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NTRODUCTION ∗† eb Wang Wenbo , lccanNetworks Blockchain † ui Niyato Dusit , income mining ategies ficulty nless sas rs the f here date case sers ical i.e., out We the on, to- er. as to to in e, f- e - - f , a eueissalbtsal hr frwr codn to according reward of share stable but small of min its block individual a rate secure an win hash can Then, to large. pool the significantly mining aggregating becomes a reward for By probability into rate. the miners, task accordin hash many miners puzzle-solution devoted the a their to works them dividing to pool assigning by mining and sub-tasks A scheduler coalitions smaller [2]. task the pools a represent mining as that as known nodes ar the miners been proxy networks in of blockchain has the rate real-world competitions by hash the of dominated mining result, growth a the [4], exponential As of the blockchain network. mine round to individual the due one for of negligible chance in head the win new scenarios, to the practical network the In as of [5]. majority confirmed the final be to block the propagated can block the sin determines first round, addition, consensus the network one only In within total P2P confirmation [3]. the block the of and network result in time) entire time unit the per propagation of oracle rate random numbe (i.e., the hash rate to hash miner’s queries a of between ratio the on depends [2]. [1], blockchain the receive of also mechanism may generation competitions mining winn the block the the for an issuers, of Except transaction round block. a the in by one named a by fees receive blockchain transaction will the transact network approving extending in the and contribution of its for majority the reward the monetary bloc In by candidate proposals. whose accepted miner block gets first the individual process, preimage awarding their a block on satisfying string based a (e.g. find condition oracle to where random function) trusted [3], hash a SHA-256 process query imple exhaustively (PoW) the miners is Proof-of-Work by the process a recognized solving the through are crypto-puzzle award mented blocks The to network. published process entire generation their reward when a una miners financially (b) attacks and Sybil make fordable, cry to computation-intensive process a solving (a) puzzle parts: two of consists nism b to recorded. guaranteed are are they protocol, blockchain once the Nakamoto majority immutable on the a data follow when transactional honestly in the words, nodes blockchain mining other able a the In is of of environment. liveness protocol and Byzantine Nakamoto persistence a the the that guarantee shown to have [3] studies h rbblt fwnigapzl-ovn competition puzzle-solving a winning of probability The mecha- incentive financial the protocol, Nakamoto the In ex-nihilo † aiaZhao Narisa , xdaon faadacrigt h token- the to according award of fixed-amount , nlg,Dla,Cia 116024 China, Dalian, hnology, est,Snaoe 639798 Singapore, versity, ∗ n igWang Ping and † ions pto- a , ce er er rs f- k g e e 1 - r 2 the ratio between its hash rate and the hash rate of the pool. is mainly determined by two factors, namely, the transmission In this paper, we study the problem of mining pool selection delay over each link and the transaction verification time at in a PoW-based blockchain network, where each mining pool each relaying node. For a block of size s, the transmission s may adopt a different arbitrary block mining strategy [6]. By delay can be modeled as τp(s)= γc [4], where γ is a network assuming that the individual miners are rational and profit- scale-related parameter, and c is the average effective channel driven, we propose a model based on the evolutionary game capacity of each link. On the other hand, since verifying a to mathematically describe the dynamic mining-pool selection transaction requires roughly the same amount of computation, process in a large population of individual miners. Considering the block verification time can be modeled as a linear function the computation power and propagation delay as the two major τv(s)= bs, where b is a parameter determined by both the factors to determine the results of mining competitions, we network scale and the average verification speed of each node. focus on how these two factors as well as the cost of the com- Then, the average propagation time for a block of size s can putation resource (mainly in electricity) impact the strategies be expressed as: of the individual miners for pool selection. Based on a case s τ(s)= τ (s)+ τ (s)= + bs. (2) study of two mining pools, we provide the theoretical analysis p v γc of evolutionary stability for the pool-selection dynamics. Our Based on (2), the incidence of abandoned (i.e., orphaning) numerical simulation results provide the evidences that support a valid block due to the propagation delay can be modeled our theoretical discoveries and further present the experimental as a Poisson process with mean 1/T , where T is maintained insight into the impact of