Fair allocation in crowd-sourced systems

Assif, Mishal; Kennedy, William; Saniee, Iraj Article Fair allocation in crowd-sourced systems Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Assif, Mishal; Kennedy, William; Saniee, Iraj (2023) : Fair allocation in crowdsourced systems, Games, ISSN 2073-4336, MDPI, Basel, Vol. 14, Iss. 4, pp. 1-18, https://doi.org/10.3390/g14040057 This Version is available at: https://hdl.handle.net/10419/330050 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Citation: Assif, M.; Kennedy, W.; Saniee, I. Fair Allocation in Crowd-Sourced Systems. Games 2023, 14, 57. https://doi.org/10.3390/ g14040057 Academic Editors: Ulrich Berger, Heinrich H. Nax and Garrison Greenwood Received: 27 June 2023 Revised: 1 August 2023 Accepted: 13 August 2023 Published: 15 August 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). games Article Fair Allocation in Crowd-Sourced Systems Mishal Assif 1, William Kennedy 2and Iraj Saniee 2,* 1Department of Electrical and Computer Engineering, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA; [email protected] 2Mathematics & Algorithms Research Group, AI Lab, Bell Labs, Nokia, Murray Hill, NJ 07974, USA *Correspondence: [email protected] Abstract: In this paper, we address the problem of fair sharing of the total value of a crowd-sourced network system between major participants (founders) and minor participants (crowd) using cooperative game theory. We use the framework of a Shapley allocation which is regarded as a fundamental method of computing the fair share of all participants in a cooperative game when the values of all possible coalitions could be quantified. To quantify the value of all coalitions, we define a class of value functions for crowd-sourced systems which capture the contributions of the founders and the crowd plausibly and derive closed-form expressions for Shapley allocations to both. These value functions are defined for different scenarios, such as the presence of oligopolies or geographic spread of the crowd, taking network effects, including Metcalfe’s law, into account. A key result we obtain is that under quite general conditions, the crowd participants are collectively owed a share between 1 2 and 2 3 of the total value of the crowd-sourced system. We close with an empirical analysis demonstrating the consistency of our results with the compensation offered to the crowd participants in some public internet content sharing companies. Keywords: fair allocation; cooperative games; shapley value; network effect; social media; value function 1. Introduction Many existing and emerging online network systems and services are designed to work semi-autonomously using cloud hardware and secure software created by a small group of founders and enabled by the mass participation of crowds. We refer to (the small number of) the former as major participants and (the large number of) the latter as minor participants. Without the hardware platform and the associated reliability and security provided by the software, there would be no platform and service. But without the participation of the crowd, there would be no data or content to enable the service. Recent examples of such platforms include Waze [ 1 ] and Helium [ 2 ] and one can count Google and Facebook as older and well-established instances of such services. There is a large body of literature in the computer science community on the design and operation of such services and a growing number of publications on reliable mechanisms that automatically account for and accumulate rewards for participants (see, for example, [ 3 ]). This literature, however, has so far not discussed substantially the concept of fair allocation of the service’s total value to the crowd participants who by constantly feeding data and information to the system are key to its success. As described in [ 3 ], one observes a trend toward rewarding crowd participants via automatically counted tokens based on such measures as the number of queries, messages, or packets each participant processes, independently of the total value that the network system and service as a whole generates. This approach particularly obscures the fact that in many such network systems, crowd participants additionally provide critical local or even private information free of charge purely as part of participation which is monetized by the service provider, adding to the total value generated by the service without direct benefit to the participants. Games 2023,14, 57. https://doi.org/10.3390/g14040057 https://www.mdpi.com/journal/games Games 2023,14, 57 2 of 18 The lack of value-based allocation to participants in large-scale crowd-sourced systems is felt keenly by the general public after over two decades since the emergence of these once novel enterprises (see, for example, [ 4 ]), but interestingly, this topic has not been addressed adequately by the research and networking communities; for notable exceptions, see recent articles [ 5 , 6 ] and with respect to the narrower Internet Service Provider settlements, see older articles [ 7 – 10 ]. Motivated by the interest in revisiting the structural aspects of crowd-sourced systems by the Web 3.0 Foundation (see [ 11 ]), and more directly, the new possibilities offered by the enterprise-sourced systems as part of Industry 4.0/5.0 (see [ 12 ]), this paper aims to help bridge this gap via a novel and formal methodology for fair