Incentivizing Crowdsourcing Systems with Network Effects

IEEE INFOCOM 2016 - The 35th Annual IEEE International Conference on Computer Communications

Yanjiao Chen‡, Baochun Li‡, Qian Zhang§

‡Department of Electrical and Computer Engineering, University of Toronto §Department of Computer Science and Engineering, Hong Kong University of Science and Technology Email: ‡{ychen, bli}@ece.toronto.edu, §[email protected]

Abstract—In a crowdsourcing system, it is important for the Such intrinsic rewards, however, do not typically remain crowdsourcer to engineer extrinsic rewards to incentivize the unchanged as the size of participant population grows. When participants. With mobile social networking, a user enjoys an a user’s behavior aligns with other users, she will obtain intrinsic benefit when she aligns her behavior with the behavior of others. Referred to as network effects, such an intrinsic benefit higher intrinsic rewards, usually due to social factors. This is becomes more significant as the number of users grows in the known as the network effects [6]. In the example of Waze, a crowdsourcing system. But should a crowdsourcer design her mobile crowdsourcing platform for sharing traffic information, extrinsic rewards differently when such network effects are taken a driver can get a better route if more users join and contribute into account? In this paper, we, for the first time, consider their local traffic data. It is intuitively conceivable that a network effects as a contributing factor to intrinsic rewards, and study its influence on the design of extrinsic rewards. Rather than growing population of the crowdsourcing platform — with assuming a fixed participant population, we show that the num- more intrinsic rewards due to network effects — would help ber of participating users evolves to a steady equilibrium, thanks reduce the amount of extrinsic rewards that a crowdsourcer to subtle interactions between intrinsic rewards due to network will need to provide. effects and extrinsic rewards offered by the crowdsourcer. Taken Unfortunately, existing works in the literature have not yet network effects into consideration, we design progressively more sophisticated extrinsic reward mechanisms, and propose new and considered the influence of intrinsic rewards, as well as the optimal strategies for a crowdsourcer to obtain a higher utility. exquisite and subtle interactions between intrinsic and extrinsic Via extensive simulations, we demonstrate that with our new rewards. Reverse auctions (e.g., [7]) and Stackelberg games strategies, a crowdsourcer is able to attract more participants (e.g., [8]) are typically used to model extrinsic rewards in with higher contributed efforts; and participants gain higher existing works. With reverse auctions, users submit bids with utilities from both intrinsic and extrinsic rewards. their desired extrinsic rewards, and the crowdsourcer chooses the users based on their bids, with a selection process that is I.INTRODUCTION typically NP-hard and impractical. With Stackelberg games, Crowdsourcing combines the collective efforts of the crowd the crowdsourcer first announces her policy on extrinsic re- to accomplish a specific task, such as collecting large volumes wards, and the users would then make decisions on their of data, which is otherwise extremely costly or even unattain- contribution levels. In both models, the user population is able. Crowdsourcing systems, such as Uber [1], Waze [2], assumed to be fixed, without considering intrinsic rewards and and Amazon Mechanical Turk (MTurk) [3], have become in- network effects. creasingly popular. Another excellent example is ResearchKit, In this paper, we bring intrinsic rewards into the spotlight, recently launched by Apple as a mobile application framework with a focus on how network effects affect the mechanism that supports a crowdsourcing platform for medical research design when a crowdsourcer provides extrinsic rewards to [4]. incentivize crowdsourcing systems. In particular, we study The success of a new crowdsourcing platform relies on the the dynamics of participation levels as a result of the inter- scale of user participation, as well as the contribution from action between intrinsic rewards incurred by network effects each individual participant. To recruit and maintain a large and extrinsic rewards from the crowdsourcer. Based on our number of users, the crowdsourcer usually provides users with analyses, we propose two extrinsic reward mechanisms, taking monetary compensation, referred to as extrinsic rewards. In advantage of the intrinsic rewards to boost user participation contrast to such extrinsic rewards offered by the crowdsourcer, and increase the crowdsourcer’s utility. a participant often enjoys a reward that is derived from a To start with, we first present a simple mechanism with fixed sense of satisfaction, social status, or honor, and such rewards extrinsic rewards (Section III), which is simple to implement. are inherently intrinsic. In ResearchKit, for example, it has In fact, mechanisms with fixed extrinsic rewards have been been reported that only a few hours after its release, over adopted by many existing crowdsourcing systems, such as 7,000 people has voluntarily enrolled in a study on Parkinson’s MTurk. Thanks to intrinsic rewards, the participation level disease without any extrinsic rewards, and the largest study is above-zero even without extrinsic rewards. Given a certain ever was only 1,700 [5]. extrinsic reward, the participation level will evolve to a stable

