion, which greatly save the costand improve the efficiency. Since its launch in 2009, Bitcoineconomy has experienced an exponential growth, and itscapital market now has reached over 70 billion dollars [2].After the success of Bitcoin, blockchain has been applied in

many applications, such as access control systems [3], smartcontracts [4], [5], content delivery networks [6], cognitiveradio networks [7], and smart grid powered systems [8], [9].

The core issue of the blockchain is a computational processcalled mining, where the transaction records are added into theblockchain via the solution of computational difficult problem,i.e., the proof-of-work puzzle. Confirming and securing theintegrity and validity of transactions are processed by a setof participants called “miners”. The security of blockchaindirectly relies on the distributed consensus mechanism main-tained by these miners [10]. The typical consensus mechanismis introduced as follows. First, the miners consider and bundlea number of transactions that are processed to form a single“block”. The miner propagates its mined block to the restof the blockchain network as soon as it solves the puzzlein order to claim the mining reward. Then, this block isverified by the majority of miners in this network, i.e., tryingto reach consensus. After the propagated block reaches theconsensus, it is successfully added into the globally-accessibledistributed public ledger, i.e., blockchain. The miner whichmines a block receives the mining reward when the minedblock is successfully added to the blockchain. This consen-sus mechanism guarantees the security and dependability ofblockchain systems [11], [12].

However, blockchain has not been adopted widely in mobileapplications [13]. This is because blockchain mining needsto solve a proof-of-work puzzle, which is expensive to im-plement in mobile devices due to the high computing powerneeded. Thus, deploying blockchain in a mobile environmentis truly challenging. In this paper, we consider the edgecomputing as a network enabler for the mobile blockchain. Inparticular, we consider the price-based resource managementin mobile blockchain, in which an Edge computing ServiceProvider (ESP) is introduced to support proof-of-work puzzleoffloading [14] by using its edge computing nodes. Further,we propose a two-stage game model, i.e., the Stackelberggame. In the first stage, the edge computing service providersets the price and obtains the revenue from charging theminers for offloading the mining task. In the second stage,the miners decide on the service demand to purchase fromedge computing service provider. Specifically, we analyze twopricing schemes, i.e., uniform pricing in which a uniform unitprice is applied to all the miners and discriminatory pricingin which different unit prices are assigned to different miners.The uniform pricing is easier to be implemented as the ESP

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does not need to keep track of information of all miners, andcharging same prices which is fair to all miners. However, itmay not yield the highest profit compared with discriminatorypricing in which the price can be adjusted for an individualminer [15].

To the best of our knowledge, this is the first work toinvestigate the mobile blockchain with resource managementand pricing using game theory. The main contributions of thiswork are summarized as follows.

1) We formulate a pricing and service demand problem toanalyze the interactions among the ESP and miners. Inparticular, we adopt the two-stage Stackelberg game tomodel their interactions to maximize the profit of the ESPand the individual utilities of miners jointly for mobileblockchain applications.

2) Through backward induction, we derive a unique Nashequilibrium point among the miners in the second stage,and investigate the uniform pricing as well as discrimina-tory pricing for profit maximization of the ESP in the firststage. The Stackelberg equilibrium is derived analyticallyfor both the pricing schemes.

3) In particular, the existence and uniqueness of Stackelbergequilibrium are validated by identifying the best responsestrategies of the miners under the uniform pricing scheme.Likewise, the Stackelberg equilibrium is proved to existand be unique by capitalizing on the Variational Inequal-ity theory under discriminatory pricing scheme.

4) We conduct extensive numerical simulations to evaluatethe performance of the proposed price-based resourcemanagement for mobile blockchain. The results show thatthe discriminatory pricing helps the ESP to encouragemore service demand from the miners and achieve greaterprofit. Moreover, under uniform pricing, the serviceprovider has an incentive to set the maximum price forthe profit maximization.

5) Our work helps to achieve the proof-of-work puzzle of-floading and guide the ESP to extract the surplus throughcharging miners strategically. Further, we perform the realexperiment on mobile blockchain mining to validate theproposed analytical model.

The rest of the paper is organized as follows. Section IIpresents a review of the related work. We describe the systemmodel and formulate the two-stage Stackelberg game in Sec-tion III. In Section IV, we analyze the optimal service demandof miners as well as the profit maximization of the ESPusing backward induction for both uniform and discriminatorypricing schemes. We present the performance evaluations inSection V. Section VI concludes the paper with summary andfuture directions.

II. RELATED WORK

Recently, there have been several studies on mining schemesmanagement for blockchain network. In [16], the authorsdesigned a noncooperative game among the miners, i.e., theplayers. The miner’s strategy is to choose the number oftransactions to be included in a block. In the model, solving

the proof-of-work puzzle for mining is modeled as a Poissonprocess. The solution of the game is the Nash equilibriumwhich was derived only for two miners in [16]. Then, theauthors in [17] modeled the mining process as a sequentialgame where the miners compete for mining reward in sequen-tially among them. In the game model, the miners are assumedto be rational, and they have to choose whether or not topropagate their solution, i.e., the mined block. It is provedin [17] that there exists a multiplicity of Nash equilibrium.Further, it is found that not propagating is an optimal strategyunder certain conditions. Similar to that in [17], the authorsin [18] formulated the stochastic game for modeling themining process, where miners decide on which blocks toextend and whether to propagate the mined block. In particular,two game models in which miners play a complete informationstochastic game are studied. In the first model, each minerpropagates immediately the mined block that it mines. Thestrategy of each miner is to select an appropriate block tomine. In the second model, the miner selects which block tomine, but it may not propagate its mined block immediately.For both models, it is proved in [18] that when the number ofminers is sufficiently small, the Nash equilibrium with respectto mining behaviors exists.

Traditionally, miners mine blocks individually, which wecall the solo mining. The advantage of solo mining is that theminer obtains all the reward when it successfully mines theblock. Recently, the pool mining is introduced as a alternativeway for miners to pool their resources together for miningthe block, in order to obtain steady reward [19]. The authorsin [20] proposed the cooperative game based blockchainmining scheme, in which the pool mining was examined. Inparticular, the cooperative game theoretic tools are used tostudy which pools that the miners want to join and how theminers in the same pool share their reward. The interactions ofminers and pool are modeled as a coalitional game. It is provedthat there is no stable way to divide the payoff among theminers, i.e., the coalition structure has an empty core. Further,the proposed scheme is applied to the real-world Bitcoinnetwork environment. It is found in [20] that under any rewardallocation schemes, some miners always have the incentiveto switch to other pools for higher expected reward. Inspiredby [20], the authors in [21] further studied optimal pool miningmechanisms, in which the utility model and social welfare ofminers are considered. It is demonstrated that the geometricpay pooling strategy [22] achieves the optimal steady-stateutility for miners. The results in [21] can also be applied toother forms of mining systems.