the arbitrary strategies on the reward by the network as a fixed average mining time (e.g., 600s outcomes of different mining pools. in ) [4]. Then, the probability of orphaning a valid candidate block of size s in one consensus round is: II. PROBLEM FORMULATION orphan − −( s +bs)/T Pr (s)=1 − e τ(s)/T =1 − e γc . (3) A. Financially Incentivized Block Mining with Proof-of-Work We consider a blockchain network adopting the Nakamoto From (1) and (3), the probability for pool i to ultimately win consensus based on Proof-of-Work (PoW) [1]. Assume that a block mining race with a block of size si can be derived as: s the network is composed of a large population of N individual win ωixi −( i +bs )/T Pr (x,ωωω,s )= e γc i . (4) miners. For each miner, the chance of mining a new block is i i M j=1 ωj xj in proportion to the ratio between its individual hash rate for solving the crypto-puzzles in PoW and the total hash rate in We assume that the transactionsP in the blockchain network the network. According to the Nakamoto consensus protocol, are issued with an invariant rate of transaction fees. When the miner of each confirmed block receives a fixed amount of the transactions are of fixed size, pool i’s mining reward blockchain tokens from the new block’s and a flexible from transaction fee collection can also be modeled as a amount of transaction fees as the reward for maintaining the linear function of the block size si. Let ρ denote the price blockchain’s consensus and approving the transactions [2]. of transaction in a unit block size [5]. Then, the reward of We consider that the individual miners organize themselves pool i from transaction fees can be written as ρsi. Let R into a set of M mining pools, namely, M = {1, 2,...,M}. denote the fixed reward from the new block’s coinbase. Then, We further consider that each mining pool may set different the expected reward for pool i can be expressed as follows: ωixi − si requirement on the hash rate contributed by an individual x ( γc +bsi)/T E{ri( ,ωωω,si)} = (R + ρsi) M e . (5) miner trying to join the pool. Let ωi denote the individual hash j=1 ωj xj rate required by pool i (i ∈ M), and x denote the miners’ i Since the process of crypto-puzzle solving in PoW is com- population fraction in pool i. Then, the probability for pool i P putationally intensive, the rational miners also have to consider to mine a block in one consensus round can be expressed as: the cost of power consumption due to hash computation in mine x ωixi the block mining process. Noting that the new blocks are Pri ( ,ωω)= M , (1) j=1 ωjxj discovered with a roughly fixed time interval, we denote the energy price for generating a unit hash query rate during that where ω = [ω ,...,ω ]⊤, x =P [x ,...,x ]⊤, x = 1 M 1 M i∈M i time interval by p. Then, we can obtain the expected payoff and , . 1 ∀i xi ≥ 0 for an individual miner in pool i as follows: After successfully mining a block, pool i broadcastsP the s R + ρsi ωixi −( i +bs )/T mined block to its neighbors in the hope that it will be y (x,ωωω,s )= e γc i − pω . (6) i i Nx M i propagated to the entire network and confirmed as the new i j=1 ωjxj head block of the blockchain. However, in the situation where more than one mining pool discover a new block at the P same time, only the block that is first disseminated to the B. Mining Pool Selection as an Evolutionary Game network will be confirmed by the network. All of the rest Consider that the individual miners are rational and aim to candidate blocks will be discarded (orphaned). According to maximize their net payoff given in (6). Then, it is nature to the empirical studies in [4], [5], the block propagation time model the process of mining pool selection in the population 3 of individual miners as an evolutionary game. Mathematically, Algorithm 1 Mining Pool Selection Following the Pairwise we can define the evolutionary game for mining pool selection Proportional Imitation Protocol. as a 4-tuple: G = hN , M, x, {yi(x;ωω,si)}i∈Mi, where 1: Initialization: ∀i ∈ N , miner i randomly selects a mining • N is the population of individual miners, |N |=N. pool to start with. • M = {1, 2,...,M} is the set of mining pools, and 2: t ← 1 3: while x has not converged and t< MAX COUNT do (wi,si) is the mining strategy preference of each pool i ∈M. 4: for i ∈ N do ⊤ 5: j ← Rand(1,M) {Randomly selects a mining pool • x = [x1,...,xM ] ∈ X is the vector of the population j ∈ M} states, where xi represents the fraction of population that x RM 6: Determine whether to switch to pool j according to choose mining pool i. X = { ∈ + : i∈M xi =1}. the following probability of pool switching ρ : • {yi(x;ωω,si)}i∈M is the set of individual miner’s payoff i,j P in each mining pool. yi(x;ωω,si) is given by (6). ρi,j = xj max(yj (x;ωω,sj ) − yi(x;ωω,si), 0). (12) We note that ωi and si form the predetermined mining strategy of pool i. Given a population state x ∈ X , we can 7: end for derive the average payoff of the individual miner in N based 8: t ← t +1 on (6) as follows: 9: end while