978-1-4673-9953-1/16/$31.00 ©2016 IEEE equilibrium. Targeting the most profitable participation level, the participation level.1 we are able to compute the corresponding optimal fixed Extrinsic rewards. The crowdsourcer provides users with an extrinsic reward. extrinsic reward of P (bi), satisfying P (0) = 0. In the fixed Taking it a step further, we proceed to design a flexible extrinsic reward mechanism, P (bi) = p, irrespective of the extrinsic reward mechanism, where the extrinsic reward of users’ effort levels; in the flexible extrinsic reward mechanism, a user is a function of her effort level (Section IV). To on the other hand, P (·) is a function of bi. specify the extrinsic reward function is challenging since User i’s utility ui is the sum of intrinsic and extrinsic we do not have any prior knowledge about its form (e.g., rewards minus the cost: linear or logarithmic). Moreover, different extrinsic reward ui = vibi + E(n) + P (bi) − cibi. (1) functions, via a complex interaction with intrinsic rewards, lead to different equilibrium participation levels. To tackle We combine vibi and cibi as they have the common term bi: this problem, we first focus on a certain participation level, ui = E(n) + P (bi) − βibi, (2) and obtain the extrinsic reward function that achieves the highest utility for the crowdsourcer under this participation in which βi = ci − vi is defined as the net cost of user level. We then choose the best participation level that yields i. For some users, vi > ci, so the net cost βi is negative. the maximum utility, and derive the corresponding optimal Even without extrinsic rewards, these self-motivated users extrinsic reward function. have incentives to participate in the crowdsourcing system. These pioneers help attract others via network effects. The The superiority of our proposed extrinsic reward mech- net cost is a user’s private information. Thus, we assume anisms is verified through extensive simulations. With the that β ∈ [β, β] is a random variable, with a cumulative help of intrinsic rewards, the crowdsourcer is able to reach a i distribution function F (β), and a probability density function higher participation level and obtain a higher utility. Stronger f(β) = F 0(β). We have β < 0 and β > 0, as users may have network effects will contribute to higher intrinsic rewards, negative or positive net costs. thus more beneficial to the crowdsourcer. Compared with Aiming at maximizing her utility in (2), a user’s optimal the fixed extrinsic reward mechanism, the flexible extrin- effort level b∗ is a function of her net cost β , i.e., b∗ = g(β ). sic reward mechanism is more efficient in soliciting more i i i i She will drop out if her utility is always negative whatever the user contributions. Interestingly, the flexible extrinsic reward effort level is. mechanism improves both the crowdsourcer’s and the users’ The crowdsourcer makes a profit from the users’ contribu- utilities, resulting in a win-win situation. This is because users’ tions, while having to pay extrinsic rewards. Her utility U is higher contributions not only earn themselves higher extrinsic the total aggregated contribution from all participants minus rewards, but also become a valuable asset to the crowdsourcer. the total extrinsic rewards: Z Z U = µ ln(1 + g(β))dF (β) − P (g(β))dF (β), (3) II.SYSTEM MODEL β β in which µ is the equivalent monetary worth of users’ con- By participating in a crowdsourcing system, a user receives tributions. Note that g(β) is a user’s effort. We use the both extrinsic rewards from the crowdsourcer, and intrinsic logarithmic function ln(·) to transform a user’s effort to the rewards due to the benefits or social status she obtains. More perceived utility by the crowdsourcer, which features the law formally, user i exerts an effort of bi. bi ∈ [b, b], in which of diminishing return: a user’s contribution increases with her b is the minimum effort, for example, a user has to register effort level but the marginal return decreases. If a user does not and fill in the basic information; b is the maximum effort due contribute any effort, the utility received by the crowdsourcer to limitations such as time, battery life and manpower. To is ln(1 + 0) = 0. contribute an effort of bi, the user incurs a cost of cibi, where The crowdsourcer’s objective is to maximize her utility ci is user i’s unit cost. in (3) by determining the optimal extrinsic rewards. In the Intrinsic rewards. On one side, a user benefits from her fixed extrinsic reward mechanism, the crowdsourcer has to own effort, for example, a healthcare crowdsourcing platform decide the optimal uniform extrinsic reward p∗; in the flexible enables a user to get a better understanding of her health extrinsic reward mechanism, the crowdsourcer has to design condition by keeping track of her diet, exercise and heart rate. the optimal extrinsic reward function P ∗(·). On the other side, a user enjoys the social advantage of a III.FIXED EXTRINSIC REWARDS large crowd owing to network effects. Let vi denote the unit value a user gets from her own effort, and E(n) denote the In this section, we first study how the interplay of a fixed network effects, where n ∈ [0, 1] represents the normalized extrinsic reward and network effects leads to participation participation level and E(·) is a concave function, satisfying levels at equilibrium, and then we derive the optimal value E(0) = 0,E0(·) > 0 and E00(·) < 0. This indicates that of the fixed extrinsic reward. network effects monotonically increase with the participation 1For simplicity, in this paper, we assume that network effects E(n) are the level, but the marginal return decreases. Therefore, a user’s same for all users. In the future, we will study the case where users obtain intrinsic reward is vibi + E(n), a function of her effort and heterogeneous rewards from network effects, i.e., Ei(n) for user i. A. Equilibrium Participation Level 0.25 p = 0.3 Given a fixed extrinsic reward, a user’s utility becomes: 0.2 p = 0.4