In [23], the authors defined a novel model, in which thepools use some of their miners to infiltrate other pools andperform such an attack. The attacked pool shares its rewardwith the attacker, and so each of its miners in the pool earnsless reward. It is proved that the decision to attack or notis the miner’s dilemma and an instance of iterative prisoner’sdilemma [24]. In [25], the authors developed a novel incentivepayment mechanism for pool mining, in which the groupbargaining solution is adopted by considering peer-to-peer

Table INOTATIONS

Symbol DefinitionN , N Set of miners, and the total number of miners, respectivelyxi Edge computing service demand of miner ix The minimum service requirement for all the minersx The maximum service demand for all the minersx The service demand profile of all the miners

x−iThe service demand profile of all other miners exceptminer i

tiThe size of block mined by miner i, i.e., the numberof transactions in its block

αi The relative computing or hash power of miner iui The utility of miner i

pThe unit price set by the edge computing serviceprovider for all the miners under uniform pricing

piThe unit price set by the edge computing serviceprovider for miner i under discriminatory pricing

p The maximum price constriantR Fixed reward of successfully mining a blockr Variable reward factorΠ The profit of edge computing service providerT The average time taken to mine a block, λ = 1/Tc The service cost factorτ The block propagation time

PiThe probability of miner i successfully mining a blockor winning the mining reward

Porphan The probability that the block is orphanedz The propagation delay factor

relationship of miners. In the proposed scheme, miners aregrouped as a miner pool based on contribution levels. There-fore, multiple groups with different computing contributions,i.e., mining pools are formulated. As such, the bargainingoccurs within individual miners in each mining pool and acrossmultiple pools simultaneously. It is proved in [25] that theunique Nash bargaining solution [26] exists in the proposedscheme. The authors in [27] proposed a new computationalpower splitting game for the blockchain mining. This gamemodel includes multiple incentivized miners to compete insolving proof-of-work puzzle in exchange for mining reward.Based on the game model, the miner with computation powerplay a game of solving the puzzle through distributing itscomputing power into different pools such that its expectedreward is maximum. However, it is shown that this game hasno pure Nash equilibrium with pure strategies. Some findingsin [27] are consistent with those in [23].

Nevertheless, all of the above works studied the miningmanagement problem by using dedicated nodes, without con-sidering the operation of blockchain with mobile devices,i.e., mobile blockchain. Therefore, this motivates us to takea step further to reconsider the mining strategies as well asresource management in mobile environment, thereby openingnew opportunities for the development of blockchain in mobileservices and applications.

III. SYSTEM MODEL AND GAME FORMULATION

In this section, we first propose the system model of themobile blockchain network under our consideration. Then, wepresent the Stackelberg game formulation for the price-based

Edge computing service provider

Data synchronization

Offloading Offloading

Edge computing

unit

Edge computing

unit

Mobile device running

blockchain app

Figure 1. System model.

edge computing resource management for mobile blockchainapplications.

A. Mobile Blockchain

Blockchain can be employed for mobile applications tosupport peer-to-peer (P2P) secure data services, e.g., P2P filetransfer and P2P direct payment [28], [29]. To create a chainof blocks, the mining needs to be done. The mining process isused to confirm and secure transactions to be stored in a block.This mining process is organized as a speed game among theminers with different computing powers. The problem is aproof-of-work puzzle, which is expensive to solve and takeshigh computing power, time, and energy. In brief, this proof-of-work puzzle includes considering a set of transactions thatare present in the network, solving a mathematical problem1

that depends on this set and propagating the result to theblockchain network for this solution to be checked and reachconsensus. Once all these steps are done successfully, the setof transactions proposed by the miner forms a block that isappended to the current blockchain. The first miner whichsuccessfully obtains the solution of the puzzle and reach theconsensus is considered to be the winner to which a certainreward2 is given. This is referred to as the speed game amongthe miners. The earlier studies, e.g., in [29], have found that thepuzzle cannot be handled efficiently using mobile devices. Inthe following, we introduce using the edge computing conceptfor offloading the mobile blockchain processing, e.g., miningprocess. The more details of blockchain networks and themining process can be found in [12].

B. Chain Mining with Edge Computing

We consider a mobile blockchain application, e.g., as pre-sented in [29]. There is a group of N mobile users, i.e., theminers, the set of which is denoted by N = {1, . . . , N}. Each

1Solving the problem needs hashing the blockchain information by theminers, and thus we use the hash power and computing power interchangeablythroughout the paper.

2The detailed discussion of reward mechanism can be found in [30].

mobile user runs mobile blockchain applications to record thetransactions performed in the group. There is an Edge com-puting Service Provider (ESP) deploying the edge computingunits/nodes for the miners. The aforementioned proof-of-workpuzzle can be offloaded to a nearby edge computing unit. Inparticular, the edge computing units offer computing resourcesto mobile users, i.e., the miners which will be priced bythe ESP. Figure. 1. shows the system model of the mobileblockchain network under our consideration. Note that weassume the link between the mobile nodes and edge computingunits are always secured which can be achieved by somesecurity solutions.

The ESP, i.e., the seller, sells the edge computing services,and the miner, i.e., the buyers, access and consume this servicefrom the nearby edge computing unit. Each miner i ∈ Ndetermines their individual service demand, denoted by xi.Additionally, we consider xi ∈ [x, x], in which x is theminimum service demand, e.g., for blockchain data synchro-nization, and x is the maximum service demand governed bythe ESP. Note that each miner has no incentive to unboundedlyincrease its service demand due to its financial burden. Then,let x ∆

= (x1, . . . , xN ) and x−i represent the service demandprofile of all the miners and all other miners except miner i,respectively. As such, the miner i ∈ N with the servicedemand xi has a relative computing power (hash power) αiwith respect to the total hash power of the network, which isdefined as follows:

αi(xi,x−i) =xi∑j∈N xj

, αi > 0, (1)

such that∑j∈N αj = 1.

In the mobile blockchain network, miners compete againsteach other in order to be the first one to solve the proof-of-work puzzle and receive the reward from the speed gameaccordingly. The occurrence of solving the puzzle can bemodeled as a random variable following a Poisson processwith mean λ = 1

600 sec [16]. Note that our model is generalthat can be applied with other values of λ easily. The setof transactions to be included in a block chosen by mineri is denoted as ti. Once the miner successfully solves thepuzzle, the miner needs to propagate its solution to the wholemobile blockchain network and its solution needs to reachconsensus. Because there is no centralized authority to verifythe validate a newly mined block, a mechanism for reachingnetwork consensus must be employed. In this mechanism, theverification needs to be processed by other miners before thenew mined block is appended to the current blockchain.

The first miner to successfully mine a block that reachesconsensus earns the reward. The reward consists of a fixedreward denoted by R, and a variable reward which is definedas rti, where r denotes a given variable reward factor and tidenotes the number of transactions included in the block minedby miner i [16], [31]. Additionally, the process of solving thepuzzle incurs an associated cost, i.e., the payment from mineri to the ESP, pi. The objective of the miners is to maximize

their individual expected utility, and for miner i, it is definedas follows:

ui = (R+ rti)Pi (αi(xi,x−i), ti)− pixi, (2)

where P (αi(xi,x−i), ti) is the probability that miner i suc-cessfully mines the block and its solutions reach consensus,i.e., miner i wins the mining reward.