M x x y( )= yi( ;ωω,si)xi. (7) In Algorithm 1, we describe the strategy evolution of the N i=1 X individual miners following the revision protocol of pairwise Then, by the pairwise proportional imitation protocol [7], proportional imitation [9]. When receiving a signal for strategy the replicator dynamics for the evolution of the population revision of choosing a new pool, an individual miner switches states can be expressed by the following system of Ordinary from it current pool to the new pool probabilistically according Differential Equations (ODEs) ∀i ∈M [7]: to (12). As the population size increases, the pairwise pro- portional imitation will asymptotically lead to the replicator x x x x˙ i(t)= fi( (t);ωω,si)= xi(t)(yi( (t);ωω,si)−y( (t))), (8) dynamics described by the ODEs in (8). where x˙ i(t) represents the growth rate of the size of pool i with respect to time t. C. A Case Study of Two Mining Pools We are interested in the Nash Equilibria (NE) of game G In this section, we consider a special case of a blockchain described by (8). Let Y (x) denote the vector of individual network with two mining pools, i.e., M = 2. Let the pop- x x x ⊤ payoffs for all the mining pools, Y ( )=[y1( ),...,yM ( )] ulation fraction of each pool be x1 = x, and x2 = 1 − x. and let E(Y ) denote the set of NE in game G. Then, E(Y ) From Definition 1 and by solving x˙ i(t)=0,i ∈ [1, 2], we can can be defined as follows [8]: obtain Theorem 1 as follows. ∗ Definition 1 (NE). A population state x ∈ X is an NE of THEOREM 1. Based on the replicator dynamics in (8), a the evolutionary game G, i.e., x∗ ∈E(Y ), if for all feasible blockchain network of two mining pools has three rest points population state x∈X the following inequality holds in the form of (x∗, 1 − x∗) with x x∗ ⊤ x∗ a − b ω ( − ) Y ( ) ≤ 0. (9) x∗ ∈ 0, 1, − 2 , (13) Np(ω − ω )2 ω − ω  1 2 1 2  It is straightforward that an NE is a fixed point of the repli- s1 −( +bs1)/T cator dynamics given by (8), namely, x where a = (R + ρs1)ω1e γc , b = (R + ∀i∈M,fi( (t);ωω,si)= s2 −( +bs2)/T a−b ω2 γc and 2 . 0 [7]. Then, we need to further investigate the stability of an ρs2)ω2e 0 < Np(ω1−ω2) − ω1−ω2 < 1 NE state x∗ ∈ E(Y ) for pool selection. Suppose that there Proof. From f (x(t))=0, ∀i ∈{1, 2}, we have exists another population state x′ trying to invade state x∗ by i attracting a small share ǫ ∈ (0, 1) in the population of miners fi(x(t)) = xi(t)(yi(x(t),ai) − y(x(t))) ′ ′ to switch to x . Then, x is an Evolutionary Stable Strategy a − b = x (t)(1−x (t)) −p(ω −ω ) . (ESS) if the following condition holds for all ǫ ∈ (0, ǫ): i i N(ω x (t)+ω (1−x (t))) 1 2  1 i 2 i  ∗ x∗ x′ ′ x∗ x′ (14) xi yi((1−ǫ) +ǫ ) ≥ xiyi((1−ǫ) +ǫ ). (10) i∈M i∈M Then, by solving fi(x(t)) = 0, we can obtain the three rest X X ∗ ∗ Based on (10), we can formally define the ESS as follows. points for the case of two mining pools as (x , 1−x ), where ∗ a−b ω2 x ∈ 0, 1, 2 − . Since from any initial Definition 2 (ESS [8]). A population state x∗ is an ESS of Np(ω1−ω2) ω1−ω2 state x(0) ∈ X , the rest point of (8) should stay in the interior game G, if there exists a neighborhood B ∈ X , such that n o of X , we have the following condition: ∀x ∈B− x∗, the condition (x − x∗)⊤Y (x∗)=0 implies that a − b ω ∗ ⊤ 2 (15) (x − x) Y (x) ≥ 0. (11) 0 < 2 − < 1. Np(ω1 − ω2) ω1 − ω2 4