ui = E(n) + p − βibi. (4) 0.15 0.1 If βi < 0, a user will deﬁnitely participate with her maximum effort b; otherwise, she will participate with her minimum (n) 0.05 Φ n effort b if a positive utility can be obtained. Given an expected 0 D n n n e C A B participation level n and the corresponding network effects −0.05 E(ne), the marginal user, who is indifferent to the choices of −0.1 participating or not, has a utility of zero. Let βn denote the net cost of the marginal user. We have: 0 0.2 0.4 0.6 0.8 1 Participation level n 1 e Fig. 1: Multiple equilibria under a certain extrinsic reward. βn(p) = E(n ) + p . (5) √ b E(n) = n, b = −1,F (·) ∼ N(1, 0.2).

βn(p) is upward sloping in p, and the curve will shift up if ne increases. This implies that the users with higher net costs will participate if either extrinsic or intrinsic rewards go up. Similarly, suppose there is a small perturbation ∆n downwards Since F (βn) = n, we have: at nA, Φ(nA −∆n) < 0, therefore, more users will leave, and the participation level will be pushed further down to 0. E(ne) + p n = F . (6) In summary, we have the following lemma to characterize b the stability of an equilibrium. At equilibrium, the expected participation level equals the e Lemma 1. Stable Equilibrium. An equilibrium participation real participation level, that is, n = n . With p > bβ − E(1), 0 n = 1 will be an equilibrium,2 but the crowdsourcer will never level is stable if Φ (n) < 0. set a p that is more than enough to achieve full participation. Proof. Since Φ(n) = 0 and Φ0(n) < 0, it can be easily proved Thus, we stipulate that p ≤ bβ − E(1). that Φ(n + ∆n) < 0 and Φ(n − ∆n) > 0. In this case, the Proposition 1. Existence of an Equilibrium Participation equilibrium is stable according to our preceding analysis. Level. For any extrinsic reward p, Equation (6) has at least one root. Proposition 2. Existence of a Stable Equilibrium Participa- tion Level. For any extrinsic reward p, there exists at least E(n)+p Proof. Define Φ(n) = F b − n. Φ(n) is continuous in one stable equilibrium participation level. In particular, the [0,1]. Φ(0) = F (p/b) > 0 and Φ(1) = F ((E(1)+p)/b)−1 ≤ highest equilibrium participation level is stable. 0. By the intermediate value theorem, 0 is a value of Φ(n) for max some n ∈ [0, 1], which is just the equilibrium participation Proof. Let n denote the highest equilibrium participation 0 max level. level. We prove that Φ (ne,max) < 0 must be true, then n is stable according to Lemma 1. Fig. 1 shows the value of Φ(n) under different participation levels, and the condition Φ(n) = 0 pinpoints the equilibria. Φ(nmax + h) − Φ(nmax) Φ(nmax + h) Given a certain extrinsic reward, there are multiple equilibria, Φ0(nmax) = lim = lim . but they have different stability attributes. h→0 h h→0 h Stable equilibria n n (1) , such as B and D in Fig. 1. Φ(nmax +h) < 0 must be true, otherwise there will be another ∆n n Suppose there is a small perturbation upwards at B, equilibrium between n + h and 1, which contradicts the Φ(n + ∆n) < 0, i.e., β b > E(n + ∆n) + p. max B nB +∆n B fact that nmax is the highest. The participation level will be pushed downwards back to n, because the net costs of the new participants are greater than their rewards, so they will leave. Similarly, suppose there is a small perturbation ∆n downwards at n , Φ(n − ∆n) > 0, B B B. Optimal Fixed Extrinsic Reward i.e., βnB −∆nb > E(nB −∆n)+p. The participation level will be pushed upwards back to n, because users whose net costs The fixed extrinsic reward p leads to the equilibrium partic- are smaller than their rewards will rejoin. ipation level, which in turn, affects the crowdsourcer’s utility.3 (2) Unstable equilibria, such as nA and nC in Fig. 1. Since users with βi ∈ [β, 0] make an effort of b and users with Suppose there is a small perturbation ∆n upwards at nA, β ∈ (0, βn] make an effort of b, the crowdsourcer’s utility Φ(nA + ∆n) > 0, which implies that more users will rush in, becomes: and the participation level will be pushed further up to nB. 3For tractability, we only consider the highest stable equilibrium participa- 2If p ≤ bβ, n = 0 will be an equilibrium. Nevertheless, β < 0, so this tion level. In the future, we will study the possibilities of other equilibria, and will not happen. their influence on the design of extrinsic rewards. 3 0.3 γ = 0.1 γ = 0.1 2.4 γ = 0.5 0.28 γ = 0.5 2.5 γ = 0.9 γ = 0.9 2.2 0.26 2 2 0.24 1.8 γ = 0.1 0.22 1.5 γ = 0.5 The crowdsourcer utility 1.6