The process of successfully mining a block consists of twosteps, i.e., the mining step and the propagation step. In themining step, the probability that miner i mines the blockis directly proportional to its relative computing power αi.Furthermore, there are diminishing chances of wining if oneminer chooses to propagate a block that propagates slowlyto other miners in the propagation step. In other words, eventhough one miner may find the first valid block, if its minedblock is large, then this block will be likely to be discardedbecause of long latency, which is called orphaning [16].Considering this fact, the probability of successful mining byminer i is discounted by the chances that the block is orphaned,Porphan(ti), which is expressed by

Pi(αi(xi,x−i), ti) = αi(1− Porphan(ti)). (3)

Using the fact that block mining times follow the Poissondistribution aforementioned, the orphaning probability is ap-proximated as [32], [33]:

Porphan(ti) = 1− e−λτ(ti), (4)

where τ(ti) is the block propagation time, which is a functionof the block size. In other words, the propagation time neededfor a block to reach consensus is dependent on its size ti, i.e.,the number of transactions in it [16], [34]. Thus, the biggerthe block is, the more time needed to propagate the block tothe whole mobile blockchain network [35]. Same as [16], weassume this time function is linear, i.e., τ(ti) = z × ti withz > 0 represents a given delay factor. Note that this linearapproximation is acceptable according to the numerical resultsfrom [16], [36]. Additionally, it would be more appropriateto add a constant term in this function [35], but apparentlythis constant term has no effect on our subsequent analyticalresults. Thus, the probability that the miner i successfullymines a block and its solution reaches consensus is expressedas follows:

Pi(αi(xi,x−i), ti) = αie−λzti , (5)

where αi(xi,x−i) is given in (1).

C. Two-Stage Stackelberg Game Formulation

The interaction between the ESP and miners can be modeledas a two-stage Stackelberg game, as illustrated in Fig. 2.The ESP, i.e., the leader, sets the price in the upper StageI. The miners, i.e., the followers, decide on their optimalcomputing service demand for offloading in the lower StageII, being aware of the price set by the ESP. By using backwardinduction, we formulate the optimization problems for theleader and followers as follows.

Lower Stage II

(followers)

Upper Stage I

(leader)

Payment Edge computing

service

Miner 1 ….Miner 1 ….

Noncooperative game

Edge Computing Service Provider Edge Computing Service Provider

Miner 2 Miner 3 Miner N

Stackelberg game

Figure 2. Two-stage Stackelberg game model of the interactions among theESP and miners in the mobile blockchain.

1) Miners’ mining strategies in Stage II: Given the pricingof the ESP and other miners’ strategies, the miner i determinesits computing service demand for its hash power maximizingthe expected utility which is given as:

ui(xi,x−i, pi) = (R+ rti)xi∑j∈N xj

e−λzti − pixi, (6)

where pi is the price per unit for service demand of miner i.The miner sub-game problem can be written as follows:

Problem 1. (Miner i sub-game):

maximizexi

ui(xi,x−i, pi)

subject to xi ∈ [x, x].(7)

2) ESP’s pricing strategies in Stage I: The profit of theESP is the revenue obtained from charging the miners forcomputing service minus the service cost. The service cost isdirectly related to the time that the miner takes to mine a block,the cost of electricity, c, and the other cost that is a function ofthe service demand xi. Therefore, the ESP decides the pricingwithin the strategy space {p = [pi]i∈N : 0 ≤ pi ≤ p} tomaximize its profit which is represented as:

Π(p,x) =∑i∈N

pixi −∑i∈N

cTxi. (8)

Note that practically the price is bounded by maximum priceconstraint that is denoted by p. Then, the profit maximizationproblem of the ESP is formulated as follows.

Problem 2. (ESP sub-game):

maximizep

Π(p,x)

subject to 0 ≤ pi ≤ p.(9)

Problem 1 and Problem 2 together form the Stackelberggame, and the objective of this game is to find the Stackelbergequilibrium. The Stackelberg equilibrium is a point where thepayoff of the leader is maximized given that the followersadopt their best responses, i.e., the Nash eqilibrium [37]. Inour problem, the Stackelberg equilibrium can be written asfollows.

Definition 1. Let x∗ and p∗ denote the optimal service de-mand vector of all the miners and optimal unit price vector ofedge computing service, respectively. Then, the point (x∗,p∗)is the Stackelberg equilibrium if the following conditions,

Π(p∗,x∗) ≥ Π(p,x∗) (10)

and

ui(x∗i ,x∗−i,p

∗) ≥ ui(xi,x∗−i,p∗),∀xi ≥ 0,∀i (11)

are satisfied, where x∗−i is the best response service demandvector for all the miners except miner i.

Note that the same or different prices can be applied tothe miners, which we refer to them as the uniform anddiscriminatory pricing schemes, respectively. In the following,we investigate these two pricing schemes for resource man-agement in mobile blockchain.

IV. EQUILIBRIUM ANALYSIS FOR EDGE COMPUTINGRESOURCE MANAGEMENT

In this section, we propose the uniform pricing and discrim-inatory pricing schemes for resource management in mobileblockchain. We then analyze the optimal service demand ofminers as well as the profit maximization of the ESP underboth pricing schemes.

A. Uniform Pricing Scheme

We first consider the uniform pricing scheme, in whichthe ESP charges all the miners the same unit price for theircomputing service demand, i.e., pi = p, ∀i. Given the payofffunctions defined in Section III, we use backward inductionto analyze the Stackelberg game.

1) Stage II: Miners’ Demand Game: Given the price pdecided by the ESP, in Stage II, the miners compete with eachother to maximize their own utility by choosing their individ-ual service demand, which forms the noncooperative Miners’Demand Game (MDG) Gu = {N , {xi}i∈N , {ui}i∈N }, whereN is the set of miners, {xi}i∈N is the strategy set, and ui isthe utility, i.e., payoff, function of miner i. Specifically, eachminer i ∈ N selects its strategy to maximize its utility functionui(xi,x−i, p). We next analyze the existence and uniquenessof the Nash equilibrium in the MDG.

Definition 2. A demand vector x∗ = (x∗1, . . . , x∗N ) is the Nash

equilibrium of the MDG Gu = {N , {xi}i∈N , {ui}i∈N }, if, forevery miner i ∈ N , ui(x∗i ,x

∗−i, p) ≥ ui(xi

′,x∗−i, p) for allxi′ ∈ [x, x], where ui(xi,x−i) is the resulting utility of the

miner i, given the other miners’ demand x−i.

Theorem 1. A Nash equilibrium exists in MDG Gu ={N , {xi}i∈N , {ui}i∈N }.

Proof. Firstly, the strategy space for each miner is defined tobe [x, x], which is a non-empty, convex, compact subset ofthe Euclidean space. From (6), ui is apparently continuousin [x, x]. Then, we take the first order and second orderderivatives of (6) with respect to xi to prove its concavity,which can be written as follows:

∂ui∂xi

= (R+ rti)e−λzti ∂αi

∂xi− p, (13)

and∂2ui∂xi2

= (R+ rti)e−λzti ∂

2αi∂xi2

< 0, (14)

xi∗ = Fi(x) =

x,

√(R+rti)

∑i6=j

xj

pe−λzti−∑i6=j

xj < x√(R+rti)

∑i6=j

xj

pe−λzti−∑i 6=j

xj , x ≤

√(R+rti)

∑i6=j

xj

pe−λzti−∑i 6=j

xj ≤ x

x,

√(R+rti)

∑i6=j

xj

pe−λzti−∑i6=j

xj > x

. (12)

where

∂αi∂xi

=

∑i 6=j

xj(∑i∈N

xj

)2 > 0, (15)

and

∂2αi∂xi2

= −2

∑i 6=j

xj(∑i∈N

xj

)3 < 0. (16)

Therefore, we have proved that ui is strictly concave withrespect to xi. Accordingly, the Nash equilibrium exists in thisnoncooperative MDG Gu [37]. The proof is now completed.