Then, the proof of Theorem 1 is completed.

∂f2(x) aω1x1 Now, we are ready to investigate the evolutionary stability =x2 pω1 + ∗ ∗ ∂x N(ω x +ω x )2 of the three fixed points. In the case of x = 0 and x = 1, 1 1 1 2 2 the population state is (0, 1) and (1, 0), respectively. We know bω (1−x ) a 1 2 (23) that the two fixed points are of the similar form, since the − 2 − , N(ω1x1 +ω2x2) N(ω1x1 +ω2x2) individual payoff functions are similar for each mining pool. ! ∗ Therefore, we only need to check the case with x1 = x =0. ∂f (x) b 2 =(1 −2x ) −pω Lemma 1. For game G with two mining pools, 1) The rest ∂x 2 N(ω x +ω x ) 2 ∗ 2 1 1 2 2 ! point with x = 0 is an ESS, if the conditions given by (16) 2 2 and (17) hold: bω2(x2 −x2) aω1x1 − 2 − 2 + pω1x1. (24) N(ω1x1 +ω2x2) N(ω1x1 +ω2x2) a − b − p(ω1 − ω2) < 0, (16) Based on (21)-(24), we have Nω2 1) After some tedious mathematical manipulations, the de- terminants of the principle minors of at ∗ should a − b b J x =0 − p(ω − ω ) pω − > 0. (17) satisfy the following conditions to guarantee the negative Nω 1 2 2 Nω  2   2  definiteness of J: 2) If the conditions given by (18) and (19) hold, the rest point a − b ∗ a−b ω2 det(J11)= −p(ω1 −ω2) < 0, (25) with 2 is an ESS. x = Np(ω1−ω2) − ω1−ω2 Nω2 a−b b c(a(ω + ω )+ ω (−2b + Npω (ω − ω ))) det (J)= −p(ω −ω ) (pω − ) > 0. (26) 1 2 1 1 2 1 < 0, (18) Nω 1 2 2 Nω (a − b)  2  2 ∗ a−b ω2 2) Similarly, at 2 , the following x = Np(ω1−ω2) − ω1−ω2 pc(−bω + aω )(a − b + Npω (ω − ω )) conditions can be obtained for the negative definiteness 1 2 1 2 1 > 0, (19) (ω1 − ω2) of J after some mathematical manipulations: c (a(ω + ω )+ω (−2b+Npω (ω −ω ))) where c = a − b + Npω (ω − ω ). 1 2 1 1 2 1 2 2 1 det(J11)= 2 N(a−b)(ω1 −ω2) Proof. According to Definition 2.6 in [9], the asymptotically < 0, (27) stable state of the ODE system given in (8) is guaranteed to be an ESS. When the replicator dynamics is continuous-time, it pc(−bω1 +aω2)(a−b+Npω1(ω2 −ω1)) det(J)= 2 > 0. is asymptotically stable if the Jacobian matrix of the dynamic N(a − b) (ω1 − ω2) (28) system at the equilibrium is negative definite, or equivalently, if all the eigenvalues of the Jacobian matrix have negative real Then, the proof to Lemma 1 is completed. parts [10]. For the replicate dynamic system given in (8), the We note that the blockchain network is comprised by a large Jacobian matrix of the replicator dynamics in a two-mining- population of individual miners in the real-world scenarios. pool network is Then, from Lemma 1, we can employ the asymptotic analysis and obtain the following theorem on evolutionary stability of ∂f (x) ∂f (x) 1 1 the rest points. ∂x ∂x J = 1x 2x . (20)  ∂f2( ) ∂f2( )  THEOREM 2. Assume that the population size N is suffi- ∗ ∗ (x1=x ,x2=1−x ) ∗  ∂x1 ∂x2  ciently large. Then, neither of the rest points with x ∈{0, 1} ∗ a−b   is evolutionary stable. The rest point with x = 2 − Further, the elements in (20) can be derived as follows: Np(ω1−ω2) ω2 is an ESS if the following conditions are satisfied: ω1−ω2 x ∂f1( ) a a − b< 0, =(1−2x1) −pω1 (29) ∂x1 N(ω1x1 +ω2x2) (bω − aω )(ω − ω ) > 0. !  1 2 2 1 aω (x −x2) bω x2 Proof. First, at the rest point with x∗ = 0, by Lemma 1, 1 1 1 2 2 (21) − 2 − 2 + pω2x2, N(ω1x1 +ω2x2) N(ω1x1 +ω2x2) we can obtain the following conditions for the Jacobian if ω1 ≤ ω2, a − b x lim det(J11)= lim − p(ω1 − ω2) ≥ 0. (30) ∂f1( ) aω2(1−x1) N→+∞ N→+∞ Nω2 =x1 pω2 − 2 ∂x2 N(ω1x1 +ω2x2) Then, the Jacobian matrix is not negative definite. Alterna- tively, if , we have bω x b ω1 >ω2 + 2 2 − , (22) N(ω x +ω x )2 N(ω x +ω x ) a − b 1 1 2 2 1 1 2 2 ! lim det(J11)= lim − p(ω1 − ω2) < 0, (31) N→+∞ N→+∞ Nω2 5 and 1 1 x 0.9 0.9 1