Optimal fixed extrinsic reward 0.2 γ = 0.9 Equilibrium participation level 1.4 1 0.18 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Network effects intensity α Network effects intensity α Network effects intensity α (a) (b) (c) Fig. 2: The ﬁxed extrinsic reward mechanism. b ∈ [1, 10],F (·) ∼ UNIF(−1, 15).

IV. FLEXIBLE EXTRINSIC REWARDS Z 0 Z βn In this section, we first analyze the users’ strategic behavior U = µ ln(1 + b)dF (β) + µ ln(1 + b)dF (β) − np in response to an extrinsic reward function, which leads to an β 0 equilibrium participation level. Given a targeted participation = µF (0) ln(1 + b) + µ n − F (0) ln(1 + b) − np level, we can then derive the extrinsic reward function which 1 + b helps the crowdsourcer achieve the highest possible utility = µF (0) ln + µ ln(1 + b) − pn. 1 + b (referred to as the conditional optimal extrinsic reward func- (7) tion). Finally, we compute the optimal participation level that The extrinsic reward p determines the equilibrium participa- maximizes the crowdsourcer’s utility, and the corresponding tion level n according to Equation (6). Therefore, finding the global optimal extrinsic reward function. optimal extrinsic reward p is equivalent to finding the optimal A. Users’ Optimal Effort Level participation level n induced by p: Given an extrinsic reward function instead of a fixed extrinsic reward, users will strategically choose their effort levels to max U ⇒ max µ ln(1 + b) + E(n) − bF −1(n)n. (8) maximize their utilities, rather than toggling between b and b. p n Proposition 4. Optimal Effort Level. The optimal effort level Proposition 3. Optimal fixed extrinsic reward. Given the of user i, based on her net cost β , i.e., b = g(β ), is implicitly optimal equilibrium participation level as the solution of (8), i i i given by the following equation: the optimal fixed extrinsic reward is: F −1(n) p∗ = bF −1(n∗) − E(n∗). (9) Z E(n) + P (bi) − βig(βi) = g(x)dx. (10) Fig. 2 shows the optimal extrinsic reward, the equilibrium βi participation level, and the crowdsourcer’s utility under the Proof. We prove this by the envelop theory [12]. Take partial fixed extrinsic reward mechanism. Exponential functions are derivative of βi on both sides of Equation (2): commonly used to model network effects in the existing ∂ui literature [6], [9], [10]. Therefore, we assume that the network = −bi. (11) ∂β effects function is E(n) = α · nγ . Stronger network effects (a i larger α and a smaller γ) yield higher intrinsic rewards. The Let βn represent the net cost of the marginal user, who crowdsourcer therefore can take advantage of this to obtain a is indifferent towards the options of participating or not. higher equilibrium participation level (Fig. 2(b)) with a lower Integrating both sides of Equation (11) from βi to βn: fixed extrinsic reward (Fig. 2(a)), and her utility rises as well Z βn (Fig. 2(c)). Neglecting network effects (α = 0) will potentially u(βn) − u(βi) = − g(x)dx. (12) βi cause a significant amount of loss to the crowdsourcer. This yields Equation (10) as u(β ) = 0 and n = F (β ). An anecdotal real-world example that substantiates the n n observations above is that, according to its co-founder Elon Musk, when PayPal Inc. was started in 1998, a fixed monetary Given a complicated extrinsic reward function P (·), it is reward was paid for each user to sign up for the service. difficult to solve Equation (10) to obtain the optimal effort As the number of users grew, the amount of the reward was level function g(β). Fortunately, we show in the following gradually reduced to zero, without affecting the growth of the section that, as the crowdsourcer intentionally designs P (·) user population due to network effects [11]. for utility maximization, g(β) has a closed-form expression. B. Optimal Extrinsic Reward Function The last term is: If a user cannot gain positive utility even with the optimal β β β effort level, she will not participate at all. Therefore, different Z n Z n Z n βn extrinsic reward functions will result in different equilibrium g(x)dxdF (β) = g(x)dx ∗ F (β) β β β β participation levels. Based on this knowledge, the crowd- βn (21) Z βn R Z βn sourcer can design the conditional optimal extrinsic reward ∂ β g(x)dx − F (β) dβ = F (β)g(β)dβ. function for a targeted participation level. β ∂β β