Further, based on the first order derivative condition, wehave

∂ui∂xi

= (R+ rti)e−λzti ∂αi

∂xi− p = 0, (17)

and we obtain the best response function of miner i bysolving (17), as shown in (12).

Theorem 2. The uniqueness of the Nash equilibrium inthe noncooperative MDG is guaranteed given the followingcondition

2(N − 1)e−λzti

R+ rti<∑i∈N

e−λzti

R+ rti(18)

is satisfied.

Proof. Let x∗ denote the Nash equilibrium of the MDG. Bydefinition, the Nash equilibrium needs to satisfy x = F (x),in which F (x) = (F1(x),F2(x), . . . ,FN (x)). In particular,Fi(x) is the best response function of miner i, given thedemand strategies of other miners. The uniqueness of the Nashequilibrium can be proved by showing that the best responsefunction of miner i, i.e., as given in (12), is the standardfunction [37].

Definition 3. A function F (x) is a standard function whenthe following properties are guaranteed [37]:(1) Positivity: F (x) > 0;(2) Monotonicity: If x ≤ x′, then F (x) ≤ F (x′);(3) Scalability: For all λ > 1, λF (x) > F (λx).

Firstly, for the positivity, under the condition in (18), wehave (from Lemma 1)∑

i 6=j

xj <R+ rtipe−λzti

<R+ rti4pe−λzti

, (19)

then we can conclude that

∑i 6=j

xj <

√√√√R+ rti∑i 6=j

xj

pe−λzti. (20)

Thus, we can prove that

Fi(x) =

√√√√R+ rti∑i6=j

xj

pe−λzti−∑i 6=j

xj > 0, (21)

which is the positivity condition. Secondly, we prove themonotonicity of (12). Let x′ > x, we can further simplifythe expression of Fi(x′) − Fi(x), which is shown in (22).In particular, we have

√∑i 6=j

x′j −√∑i 6=j

xj > 0, and we can

easily verify that√R+ rtipe−λzti

−√∑

i 6=j

x′j −√∑

i 6=j

xj ∈√R+ rtipe−λzti

− 2

√∑i6=j

x′j ,

√R+ rtipe−λzti

− 2

√∑i 6=j

xj

. (24)

Under the condition in (32), we can prove that√R+ rtipe−λzti

− 2

√∑i 6=j

xj > 0,∀xj . (25)

Thus, the best response function of miner i in (12) is alwayspositive.

At last, as for scalability, we need to prove that λF (x) >F (λx), for λ > 1. The steps of proving the positivity ofλF(x) − F(λx) are shown in (23). Therefore, λF (x) >F (λx) is always satisfied for λ > 1. Until now, we haveproved that the best response function in (12) satisfies threeproperties described in Definition 2. Therefore, the Nashequilibrium of MDG Gu = {N , {xi}i∈N , {ui}i∈N } is unique.The proof is now completed.

Fi(x′)−Fi(x) =

√√√√R+ rti∑i 6=j

x′j

pe−λzti−∑i 6=j

x′j −

√√√√ (R+ rti)∑i 6=j

xj

pe−λzti−∑i6=j

xj

=

√R+ rtipe−λzti

√∑i 6=j

x′j −√∑

i6=j

xj

−∑i 6=j

x′j −∑i6=j

xj

=

√R+ rtipe−λzti

−√∑

i 6=j

x′j −√∑

i6=j

xj

√∑i 6=j

x′j −√∑

i 6=j

xj

. (22)

λFi(x)−Fi(λx) = λ

√√√√ (R+ rti)∑i6=j

xj

pe−λzti− λ

∑i 6=j

xj −

√√√√ (R+ rti)∑i 6=j

λxj

pe−λzti−∑i 6=j

λxj

=(λ−√λ)√√√√ (R+ rti)

∑i 6=j

xj

pe−λzti> 0,∀λ > 1. (23)

Theorem 3. The unique Nash equilibrium for miner i in theMDG is given by

xi∗ =

N − 1∑j∈N

pe−λztj

R+rtj

−

N − 1∑j∈N

pe−λztj

R+rtj

2

pe−λzti

R+ rti,∀i, (26)

provided that the condition in (18) holds.

Proof. According to (13), for each miner i, we have themathematical expression∑

i 6=jxj( ∑

j∈Nxj

)2 =pe−λzti

R+ rti. (27)

Then, we calculate the summation of this expression for allthe miners as follows:

(N − 1)∑j∈N

xj( ∑j∈N

xj

)2 =∑i∈N

pe−λzti

R+ rti, (28)

which means (N−1)∑j∈N

xj=∑i∈N

pe−λzti

R+rti. Thus, we have

∑j∈N

xj =N − 1∑

i∈N

pe−λztiR+rti

. (29)

Recall from (12), according to the first order derivative con-dition, we have

∑j∈N

xj =

√√√√ (R+ rti)∑i 6=j

xj

pe−λzti. (30)

By substituting (30) into (29), we have

N − 1∑i∈N

pe−λztiR+rti

=

√√√√√√R+ rtipe−λzti

N − 1∑i∈N

pe−λztiR+rti

− xi

. (31)

After squaring both sides, we have

N−1∑i∈N

pe−λztiR+rti

2

=

R+rtipe−λzti

N−1∑i∈N

pe−λztiR+rti

− xi

. With simple transformations,

we obtain the Nash equilibrium for miner i as shownin (26).

Lemma 1. Given

2(N − 1)e−λzti

R+ rti<∑i∈N

e−λzti

R+ rti, (32)

the following condition∑i 6=j

xj <R+ rti4pe−λzti

(33)

is satisfied.

Proof. According to (26) and (29), we can obtain

∑j 6=i

xj =

N − 1∑j∈N

pe−λztj

R+rtj

2

pe−λzti

R+ rti. (34)

After substituting (32) into (34), we have

2(N − 1)pe−λzti

R+ rti<∑i∈N

pe−λzti

R+ rti, (35)

which means that the condition in (32) needs to be ensured. Onthe contrary, if the condition in (32) holds, then, the conditionin (34) is satisfied. The proof is now completed.

Generally, we can use the best-response dynamics for ob-taining the Nash equilibrium of the N-player noncooperativegame in Stage II [37]. In the following, we analyze the profitmaximization of the ESP in Stage I under uniform pricing.

2) Stage I: ESP’s Profit Maximization: Based on theNash equilibrium of the computing service demand in theMDG Gu = {N , {xi}i∈N , {ui}i∈N } in Stage II, the leader ofthe Stackelberg game, i.e., the ESP, can optimize its pricingstrategy in Stage I to maximize its profit defined in (8). Thus,the optimal pricing can be formulated as an optimizationproblem. By substituting (26) into (8), the profit maximizationof the ESP is simplified as follows:

maximizep>0

Π(p) = (p− cT )N − 1∑

j∈N

pe−λztj

R+rtj

subject to 0 ≤ p ≤ p.

(36)

Theorem 4. Under uniform pricing, the ESP achieves theglobally optimal profit, i.e., profit maximization, under theunique optimal price.

Proof. From (36), we have

Π(p) =p− cTp

N − 1∑j∈N

e−λztj

R+rtj

. (37)

The first and second derivatives of profit Π(p) with respect toprice p are given as follows:

dΠ(p)

dp=cT

p2

N − 1∑j∈N

e−λztj

R+rtj

(38)

andd2Π(p)

dp2= −2cT

p2

N − 1∑j∈N

e−λztj

R+rtj

< 0. (39)

Due to the negativity of (39), the strict concavity of theobjective function is ensured. Thus, the ESP is able to achievethe maximum profit with the unique optimal price. The proofis now completed.