0.8 0.8 x a−b b 2 lim det(J)= lim ( −p(ω1−ω2))(pω2− )<0. 0.7 0.7 → ∞ → ∞ N + N + Nω2 Nω2 0.6 0.6

(32) 0.5 0.5 Again, the Jacobian matrix cannot be negative definite. Then, 0.4 0.4 the rest point with x∗ =0 is not an ESS. Following the same 0.3 0.3 ∗ 0.2 0.2 procedure, we can show that the rest point with x =1 is not 0.1 0.1

0 0 evolutionary stable, either. 0 100 200 300 400 500 600 700 800 900 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 By [10], we know that any rest point in the interior of X ∗ a−b ω2 (a) (b) is an NE. Then, for the NE with 2 , x = Np(ω1−ω2) − ω1−ω2 following Lemma 1, we obtain Fig. 1. (a) Evolution of the miners’ population states over time with two mining pools. (b) Replicator dynamics of the pool-selection strategies and the x . , . lim det(J11) evolution trajectory starting from (0)=(0 75 0 25). N→+∞ (a − b + Npω (ω − ω ))a(ω + ω ) 1 0.1 2 2 1 1 2 0.9 0.08 = lim 2 + N→+∞ N(a − b)(ω − ω ) 0.8 0.06 1 2 x 1 0.7 0.04 x (a − b + Npω2(ω2 − ω1))ω1(−2b + Npω1(ω2 − ω1)) 0.6 2 0.02 2 N(a − b)(ω1 − ω2) 0.5 0 2 0.4 -0.02 Np ω1ω2 -0.04 = lim , (33) 0.3 N→+∞ a − b 0.2 -0.06 0.1 -0.08