Proposition 5. Conditional optimal extrinsic reward func- Therefore, U = R βn I(β)dF (β), in which: tion. Given a targeted participation level n and the cor- β responding marginal user’s net cost β = F −1(n), under n F (β) conditions that βe > 0 and E(n) ≤ bβn, the crowdsourcer’s I(β) = µ ln(1 + g(β)) − g(β)β + E(n) − g(β). (22) conditional optimal extrinsic reward function, and the users’ f(β) optimal effort level are given as follows, in which βb satisfies Maximizing the crowdsourcer’s utility U via the extrinsic F (βb) µ F (βe) µ reward function P (·) is equivalent to inducing the optimal βb + = 1+b , and βe satisfies βe + = . f(βb) f(βe) 1+b effort level b = g(β) through P (·). Take the first and second • If βn ∈ [β, βe), the conditional optimal extrinsic reward derivatives of I with respect to g(β): function is: P (b) = β b − E(n). (13) n ∂I µ F (β) = − β − , The users’ optimal effort level is: ∂g(β) 1 + g(β) f(β) (23) ∂2I µ g(β) = b, β ∈ [β, βn]. (14) = − < 0. ∂g2(β) (1 + g(β))2 • If βn ∈ [β,e βb), the conditional optimal extrinsic reward function is: If β < βe, ∂I/∂g(β) > 0, meaning that I monotonically

Z g(βn) increases with b, thus g(β) = b. If β > βb, ∂I/∂g(β) < P (b) = g−1(b)b − E(n) + xdg−1(x). (15) 0, meaning that I monotonically decreases with b, hence b g(β) = b. Otherwise, the optimal effort level b is the solution The users’ optimal effort level is: to ∂I/∂b = 0. In this way, we get (14), (16) and (18). ( Substituting g(β) in (10) with (14), (16) and (18), we can get b, β ∈ [β, βe], g(β) = (16) the optimal extrinsic reward functions (13), (15) and (17) for µf(β) − 1, β ∈ [β, β ]. βf(β)+F (β) e n different targeted participation levels. Special cases of g−1(b) −1 In particular, g−1(b) = βe. and g (b) are given in the appendix. • If βn ∈ [β,b β], the conditional optimal extrinsic reward function is: Corollary 1. The conditional optimal extrinsic reward func- Z g(βn) tion P (·) is non-negative and monotonically increasing. −1 −1 P (b) = g (b)b−E(n)+ xdg (x)+b(βn −βb). b Proof. If βn ∈ [β, βe), P (b) = βnb−E(n) > 0. If βn ∈ [β,e β], (17) the ﬁrst derivative of P (·) with respect to b is: The users’ optimal effort level is:  ∂P (b) −1 0 −1 −1 0 −1  b, β ∈ [β, βe], = (g (b)) b + g (b) − (g (b)) b = g (b) > 0.  µf(β) ∂b g(β) = βf(β)+F (β) − 1, β ∈ [β,e βb], (18) (24)   b, β ∈ [β,b βn]. Therefore, P (b) monotonically increases with b, and the minimum reward is P (b) = β b − E(n) ≥ 0. In particular, g−1(b) = βe and g−1(b) = βb. n Proof. The crowdsourcer’s utility is: In Proposition 5, conditions βe > 0 and E(n) ≤ bβn are both reasonable. β is the threshold net cost, below which a user will Z βn e U = µ ln(1 + g(β)) − P (g(β))dF (β). (19) make the maximum effort b. Users with negative net costs — β and those with low positive net costs as well — will exert the Substitute P (b(β)) with (10): maximum effort, with certain extrinsic rewards. Therefore, βe is assumed to be positive. E(n) ≤ bβn indicates that network