Note that the profit maximization defined in (36) is a convexoptimization problem, and thus it can be solved by stan-dard convex optimization algorithms, e.g., gradient assistedbinary search. Under uniform pricing, we have proved thatthe Nash equilibrium in Stage II is unique and the optimalprice in Stage I is also unique. Thus, we can conclude thatthe Stackelberg equilibrium is unique and accordingly thebest-response dynamics algorithm can achieve this uniqueStackelberg equilibrium [37].

B. Discriminatory Pricing Scheme

Then, we consider the discriminatory pricing scheme, inwhich the ESP is able to set different unit prices of servicedemand for different miners. Again, we use the backwardinduction to analyze the optimal service demand of minersand the profit maximization of the ESP.

1) Stage II: Miners’ Demand Game: Under discriminatorypricing scheme, the strategy space of the ESP becomes {p =[pi]i∈N : 0 ≤ pi ≤ p}. Recall that we prove the existenceand uniqueness of MDG Gu = {N , {xi}i∈N , {ui}i∈N }, giventhe fixed price from the ESP. Thus, under discriminatorypricing, the existence and uniqueness of the MDG can be stillguaranteed. With minor change from Theorem 3, we have thefollowing theorem immediately.

Theorem 5. Under uniform pricing, the unique Nash equilib-rium demand of miner i can be obtained as follows:

xi∗ =

N − 1∑j∈N

pje−λztj

R+rtj

−

N − 1∑j∈N

pje−λztj

R+rtj

2

pie−λzti

R+ rti,∀i, (40)

if the following condition

2(N − 1)pie−λzti

R+ rti<∑j∈N

pje−λztj

R+ rtj(41)

holds.

Proof. The steps of proof are similar to those in the case ofuniform pricing as shown in Section IV-A1, and thus we omitthem for brevity.

We next analyze the profit maximization of the ESP inStage I under discriminatory pricing to further investigate theStackelberg equilibrium.

2) Stage I: ESP’s Profit Maximization: Similar to that inSection IV-A2, we analyze the profit maximization with theanalytical result from Theorem 5, i.e., the Nash equilibriumof the computing service demand in Stage II. After substitut-ing (40) into (8), we have the following optimization,

maximizep>0

Π(p) =∑i∈N

pi − cT N − 1∑j∈N

pje−λztj

R+rtj

subject to 0 ≤ pi ≤ p, ∀i.

(42)

Theorem 6. Π(p) is concave on each pi, when∑i6=j

(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)≤ 0, and decreasing on

each pi when∑i 6=j

(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)> 0, provided that

the following condition

piai≥

∑j∈N

pjaj

(N − 1)2 (45)

g(p) =

∑j 6=h

ah N−1∑

h∈N

phah

−

(N−1∑

h∈N

phah

)2

phah

N−1∑h∈N

phah

−

(N−1∑

h∈N

phah

)2

pjaj

(

N−1∑j∈N

pjaj

)2

=∑j 6=h

ah1− ph

ah

N − 1∑h∈N

phah

1− pjaj

N − 1∑h∈N

phah

. (43)

∂g(p)

∂pi=∑j 6=i

(ai + aj)

−N−1

ai

∑h6=i

phah( ∑

h∈N

phah

)2

1− N − 1∑h∈N

phah

pjaj

+

N−1ai

pjaj( ∑

h∈N

phah

)2

1− N − 1∑h∈N

phah

piai

. (44)

is satisfied, where ai = (R+ rti)e−λzti .

Proof. We firstly decompose the objective function in (42) intotwo parts, namely,

∑i

cTx∗i and∑i

pix∗i . Then, we analyze the

properties of each part. We define

f(p) = −cTx∗i = −cT N − 1∑j∈N

pje−λztj

R+rtj

. (46)

Let aj = (R + rtj)e−λztj , and we have f(p) = −cT (N−1)∑

j∈N

pjaj

.

Then, we obtain the first and the second partial derivativesof (46) with respect to pi as follows.

∂f(p)

∂pi=

(N − 1)cT

ai

( ∑j∈N

pjaj

)2 , (47)

∂2f(p)

∂pi2=−2(N − 1)cT

ai2

( ∑j∈N

pjaj

)3 . (48)

Further, we have

∂f(p)

∂pipj=−2(N − 1)cT

aiaj

( ∑j∈N

pjaj

)3 . (49)

Thus, we can obtain the Hessian matrix of f(p), which isexpressed as:

∇2f(p) =−2(N − 1)cT( ∑

j∈N

pjaj

)3

1a12

1a1a2

· · · 1a1aN

1a2a1

1a22 · · · 1

a2aN...

.... . .

...1

aNa11

aNa2· · · 1

aN 2

.(50)

For each i ∈ N , we have 1ai2

> 0. Thus, the diagonal elementsof the Hessian matrix are all larger than zero, and the principle

minors are equal to zero. Therefore, the Hessian matrix off(p) is semi-negative definite.

Then, we analyze the properties of∑i

pix∗i . We first define

g(p) =∑i∈N

pixi∗ =

∑j 6=i

aixixj(∑j 6=i

xj

)2 . (51)

By substituting (40) into (51), we can obtain the finalexpression for g(p), which can be rewritten as in (43).Then, we derive the first order and the second partialderivatives of (43) with respect to pi as shown in (44)

and (53). Since we have xi = N−1∑h∈N

phah

− piai

(N−1∑

h∈N

phah

)2

=

N−1∑h∈N

phah

(1− N−1∑

h∈N

phah

piai

)> 0, 1− N−1∑

h∈N

phah

piai

> 0. When

∑i6=j

(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)≤ 0, it is observed that ∂

2g(p)∂pi2

<

0, i.e., g(p) is concave on each pi. Now we prove that Π(p) isa monotonically decreasing function with respect to pi, when∑i6=j

(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)> 0. The steps are shown in (52),

where pmin = min{p1, p2, . . . , pN}. Practically, pmin > cT .Thus, with some manipulations, we can prove ∂Π

∂pi< 0 when∑

i6=j(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)> 0, if the condition in (45)

holds. The proof is now completed.

Theorem 7. Under discriminatory pricing, the ESP achievesthe profit maximization by finding the unique optimal pricingvector.