allocation of the total value generated by a crowd-sourced system. Our key result is the derivation of the share of crowd participants as a percentage of the total value generated by the service platform. It is instructive here to draw an analogy between this share (of the participants in a crowd-sourced system) and the share of labor as part of the national income in macroeconomics [ 13 , 14 ]. In the latter, labor is contrasted with capital, and in the former, crowd with the system. Interestingly, we observe essentially the same split: our derivation shows a share for the crowd between 1 2 and 2 3 of the total value and macroeconomic data consistently exhibit a labor share between 50% and 70% of the national income; see charts in [ 13 , 14 ] and numerous other national statistical surveys. Thus, additionally, our result may be regarded as a theoretical explanation of the well-known split of shares between labor and capital in large-scale economic systems. The paper is organized as follows. In Section 2, we describe the role and requirements of value functions that are necessary for the computation of the Shapley value in a cooperative game. Our aim is to make these functions as simple as possible while capturing the key contributions of major and minor participants in crowd-sourced systems. In Section 3, we consider various models for a single crowd-sourced system consisting of a major participant or founder, and a large number of minor participants or a crowd. We derive close-formed expressions for Shapley allocations to both for a general class of value functions under different regimes. A key result here is that under quite general and plausible conditions, crowd participants collectively receive a payoff of at least 1 2 of the total value of the crowd-sourced system. We next consider a broad extension of our methodology to oligopolies of crowdsourced network systems in Section 4whereby distinct collections of single-founder crowdsourced systems agree to cooperate via pair-wise agreements. We show similar results hold in terms of the fair share of participants. Interestingly, and in contrast to the single crowd-sourced system, the ratio of the fair share of major to minor participants now increases with the number of inter-crowd connections each major participant contributes to. In Section 5, we present a model of geographic crowd-sourced systems and obtain closed-form expressions for the Shapley allocation to each community which continues to exhibit the 1 2 to 2 3 allocation to the crowd participants. We close with an empirical study of crowd-sourced systems whose public financial statements help estimate the actual revenue share of the crowd which we observe to be consistent with our model’s predictions. 2. Fair Allocation in Cooperative Games A formal derivation of fair allocation in cooperative games was first introduced by Shapley [ 15 ] and numerous extensions of it have been considered since. A good introduction is found in [ 16 ]. To summarize the main concept, we consider the setting where a group of participants N are cooperating towards a common goal and generate value ν(N) as a result, and this value needs to be distributed among all the participants in a fair way. Suppose that there is a value function ν: 2 N→R+ that takes any subset S of N as input and outputs the value that would have been generated by the coalition formed by only the participants in S , as opposed to all of N . Now, we denote by φi(ν) the payoff that participant ireceives as part of the grand coalition, i.e., when S=N. Fairness. There are a few natural properties that a fair allocation scheme must satisfy: Games 2023,14, 57 3 of 18 1. ∑ i∈N φi(ν) = ν(N), 2. ν(S∪ {i}) = ν(S∪ {j})) for all S=⇒φi(ν) = φj(ν), 3. ν(S∪ {i}) = ν(S)for all S=⇒φi(ν) = 0, 4. φi(ν1+ν2) = φi(ν1) + φi(ν2). (1) Property (3) above null player states that any player i that adds no value to any coalition S does not receive any payoff in the grand coalition, while Property (2) symmetry states that any two players i and j that add the same value to every coalition S should receive the same payoff. Participants are rewarded only based on the value they add and no other factors. Property (1) Efficiency states that the total value generated ν(N) is completely distributed among the participants, and Property (4) linearity says that for two independent games, the fair payoff to each participant must be the sum of the fair payoffs in each independent game. From these conditions, Shapley derives the following expression for the unique payoff φi(ν)to each participant i: φi(ν) = ∑ S⊆N−{i} |S|!(|N|−|S| − 1)! |N|!(ν(S∪ {i})−ν(S)). (2) Here, the expression in brackets on the extreme right hand side is the marginal value that agent i adds to coalition S and this marginal value is averaged out for all possible 2|N|−1coalitions. The payoff φi(ν)in (2) is known as the Shapley allocation to participant i. 3. A Single Crowd-Sourced System We think of a crowd-sourced system ( CSS from here on) as a cooperative game in which one entity (the founder) provides the main infrastructure, which could be physical (hardware) or virtual (software), of a value-generating service that can only succeed if a large number of spatio-temporal agents participate to enable local match of supply and demand for the said infrastructure service. In this setting, we consider a collection of participants denoted by N={g , u1 , u2 , . . . , un} where g is the