Z βn effects cannot fully cover the cost of the marginal user who U = µ ln(1 + g(β)) − g(β)β + E(n) makes the minimum effort; otherwise, the crowdsourcer will β (20) not provide any extrinsic rewards to them. Z βn Note that the conditional optimal extrinsic reward functions − g(x)dx dF (β). (13), (15), and (17) are functions of users’ effort level b, β which is observable by the crowdsourcer, but not the users’ 10 n* net cost β, which is private information. P (·) also depends on 3 n* βn, which is known by the crowdsourcer as the participation 8 2 n* level n is set as the target by the crowdsourcer. Furthermore, 1 the crowdsourcer is aware of the users’ response to the 6 extrinsic reward function, and can use Proposition 5 to derive function g(·), which is indispensible in determining P (·). In 4 the conditional optimal extrinsic reward function, the term βnb α=2 −1 α or g (b)b can be regarded as the compensation for the users’ The crowdsourcer utility U 2 =4 cost; the term −E(n) shows how network effects help curtail α=6 0 the crowdsourcer’s payment to users; the rest of the terms 0 0.2 0.4 0.6 0.8 1 are necessary to realize the targeted participation level. More Participation level n specifically, it is ensured that ui > 0, ∀βi < βn, ui < 0, ∀βi > Fig. 4: The crowdsourcer’s utility under different participation βn and βi = 0, βi = βn. Interestingly, a user’s optimal effort levels with a conditional optimal extrinsic reward function. level is not affected by network effects, which are counteracted γ = 1/2, b ∈ [1, 10],F (·) ∼ UNIF(−1, 15). by the second term of extrinsic reward functions. Corollary 2. Strict individual rationality. With the extrinsic reward functions given by Proposition 5, every participant  n µ ln(1 + b) − bβn + E(n) , n ∈ [0,F (βe)), receives strictly positive utility, i.e., ui > 0, ∀βi ∈ [β, βn).   R βn  UA + µ ln(1 + g(β))− In particular, the marginal user’s utility is zero, i.e., ui = βe U = F (β) 0, βi = βn. (β + )g(β)dF (β) + E(n)n, n ∈ [F (β),F (β)),  f(β) e b  The proof of Corollary 2 is given in the appendix. UB + n µ ln(1 + b) + E(n) − bβn , n ∈ [F (βb), 1]. As the net cost increases, a user’s optimal effort level (25) in which U = µ ln(1 + b) − bβF (β), U = F (β)µ ln(1 + declines, as shown in Fig. 3. µ reflects the crowdsourcer’s A e e B e R βb appreciation for users’ contributions. If µ is higher, the crowd- b) − bβ − F (β) µ ln(1 + b) − bβ + µ ln(1 + b(β)) − e b b βe sourcer is willing to elicit more user contributions with higher b(β)β − F (β) g(β)dF (β). extrinsic rewards. f(β) The key idea of Proposition 6 is to achieve the participation level that maximizes the crowdsourcer’s utility. Fig. 4 illus- 10 µ=10 trates the attainable utility under each participation level with 9 µ=20 the conditional optimal extrinsic reward function. The optimal 8 µ=30 participation level rests at the peak of each curve. The detailed 7 proof of Proposition 6 is in the appendix. 6 It is difficult to solve the maximization problem in Propo- 5 sition 6 since the objective function (25) contains an integral. 4 To address this problem, we propose an efficient analytical

Optimal effort level b 3 approach. We ﬁrst transform the integral in (25) when n ∈ 2 [F (βe),F (βb)) into a quadratic sum as: 1 K 0 5 10 15 X Net cost β UK =UA + E(nK )nK + f(βk)∆β µ ln(1 + g(βk)) Fig. 3: The user’s optimal effort level g(β). b ∈ [1, 10],F (·) ∼ k=0 UNIF(−1, 15). F (βk) − g(βk)βk − g(βk) , f(βk) (26)

−1 C. Optimal Participation Level in which βk = βe+ k∆β, nk = F (βk). UK+1 can be easily calculated as: Proposition 5 gives the conditional optimal extrinsic reward function for a targeted participation level. By comparing the crowdsourcer’s utility under each participation level with the UK+1 =UK − E(nK )nK + E(nK+1)nK+1 conditional optimal extrinsic reward function, we can ﬁnd the + f(β )∆βµ ln(1 + g(β )) K+1 K+1 (27) most lucrative participation level, and the corresponding global F (βK+1) optimal extrinsic reward function. − g(βK+1)βK+1 − g(βK+1) . f(βK+1) Proposition 6. Optimal participation level. The optimal par- Fig. 5 shows the gap between the approximated utility in ∗ ticipation level is n = arg maxn U, in which: (26) and the ground truth utility in (25), when the participation 8 is, a user’s extrinsic reward increases with her effort level, but the marginal return decreases. Similar to Fig. 2, network 7 effects boost the equilibrium participation level and the crowdsourcer’s utility, as shown in Fig. 6(b) and (c). 6 D. Fixed vs. Flexible Extrinsic Reward Mechanisms