Proof. From Theorem 6, we know that Π(p) is concave

on each pi, when∑i 6=j

(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)≤ 0, and

∂Π(p)

∂pi=∑j 6=i

(ai + aj)

N−1ai

∑h 6=i

phah( ∑

h∈N

phah

)2

1−N − 1∑

h∈N

phah

pj

aj

+

N−1aiaj

pj( ∑h∈N

phah

)2

1−N − 1∑

h∈N

phah

pi

ai

+

N−1ai

cT( ∑h∈N

phah

)2

=

N−1ai( ∑

h∈N

phah

)2

∑j 6=i

(ai + aj)

−∑h 6=i

ph

ah

1−N − 1∑

h∈N

phah

pj

aj

+pj

aj

1−N − 1∑

h∈N

phah

pi

ai

+ cT

≤N−1ai( ∑

h∈N

phah

)2

∑j 6=i

(ai + aj)

−∑h∈N

ph

ah

1−N − 1∑

h∈N

phah

pj

aj

+pj

aj

1−N − 1∑

h∈N

phah

pi

ai

+ cT

=

N−1ai( ∑

h∈N

phah

)2

∑j 6=i

(ai + aj)

−∑h∈N

ph

ah

1−N − 1∑

h∈N

phah

pj

aj

︸ ︷︷ ︸

>0

−N − 1∑

h∈N

phah

pi

ai

pj

aj

+ cT

= −N−1ai( ∑

h∈N

phah

)2

∑j 6=i

(ai + aj)

∑h∈N

ph

ah

1−N∑

h∈N

phah

pj

aj

︸ ︷︷ ︸<0

+

N−1ai( ∑

h∈N

phah

)2

cT −∑j 6=i

(ai + aj)N − 1∑

h∈N

phah

pi

ai

pj

aj

= −N−1ai( ∑

h∈N

phah

)2

∑j 6=i

(ai + aj)

−∑h∈N

ph

ah

1−N∑

h∈N

phah

pj

aj

+

N−1ai( ∑

h∈N

phah

)2

cT −∑j 6=i

ai + aj

aj︸ ︷︷ ︸<1

N − 1∑h∈N

phah

pipj

ai

≤ −N−1ai( ∑

h∈N

phah

)2

∑j 6=i

(ai + aj)

−∑h∈N

ph

ah

1−N∑

h∈N

phah

pj

aj

+

N−1ai( ∑

h∈N

phah

)2

cT − pminN − 1∑

h∈N

phah

N − 1pi

ai

= −N−1ai( ∑

h∈N

phah

)2

∑j 6=i

(ai + aj)

−∑h∈N

ph

ah

1−N∑

h∈N

phah

pj

aj

︸ ︷︷ ︸<0

+

N−1ai( ∑

h∈N

phah

)2

cT − pmin(N − 1)2∑h∈N

phah

pi

ai

︸ ︷︷ ︸

<0

< 0. (52)

decreasing on each pi when∑i 6=j

(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)> 0.

In other words, when Π(p) is concave on pi, pi needs tobe smaller than a certain threshold, and Π(p) is decreasingon pi when pi is larger than this threshold. Then, it can beconcluded that if the price is higher than the threshold, theminer is not willing to purchase the computing service fromthe ESP. Therefore, we know that the optimal value of profitof the ESP, i.e., Π∗(p) is achieved in the concave parts when∑i 6=j

(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)≤ 0. Clearly, the maximization

of profit Π(p) is achieved either in the boundary of domainarea or in the local maximization point. Since we know thatthe optimal value of profit, i.e., Π∗(p) is achieved in theinterior area, and thus p∗ exists. In the following, we prove thatthere exists at most one optimal solution by using Variational

Inequality theory [38], from which the uniqueness of theoptimal solution, i.e., the Stackelberg equilibrium, follows.

Let the set K =

{p =

[p1, . . . , pN ]>∣∣∣∣∑i 6=j

(ai + aj)

(1−

Npjaj∑

j∈N

pjaj

)≤ 0,∀i ∈ N

}.

The constraint can be rewritten as follows:

∑i 6=j

((ai + aj)

(∑h∈N

phah−N pj

aj

))≤ 0. (56)

Thus, we redefine the set K as{p = [p1, . . . , pN ]>∣∣∣∣∑

i 6=j

((ai + aj)

( ∑h∈N

phah−N pj

aj

))≤ 0,∀i ∈ N

}. Then,

∂2g(p)

∂pi2=

∑j 6=i

(ai + aj)

2N−1

ai2

∑h6=i

phah( ∑

h∈N

phah

)3

1−N − 1∑

h∈N

phah

pj

aj

− 2

N−1ai

∑h 6=i

phah( ∑

h∈N

phah

)2

N−1ai

pjaj( ∑

h∈N

phah

)2

−2N−1

ai2pjaj( ∑

h∈N

phah

)3

1−N − 1∑

h∈N

phah

pi

ai

=∑j 6=i

(ai + aj)

2N−1

ai2

∑h6=i

phah( ∑

h∈N

phah

)3

1− 2N − 1∑

h∈N

phah

pj

aj

− 2N−1ai2

pjaj( ∑

h∈N

phah

)3

1−N − 1∑

h∈N

phah

pi

ai

. (53)

∂2g(p)

∂pi2=

2N−1ai2

∑h6=i

phah( ∑

h∈N

phah

)3

∑j 6=i

(ai + aj)

1− 2N − 1∑

h∈N

phah

pj

aj

− 2N−1

ai2( ∑h∈N

phah

)3

∑j 6=i

(ai + aj)pj

aj

1−N − 1∑

h∈N

phah

pi

ai

≤2N−1

ai2

∑h6=i

phah( ∑

h∈N

phah

)3

∑j 6=i

(ai + aj)

1−N∑

h∈N

phah

pj

aj

︸ ︷︷ ︸≤0

−2N−1

ai2( ∑h∈N

phah

)3

∑j 6=i

(ai + aj)

pj

aj

1−N − 1∑

h∈N

phah

pi

ai

︸ ︷︷ ︸

≥0

. (54)

∑i 6=j

((ai + aj)

(∑h∈N

λp′

h + (1− λ)p′′

h

ah−N

λp′

j + (1− λ)p′′

j

aj

))

=∑i 6=j

((ai + aj)

(λ∑h∈N

p′

h

ah− (1− λ)

∑h∈N

p′′

h

ah− λN

p′

j

aj− (1− λ)N

p′′

j

aj

))

= λ∑i 6=j

((ai + aj)

(∑h∈N

p′

h

ah−N

p′′

j

aj

))+ (1− λ)

∑i 6=j

((ai + aj)

((1− λ)

∑h∈N

p′

h

ah−N

p′′

j

aj

))≤ 0. (55)

we formulate an equivalent problem to (42) as follows:

minimizep>0

−Π(p)

subject to p ∈ K.(57)

Let F (p) = ∇ (−Π(p)) = −[∇piΠ]>i∈N . Accordingly, the

optimization problem in (57) is equivalent to find a point setp∗ ∈ K, such that (p− p∗)F (p∗) ≥ 0,∀p ∈ K, which is theVariational Inequality (VI) problem: VI(K, F ).

Definition 4. If F is strictly monotone on K, then VI(K, F )has at most one solution, where K ∈ RN is a convex closedset, and the mapping F : K 7→ RN is continuous [38].

Let λ ∈ (0, 1), p′,p′′ ∈ K, it can be concluded that λp′ +(1 − λ)p′ ∈ K, which is shown in (55). Accordingly, K is aconvex and closed set. To prove that the mapping F : K 7→RN is strictly monotone on K, we check the positivity of(p′ − p′′)>(F (p′) − F (p′′)),∀p′,p′′ ∈ K and p′ 6= p′′. Weknow

(p′ − p′′)>(F (p′)− F (p′′)) =∑i∈N

((p′i − p′′i )

(− ∇piΠ|pi=p′i + ∇piΠ|pi=p′′i

)), (58)

and from Theorem 6, we have

∂2Π(p)

∂pi2=∂2(f(p) + g(p))

∂pi2< 0. (59)

Thus,∇piΠ is decreasing on each pi, and −∇piΠ is increasingon each pi. It can be concluded that

− ∇piΠ|pi=p′i + ∇piΠ|pi=p′′i =

{≥ 0, p′i ≥ p′′i< 0, p′i < p′′i .