founder and ui , 1 ≤i≤n are the crowd member identities. For a complete list of variables, see the Abbreviation section. There is much symmetry in this setting which we exploit fully in the sequel. It is assumed that the value ν(S) of any subset S of N is zero unless it includes the founder, and in the case where the founder is in S , its value is equal to a function of the size of the crowd. This is a restatement of the notion of a CSS (crowd-sourced system) in that it offers much credit to a founder for imagining and initiating a distributed enterprise enabled by a large number of uncoordinated participants who constantly feed the system. This form of the value function also makes it possible to derive closed-form expressions for the Shapley payoff of all participants in various settings, as we observe below. We use a power of the size of the crowd as its value: a linear function corresponds to standard scaling. The quadratic function has been widely discussed in the networking literature as Metcalfe’s law Remark 1, and higher, super quadratic, powers are included for benchmarking purposes. 3.1. Coalition Value Based on Revenue 3.1.1. Case 1 (Identical Participants) The first situation we consider is one where all the potential participants are identical. Any coalition that does not contain the major participant g is treated as one with no value, while the value of a coalition including g grows as a power of its size. The motivation for this model comes from the network effect, sometimes referred to as Metcalfe’s law, which states that the value of any network grows proportional to the square of its number of participants. This empirical law, originally devised in the context of Ethernet networks [ 17 ], seems to also hold remarkably well in many modern situations such as cryptocurrency networks [ 18 ] and Games 2023,14, 57 4 of 18 social networks [ 19 ]. The model can be formally written as N={g , u1 , . . . , un} , ν: 2 N→R and ν(S):=(0 if g/∈S ρ|S− {g}|kif g∈S,(3) where ρis some constant. Metcalfe’s law is simply a special case of our model with k=2. Here, we list a few key properties of ν(S) as defined above and explain why we believe this simple form captures the key aspects of a crowd-sourced system valuation. We observe that (1) the founder g plays a critical role in the definition of the value function in that any subset which does not include it has value 0. It is not hard to show that this value function ν defined in (3) is supermodular, and the Shapley values derived in the sequel are stable; see the end of Section 2; (2) for any specific crowd participant, there is no special incentive to join any of the numerous coalitions of size S , as the value function only depends on the size of the coalition and not its constitution. This means that the basic assumption in (2), stating that all coalitions of size Sare equivalent, is met; (3) the simple form of ν we assume—the linear, quadratic or higher powers—does not need a saturation effect, since the more crowd participants, the larger the total value of a coalition. This is in contrast with saturating systems where (say) the first 100 crowd participants provide a higher value than the last 100. (4) we note that in the analysis in the following pages, relatively small values of S offer us the asymptotic results with respect to the exponent k , which equals one and two, corresponding to linear and quadratic value functions. Our first result characterizes the Shapley value associated with each participant as a fraction of the value of the total coalition. We now look at a general computation that is useful later as well. Lemma 1. Consider the coalition game with the set of agents N={g , u1 , . . . , un} and value ν given by ν(S):=(0if g /∈S f(|S|)if g ∈S. The Shapley values of the game satisfy φg(ν) = 1 n+1 n ∑ s=0 f(s),φui(ν) = ν(N)−φg(ν) n. Proof. See Appendix A.1. Theorem 1. Consider the coalition game with the set of agents N={g , u1 , . . . , un} and value ν given by (3). The associated Shapley values satisfy φg(ν) = ρnk k+1+O(nk−1),φui(ν) = kρnk−1 k+1−O(nk−2)(4) and lim n→∞ φg(ν) ν(N)=1 k+1, lim n→∞ ∑n i=1φui(ν) ν(N)=k k+1. (5) Games 2023,14, 57 5 of 18 Proof. Applying the results of Lemma 1to the value (3), we obtain φg(ν) = ρ n+1 n ∑ s=0 sk=1 n+1 ρnk+1+Onk k+1 ≈ρnk k+1=1 k+1ν(N)for large n, nφui(ν) = ν(N)−φg(ν)≈ρk k+1ν(N). Remark 1. The results obtained in Theorem 1tell us that when k= 1, i.e., when the power of a network grows proportional to the number of participants, the value of the grand coalition is shared equally between the major participant and the collection of minor participants. In other words, φg(ν)≈ν(N) 2and ∑n i=1φui(ν)≈ν(N) 2. In the popular case of Metcalfe’s law where k= 2, the theorem says that the major participant should receive 1/3 of the value generated while 2/3 should be distributed equally among the minor participants. 