5 The flexible extrinsic reward mechanism is more efficient Ground truth ∆β=0.01 than the fixed extrinsic reward mechanism, as verified by

The crowdsourcer utility U 4 ∆β=0.05 Fig. 7. Although the flexible mechanism requires more dis- ∆β=0.1 bursement from the crowdsourcer than the fixed mechanism 3 0.1 0.15 0.2 0.25 0.3 in order to provide the extrinsic rewards to incentivize users Participation level n (Fig. 7(a)), it induces a higher user contribution level in return. Fig. 5: Algorithm 1 verification. γ = 1/2, b ∈ [1, 10],F (·) ∼ With the fixed mechanism, most participants will only provide UNIF(−1, 15). a minimum level of effort; while with the flexible mechanism, participants are stimulated to work harder in exchange for higher extrinsic rewards. As a result, the crowdsourcer has level is within the range [F (βe),F (βb)). When ∆β = 0.01, the a higher overall utility with the flexible mechanism, as shown quadratic sum can approximate the integral to near perfection. in Fig. 7(b). Interestingly, users also have a higher aggre- The largest approximation error is 0.72% when ∆β = 0.05, gate utility with the flexible mechanism, since the flexible and 1.65% when ∆β = 0.1. Selecting a proper step size mechanism gives a higher payment and motivates more users ∆β, we can compute the participation level within the range to participate. Both extrinsic rewards and intrinsic rewards [F (βe),F (βb)) that yields the highest utility, then we can find induced by network effects are augmented. This suggests that the optimal participation level over the entire range [0, 1]. the interests of users and the crowdsourcer are not necessarily Algorithm 1 summarizes the detailed process. in conflict with each other. Quite the contrary, a thriving crowdsourcing system with a higher participation level and Algorithm 1 The Optimal Participation Level a higher user contribution level will be valuable to both users Input: Network effects E(·), probability density function and the crowdsourcer. f(·), the users’ optimal effort level g(·), equivalent monetary worth of users’ contributions µ, and the step size V. RELATED WORK ∆β. Extrinsic rewards in crowdsourcing. Existing works have Output: The optimal participation level n∗. used reverse auctions and Stackelberg games to model extrin- 1: βK = β,e Umax = 0. sic rewards. With reverse auctions, the crowdsourcer selects 2: Calculate UK according to (26). users based on the bids, which reflect users’ anticipated extrin- 3: while βK ≤ βb do sic rewards [7], [13], [14]. The complexity of reverse auctions 4: if UK > Umax then is high, making them impractical in real-world implementa- 5: Umax = UK . tions. With Stackelberg games, the crowdsourcer determines 6: nmax = F (βK ). the optimal extrinsic rewards, while users compete for these 7: end if rewards by making strategic decisions on their contribution 8: βK = βK + ∆β. levels [15], [16]. Different from these two multi-winner mech- 9: Update UK according to (27). anisms, in [8], [17], the authors proposed a winner-take-all 10: end while mechanism, where a single best or designated user gets all 0 11: Find the best participation level nmax in the range the extrinsic rewards. In contrast, we focused on the interplay [0,F (βe)) and range [F (βb), 1] using fminbnd in MAT- between extrinsic and intrinsic rewards in this paper, with a 0 LAB, with the corresponding utility Umax. focus on the impact of network effects. 0 12: if Umax > Umax then Network effects. Network effects have been extensively ∗ 13: n = nmax. discussed in telecommunication networks [9], the open-source 14: else software community [10], and social networks [18]. As the ∗ 0 15: n = nmax. crowdsourcing systems connect a large number of participants, 16: end if network effects can be observed [19]. Due to network effects, users obtain higher intrinsic rewards when the total number of Fig. 6 shows the optimal extrinsic reward function, the participants increases, thus requiring less extrinsic rewards to equilibrium participation level, and the crowdsourcer’s utility compensate their costs. However, there is a lack of existing under the flexible extrinsic reward mechanism. The flexible works that take advantage of network effects for more efficient extrinsic reward mechanism remunerates users for different extrinsic rewards design. levels of contributions, as shown in Fig. 6(a). It can be Empirical studies on intrinsic and extrinsic rewards. User observed that the extrinsic reward function is concave, that behavior under the influence of extrinsic and intrinsic rewards 5 0.38 4 γ = 0.1 γ = 0.1 0.36 γ = 0.5 4 γ = 0.5 3.5 γ = 0.9 0.34 γ = 0.9 3 0.32 3 2 0.3 γ = 0.1 0.28 2.5 The crowdsourcer utility Optimal extrinsic rewards 1 γ = 0.5 γ = 0.9 Equilibrium participation level 0.26