(60)

Then, we have((p′i − p′′i )

(− ∇piΠ|pi=p′i + ∇piΠ|pi=p′′i

))≥ 0,∀i ∈ N ,

(61)and we know p′ 6= p′′, and accordingly there exists atleast one j ∈ N which satisfies the constraint in (61).Therefore, we have proved that F is strictly monotone on Kand continuous. Until now, we have proved that VI(K, F ) hasat most one solution according to Definition 4 in [38]. Thus,the equivalent problem admits at most one optimal solution.Since we know the existence of a single optimal solution, andthus the uniqueness of the optimal solution is validated. Theproof is now completed.

Figure 3. Real mobile blockchain mining experimental setup with Ethereumwhich is a popular open ledger.

Similar to that in Section IV-A, we can apply the low-complexity gradient based searching algorithm to achieve themaximized profit Π(p) of the ESP. In particular, we adoptAlgorithm 1 to obtain the unique Stackelberg equilibrium, un-der which the ESP achieves the profit maximization accordingto Theorem 7. The basic description is explained as follows:for the given prices imposed by the ESP, the followers’ sub-game is solved first. After substituting the best responses ofthe followers’ sub-game into the leader sub-game, the optimalprices can be obtained by a gradient-based algorithm. Thesimilar algorithm can be used for uniform pricing as well.

Algorithm 1 Gradient iterative algorithm to find Stackelbergequilibrium under discriminatory pricing

1: Initialization:Select initial input p = [pi]i∈N where pi ∈ [0, p], k ← 1,precision threshold ε;

2: repeat3: Each miner i decides its computing service demand x[k]

i

based on (12);4: ESP updates the prices using a gradient assisted search-

ing algorithm, i.e.,

p(t+ 1) = p(t) + µ∇Π(p(t)), (62)

where µ is the step size of the price update andµ∇Π(p(t)) is the gradient with ∂Π(p(t))

∂p(t) . The priceinformation is sent to all miners;

5: k ← k + 1;6: until ‖

p[k]−p[k−1]‖1

‖p[k−1]‖1

< ε

7: Output: optimal demand x∗[k] and optimal price p∗[k].

Edge computing service demand10 20 30 40 50 60 70 80 90

The

pro

babi

lity

of m

inin

g su

cces

sful

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Real Data (3 miners case)Proposed Model (3 miners case)Real Data (4 miners case)Proposed Model (4 miners case)

Figure 4. The comparison of real experiment results with our proposed model.

V. PERFORMANCE EVALUATION

In this section, we first perform the real experiment onmobile blockchain mining to validate the proposed utility func-tion of the miner. Then, we conduct the extensive numericalsimulations to evaluate the performance of our proposed price-based resource management for mobile blockchain.

A. Environmental Setup

We first set up the real mobile blockchain mining exper-iment, as illustrated in Fig. 3. The experiment is performedon a workstation with Intel Xeon CPU E5-1630, and androiddevices (mobile node) installing a mobile blockchain clientapplication. The mobile blockchain client application is im-plemented by the Android Studio and Software DevelopmentKits (SDK) tools [39]. All transactions are created by themobile blockchain client application. Each miner’s workingenvironment has one CPU core as its processor. The miner’sprocessor and its CPU utilization rate are generated andmanaged by the Docker platform [40]. The mobile device ofeach miner has installed Ubuntu 16.04 LTS (Xenial Xerus) andGo-Ethereum [41] as the operation system and the blockchainframework, respectively.

In Fig. 3, from Box 1 and 2, the screen of computerterminal shows that the Ethereum is running on the host, i.e.,edge computing node (Box 5). The mobile devices in Box 4are connected to the edge computing node through networkhub (Box 3) using mobile blockchain client application. Thebasic steps can be implemented as follows. The mobile users,i.e., miners use the Android device to connect to the edgecomputing node through network hub, i.e., access point. Then,the miners can request the service from edge node, and minethe block with the assistance of Ethereum service providedaccordingly.

We create 1000 blocks employing Node.js [42] and use themobile device to mine these blocks in the experiment. We

Number of miners50 60 70 80 90 100 110 120 130 140 150

Nor

mal

ized

ave

rage

opt

imal

pric

e

88

90

92

94

96

98

100

Uniform pricing, maximum price constraint: 100Discriminatory pricing, maximum price constraint: 100Uniform pricing, maximum price constraint: 95Discriminatory pricing, maximum price constraint: 95Uniform pricing, maximum price constraint: 90Discriminatory pricing, maximum price constraint: 90

Figure 5. Normalized average optimal price versus the number of miners.

consider two cases with three miners and four miners. In thethree-miner case, we first fix the other two miners’ servicedemand (CPU utilization) at 40 and 60, and then vary oneminer’s service demand. In the four-miner case, we first fixother three miners’ service demand as 40, 50 and 60, andthen vary one miner’s service demand. For our experiment,the number of transactions in each mined block is 10, i.e.,the size of block is the same. The comparison of the realexperimental results and our proposed analytical model isshown in Fig. 4. As expected, there is not much differencebetween the real results and our analytical model. This isbecause the probability that the miner successfully minesthe block is directly proportional to its relative computingpower when the block size are identical. Note that the delayeffects are negligible. In the sequel, we present the numericalresults to evaluate the performance of the proposed price-basedresource management for mobile blockchain applications.

B. Numerical Results

To illustrate the impacts of different parameters from theproposed model on the performance, we consider a groupof N miners, i.e., mobile users in a blockchain network. Weassume the size of a block mined by miner i follows the normaldistribution N (µt, σ

2). The default parameter values are setas follows: x = 10−2, x = 100, p = 100, µt = 200, σ2 = 5,R = 104, r = 20, z = 5 × 10−3, c = 10−3 and N = 100.Further, we employ the ‘fix’ function in MATLAB to roundeach ti to the nearest integer toward zero. Note that someof these parameters are varied according to the evaluationscenarios. We evaluate the performance of uniform pricingand discriminatory pricing in the following.

1) Investigation on total service demand of miners and theprofit of the ESP:

a) The comparison of uniform pricing and discriminatorypricing: We first address the comparison of uniform pricing

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Figure 6. Normalized total service demand of miners and the profit of theESP versus the number of miners.

and discriminatory pricing schemes. Figure. 5 demonstratesthe comparison of the normalized average optimal price undertwo proposed pricing schemes. It is worth noting that theoptimal price under uniform pricing is the same as the maxi-mum price, which can be explained by (38). Specifically, theexpression in (38) is always positive, and thus the profit of theESP increases with the increase of price. This means that themaximum price is the optimal value for profit maximization ofthe ESP under uniform pricing. Thus, we have the followingconclusion: the ESP intends to set the maximum possible valueas optimal price under uniform pricing. This conclusion isstill useful even when the ESP does not have the completeinformation about the miners.

Further, we find that the average optimal price of discrim-inatory pricing is slightly lower than that of uniform pricing.The intuition is that, under under discriminatory pricing,the ESP can set different unit prices of service demand fordifferent miners. For the details of operation of discriminatory

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Figure 7. Normalized total service demand of miners and the profit of theESP versus the variable reward factor.

pricing, we conduct the case study in Section V-B2. In thiscase, the ESP can significantly encourage the higher totalservice demand from miners and achieve greater profit gainunder discriminatory pricing, which is also consistent with thefollowing results. As shown in Figs. 6-8, in all cases, the totalservice demand from miners and the profit of the ESP underthe uniform pricing scheme is slightly smaller than that underthe discriminatory pricing scheme.