0 -0.1 and 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p(a − b + Npω2(ω2 − ω1)) (a) (b) lim det (J)= lim 2 · N→+∞ N→+∞ N(a − b) (ω1 − ω2) 1 (−bω1 + aω2)(a − b + Npω1(ω2 − ω1)) Fig. 2. (a) Population state at the ESS vs. varying delay coefficient γc + b. 2 (b) Average payoff of the miners at the ESS vs. varying delay coefficient N(a − b) (ω1 − ω2) 1 γc + b. Np3ω ω (bω − aω )(ω − ω ) = lim 1 2 1 2 2 1 . (34) N→+∞ (a − b)2 Further, we analyze the influence of the network condition By (33) and (34), the Jacobian matrix is negative definite if the ∗ ∗ on the pool-selection strategies of the individual miners. In conditions given in (29) are satisfied, hence the NE (x , 1−x ) Figure 2, we show the evolution of the stable population is an ESS. Then, the proof to Theorem 2 is completed. states and the corresponding average payoff of the individual 1 miners with respect to varied delay coefficient γc + b. In the III. EVOLUTION ANALYSIS simulation, we adopt the same mining strategies as in Figure 1. In this section, we conduct several numerical simulations Figure 2(a) shows that as the propagation delay coefficient and provide the performance evaluation of the individual increases, more miners will tend to join the pool with a miners’ pool-selection strategies in different situations. We smaller hash rate requirement (ω2 = 20). Jointly considering first consider a blockchain network with N = 5000 individual the payoffs at NE shown in Figure 2(b), we know that a larger miners, which evolve to form two mining pools (i.e., M =2). delay coefficient leads to a higher probability of orphaning For the purpose of demonstration, we set the block generation blocks of the same size. As a result, the miners prefer to join 1 parameters as λ =1/600, γc + b =0.005, R = 1000, ρ =2 the pool that induces lower mining cost. We can also observe and p = 0.01. We also set the initial population state as in Figure 2(b) that the payoffs of the mining pool remain x = [0.75, 0.25]. We first consider that the two pools adopt unchanged at zero. This phenomenon can be interpreted as their mining strategies with the same block size, s1 = s2 = a situation of market equilibrium where the demand for the 100, and different computation power contribution, ω1 = 30 hash rate exactly meets the supply with the current settings of and ω2 = 20. By Theorem 2, we know that such strategy reward parameters. adaptation satisfies the condition for an ESS in the interior of Finally, we consider a more general situation with four the simplex X . Figure 1(a) demonstrates the evolution of the mining pools, where each pool adopts in their mining strategy miners’ pool-selection strategies. According to Figure 1(b), the the same block size as si = 100 (1 ≤ i ≤ 4) and different strategies converge to a global ESS of (0.4, 0.6), which is in requirement on the hash rate contribution as ω1 =10, ω2 =20, accordance with our theoretical prediction. We also observe ω3 =30 and ω4 =40. The evolution of the miner population that relatively fewer miners choose to join the pool requiring states is presented in Figure 3(a). In the considered case, a higher hash rate (i.e., pool 1) at the ESS. This is because a we observe that when the miners’ pool-selection strategies higher computation power requirement will lead to an increase converge to the equilibrium, selecting pool 1 becomes a in the mining cost, which exceeds the profit improvement that dominating strategy since by contributing a higher hash rate, the miner can obtain in that pool. the profit gain is unable to cover the power consumption cost 6

0.9 0 [10] R. Cressman, Evolutionary dynamics and extensive form games. MIT 0.8 x 1 Press, 2003, vol. 5. 0.7 x x -0.05 2 1 x 3 0.6 x 2 x 4 x 3 0.5 x -0.1 4 0.4

0.3 -0.15

0.2

0.1 -0.2

0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000

(a) (b)

Fig. 3. (a) The population states evolution with respect to different delay 1 1 2 coefficient γc + b, where mining strategy variables are s = 100, s = 120 and ω1 = 20, ω2 = 20. (b) Payoff evolution with respect to different delay 1 coefficient γc + b. for each miner. Then, the individual miners prefer to decrease their dedicated hash rate since they are sensitive to the mining cost. Figure 3(b) show that the payoffs of joining a pool evolves from negative value to zero. Again, this indicates a situation where the block mining business becomes a perfect competition market with an NE payoff of zero, and no miner can switch its pool selection without undermining some other miner’s payoff at the equilibrium.

IV. CONCLUSION In this paper, we have investigated the dynamic mining pool- selection problem in a blockchain network using Nakamoto consensus protocol. We model the dynamics of the individual miner’s pool-selection strategies as an evolutionary game. In particular, we have considered the computation power and propagation delay as two major factors that determine the outcome of the block mining competition. Furthermore, we have theoretically analyzed the evolutionary stability of the pool selection dynamics based on a case study of two mining pools. For the case of two mining pools, we have shown that the blockchain network conditionally admits a unique evolutionary stable state. Our simulation results have provided the numerical evidence for our theoretical discoveries.

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