0 0.24 2 2 4 6 8 10 0 1 2 3 4 0 1 2 3 4 Effort level Network effects intensity α Network effects intensity α (a) (b) (c) Fig. 6: Flexible extrinsic reward mechanism. b ∈ [1, 10],F (·) ∼ UNIF(−1, 15).

20 20 12 Fixed extrinsic reward mechanism Fixed extrinsic reward mechanism Flexible extrinsic reward mechanism 10 Flexible extrinsic reward mechanism 15 15 8

10 10 6

Total user utility 4

Total extrinsic rewards 5 5 The crowdsourcer utility Fixed extrinsic reward mechanism 2 Flexible extrinsic reward mechanism

0 0 0 20 25 30 35 40 20 25 30 35 40 20 25 30 35 40 Equivalent monetary worth of user contribution µ Equivalent monetary worth of user contribution µ Equivalent monetary worth of user contribution µ (a) (b) (c) Fig. 7: Fixed VS. flexible extrinsic reward mechanisms. α = 1, γ = 1/2, F (·) ∼ UNIF(−1, 20). have been explored by some empirical studies. Schweizer et the crowdsourcer. We have presented an optimal extrinsic al. [20] have used the feedback for crowdsourcing tasks as reward function in closed form, which can be easily used a potential intrinsic reward. Anawar et al. [21] have adopted by the crowdsourcer to determine the extrinsic reward to an self-determination theory to explain intrinsic motivations in individual user. Extensive simulation results have verified that a weight loss crowdsourcing system. Competitive extrinsic the proposed extrinsic reward mechanisms have outperformed rewards are found to be more efficient than fixed extrinsic existing ones that did not take network effects and the corre- rewards in [22]–[24]. However, intrinsic rewards incurred by sponding intrinsic rewards into consideration. network effects, as well as how intrinsic and extrinsic rewards interact with each other, have not been examined in existing VII.ACKNOWLEDGEMENT empirical studies. The co-authors would like to acknowledge the generous research support from a NSERC Discovery Research Program, VI.CONCLUSION a NSERC Strategic Partnership Grant titled “A Cloud-Assisted Crowdsourcing Machine-to-Machine Networking Platform for In this paper, we have proposed a new framework for Vehicular Applications” at the University of Toronto, grants extrinsic reward design in crowdsourcing systems, which, for from the 973 project 2013CB329006, RGC under the contracts the first time, exploits network effects and intrinsic rewards. CERG MHKUST609/13, 16212714, and 16203215. Instead of assuming a fixed participant population, we have shown how user participation levels evolve as a result of the VIII.APPENDIX interactions between extrinsic rewards and network effects. A. Proof of Proposition 5 Based on our analyses, we have proposed a fixed and a flexible extrinsic reward mechanism, designed to help a crowdsourcer 1) If βn ∈ [β, βe], Equation (10) becomes: β to enlist more users and attain a higher payoff by considering Z n E(n) + P (b) − βb = bdy, network effects. In particular, our flexible mechanism adjusts β the value of extrinsic rewards according to the contributions which yields P (b) = bβn − E(n). from the users, and improves the utilities of both users and 2) If βn ∈ [β,e βb], when b ∈ (b, b), Equation (10) becomes: Z βn P (b) = βb − E(n) + g(y)dy REFERENCES β [1] “Uber,” https://www.uber.com/. Z g(βn) −1 −1 [2] “Waze,” https://www.waze.com/. = g (b)b − E(n) + xdg (x). [3] “Amazon mechanic turk,” https://www.mturk.com/mturk/welcome. When b = b, combine (10) andb (2): [4] “ResearchKit,” http://www.apple.com/researchkit/?sr=hotnews.rss. 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− b(β)β + E(n) − b(x)dx − b(βn − βb) dF (β) β Z βn + µ ln(1 + b) − bβb + E(n) − b(βn − βb) dF (β) βb = UB + (µ ln(1 + b) + E(n) − bβn)F (βn).