From Fig. 6, we find that when σ2 decreases, the resultsunder uniform pricing scheme is close to that under discrim-inatory pricing. This is because the heterogeneity of minersin blockchain is reduced as σ2 decreases. We may considerone symmetric case, where the miners are homogeneous withthe same size of blocks to mine, i.e., σ2 = 0. In this case, thediscriminatory pricing scheme yields the same results as thoseof the uniform pricing scheme.

b) The impacts of the number of miners: We nextevaluate the impacts brought by the number of miners, and

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Figure 8. Normalized total service demand of miners and the profit of theESP versus the propagation delay factor.

the results are shown in Fig. 6. From Fig. 6, we find thatthe total service demand of miners and the profit of theESP increase with the increase of the number of miners inmobile blockchain. This is due to the fact that having moreminers will intensify the competition among the miners, whichpotentially motivates them to have higher service demand.Further, the coming miners have their service demand, andthus the total service demand from miners is increased. Inturn, the ESP extracts more surplus from miners and therebyhas greater profit gain. Additionally, it is observed that therate of service demand increment decreases as the number ofminers increases. This is from the fact that the incentive ofminers to increase their service demand is weakened becausethe probability of their successful mining is reduced when thenumber of miners is increasing. Comparing different results,it is also observed that the total service demand of minersand the profit of the ESP increase as µt increases. Thisis because when µt increases, i.e., the average size of one

block becomes larger, the variable reward for each miner alsoincreases. The potential incentive of miners to increase theirservice demand is improved, and accordingly the total servicedemand of miners increases. Consequently, the ESP achievesgreater profit gain.

c) The impacts of reward for successful mining: Then,we investigate the impacts of variable reward and fixed rewardon miners and the ESP, which are shown in Fig. 7. It isobserved that with the increase of variable reward factor, boththe total service demand of miners and the profit of the ESPincrease. This is from the fact that the increased variablereward enhances the motivation of miners for higher servicedemand, and the total service demand is enhanced accordingly.As a result, the ESP achieves greater profit gain. Further, bycomparing curves with different value of fixed reward, we findthat as the fixed reward increases, the total service demand ofminers and the profit of the ESP also increase. Similarly, thisis because the increased fixed reward induces greater incentiveof miners, which in turn improves the total service demand ofminers and the profit of the ESP.

d) The impacts of propagation delay: At last, we exam-ine the impact of propagation delay on miners and the ESP,as illustrated in Fig. 8. It is observed that as the propagationdelay factor increases, the total service demand and the profitof the ESP increase. This is because when the propagationdelay effects are strong, the miners with larger mined blockneed to have higher service demand to reduce the propagationdelay of their propagated solutions. At the same time, a minerwith smaller mined block is also incentivized from the demandcompetition with the other miners. Therefore, the total servicedemand increases, which in turn improves the profit of theESP. Additionally, we observe that as the value of servicecost factor increases, the total service demand decreases underdiscriminatory pricing and remains unchanged under uniformpricing. On the contrary, the profit of the ESP increases in bothschemes. Recall from Fig. 5, the reason is that the optimalprice under uniform pricing remains unchanged from varyingthe value of service cost factor, and thus the service demandremains unchanged under uniform pricing. Correspondingly,the ESP achieves greater profit gain from the lower cost underuniform pricing. However, under discriminatory pricing, whenthe service cost decreases, the ESP has an incentive to setlower price for some miners to encourage higher total servicedemand. On the contrary, when the value of service cost factorincreases, the ESP has no incentive to set lower price forthese miners, since the higher total service demand resultsin higher cost for the ESP. Therefore, as the value of servicecost factor decreases, the total service demand and the profitof ESP increase.

2) Investigation on optimal price under uniform and dis-criminatory pricing schemes: Then, to explore the impacts ofdiscriminatory pricing on each specific miner, we investigatethe optimal price and resulting individual edge computingservice demand from miners. We conduct a case study forthree-miner mining with the following parameters: t1 = 100,t2 = 200, t3 = 300, x = 10−2, x = 100, p = 100, R = 104,

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Figure 10. Normalized optimal price versus the variable reward factor underdiscriminatory pricing.

r = 20, z = 5× 10−3, and c = 10−3.As expected, we observe from Figs. 9 and 10 that the

optimal price charging to the miners with the smaller block islower, e.g., miners 1 and 2. This is because the variable rewardof miners 1 and 2 for successful mining is smaller than thatof miner 3. Thus, the miners 1 and 2 have no incentive to paya high price for their service demand as miner 3. In this case,the ESP can greatly improve the individual service demandof miners 1 and 2 by setting lower prices to attract them,as illustrated in Figs. 11 and 12. Due to the competition fromother two miners, the miner 3 also has the potential incentive toincrease its service demand. However, due to the high serviceunit price, as a result, the miner 3 reduces its service demandfor saving cost. Nevertheless, the increase of service demand

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Figure 11. Normalized individual demand versus the fixed reward for miningsuccessful.

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Figure 12. Normalized individual demand versus the variable reward factor.

from miners 1 and 2 are greater. Therefore, the total servicedemand and the profit of the ESP are still improved underdiscriminatory pricing compared with uniform pricing.

Further, from Fig. 9, we observe that the optimal prices forminers 1 and 2 increase with the increase of fixed reward.This is because as the fixed reward increases, the incentivesof miners 1 and 2 to have higher service demand is greater. Inthis case, the ESP is able to raise the price and charge morefor higher revenue, and thus achieves greater profit. Therefore,for each miner, the individual service demand increases asthe fixed reward increases, as shown in Fig. 11. Additionally,we observe from Fig. 10 that the optimal prices for miners1 and 2 decrease as the variable reward factor increases.This is because when the variable reward factor increases,the incentive of each miner to have higher service demand

is greater. However, the incentives of the miners with smallerblock to mine, i.e., the miners 1 and 2 are still not much as thatof miner 3, and become smaller than that of miner 3 as thevariable reward factor increases. Therefore, the ESP intendsto set the lower price for miners 1 and 2 which may inducemore individual service demand as shown in Fig. 12.

VI. CONCLUSION

In this paper, we have investigated the price-based edgecomputing resource management, which for the first time, tosupport offloading mobile blockchain process. In particular, wehave adopted the two-stage Stackelberg game model to jointlystudy the profit maximization of edge computing serviceprovider and the utility maximization of miners. Throughbackward induction, we have derived the unique Nash equi-librium point of the game among the miners. The optimalresource management schemes including the uniform anddiscriminatory pricing for the edge computing service providerhave been presented and examined. Further, the existence anduniqueness of the Stackelberg equilibrium have been provedanalytically for both pricing schemes. We have performed thereal experiment to validate the proposed analytical model.Additionally, we have conducted the numerical simulationsto evaluate the network performance, which help the edgecomputing service provider to achieve optimal resource man-agement and gain the highest profit.

For the future work, we will further investigate the poolingschemes in mobile blockchain, where the miners in the samepool can collaborate to mine the block and thereby earn thereward that are appropriately split among all pool miners [20],[21]. Another natural extension is to consider the oligopolymarket with multiple edge computing service providers, wheredifferent service providers compete with each other for sellingcomputing services to miners.

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