3.1.2. Case 2 (Non-Identical Participants) The assumption that all participants are (nearly) identical might not hold in many settings; for example, when some participants contribute significantly more to the value of the system than others due to their influence in a social network. A slightly more refined model for the value would be ν(S):=(0 if g/∈S ρ∑i∈SWα ikif g∈S,(6) where Wi refers to some notion of the amount of work performed by participant i , e.g., the number of messages routed by i . We denote by fi:=Wα i ∑jWα j the share of total work performed by i. Theorem 2. Consider the coalition game with the set of agents N={g , u1 , . . . , un} and value ν offered by (6)where k =2. The associated Shapley values satisfy lim n→∞ φg(ν) ν(N)=1 3+∑ i f2 i 6, lim n→∞ φui(ν) ν(N)=2 3fi−f2 i 6. (7) Proof. See Appendix A.1. Remark 2. The results obtained in Theorem 1under the identical participant model can be recovered from the above theorem by assuming Wi= 1. In fact, in the case where the workload is more or less uniformly distributed and not too concentrated, as formalized by the condition lim n→∞∑ i f2 i=0, the above theorem tells us that φg(ν) ν(N)≈1 3 and φui(ν) ν(N)≈2 3fi , which is equivalent to dividing the value between the founder and the crowd in the same way as in the identical participant model, and then dividing the value among the crowd proportional to the share of work performed. Games 2023,14, 57 6 of 18 Remark 3. In Theorem 2, we treated the case k= 2. However, analogous results hold true for higher values of k as well. It can be shown that lim n→∞ φg(ν) ν(N)=1 k+1+O ∑ i f2 i! lim n→∞ φui(ν) ν(N)=k k+1fi−Of2 i. 3.2. Coalition Value Based on Profit A legitimate concern regarding the use of revenue as a proxy for value is that the founder could be incurring significant costs for infrastructure which the minor participants do not. In this case, the resulting profit, revenue minus cost, is a more natural proxy for value. A question which our Shapley allocation derivation answers in the positive is the observable impact of the large infrastructure cost incurred by the founder compared to the small cost incurred by the crowd members. We modify the model to take into account such costs. We assume that each minor participant incurs a fixed constant cost ku and the major participant pays a cost of Kg per minor participant with Kg>> Ku . Formalizing all this, we obtain a new value function ν: 2N→R ν(S):=     0 if g/∈S ρ|S− {g}|k−Kg|S− {g}|− ku|S− {g}| if g∈S. (8) We compute the Shapley values in this case by appealing to Lemma 1again. Theorem 3. Consider the coalition game with the set of agents N={g , u1 , . . . , un} and value ν given by (8). The associated Shapley values satisfy φg(ν) ν(N)≈ ρnk k+1−Kgn 2−kun 2 ρnk−Kgn−kun ∑n i=1φui(ν) ν(N)≈ ρknk k+1−Kgn 2−kun 2 ρnk−Kgn−kun (9) for large values of n. Proof. Using the results of Lemma 1, the exact same computation in the proof of Theorem 1 delivers this result. Remark 4. Even though in the limit n→∞ , the results of Theorem 3collapse to that of Theorem 1, the value of Kg might be large enough that the contribution of Kgn cannot be ignored for the moderately large values of n that typically arise. Remark 5. In this case, the ratio of the share of the founder to the share of the crowd is φg(ν) ∑n i=1φui(ν)= ρnk k+1−Kgn 2−kun 2 ρknk k+1−Kgn 2−kun 2 . This shows that when k> 1, as the cost of the system infrastructure Kg increases, the share of the founder compared to the crowd of participants decrease. This may seem at first counterintuitive or unfair, especially compared to the case of k= 1 where the share is constant regardless of the cost of the infrastructure. This phenomenon occurs because the revenue should be divided according to 1 k+1and k k+1between the founder and the crowd, whereas the total cost is split equally. Thus, the superlinear scaling of revenue and the Games 2023,14, 57 7 of 18 linear scaling of cost offer a decreasing proportional share to the founder even though the absolute value of the founder share is still increasing. Another observation regarding (6) is that while the share of the founder φg(ν) is increasing in k , the founder’s profit decreases as a percentage of the total profit. This again is a consequence of the revenue of the network growing as a power of its size. 4. Oligopolies of Crowd-Sourced Systems We now consider the case of multiple crowd-sourced systems where major participants or founders are willing to cooperate, thus inter-working CSS s for a higher total payoff to all. Let us assume that each major participant v has nv minor participants in its CSS(v) , crowd-sourced system affiliated with v . The major participants now agree to cooperate for higher payoff by inter-networking their disparate crowds. This is achieved through a series of bilateral agreements between pairs of major players. The key consequence of such cooperation between major participants i and j is joint creation of a payoff which is proportional to (ni+nj)2 for the Metcalfe value model we considered previously, in contrast to n2 i+n2 j for the case i and j do not cooperate. The question we wish to answer is fair allocation to each CSS , and each major and minor participants in each CSS assuming global cooperation represented by a graph Gas defined below. 4.1. Shapley Value of an Oligopoly: A Coarse Grain Model We now assume that major participants form a set of pairwise, bilateral agreements amongst themselves. We represent this by a connected graph G= (V , E) whose vertices v∈V denote CSS s associated with major participants and each edge in E denotes a bilateral agreement. A natural extension of the notion of Metcalfe value to a graph Gis given by ν(G) = ∑ v∈V n2 v+∑ (u,w)∈E nunw. (10) Here, n2 v is the Metcalfe law restricted to the system v and each edge (u , w) adds an additional nunwto the value of Gdue to the inter-networking between vertices uand w. We assume that the crowd-sourced systems associated with major participants v∈V form the set of agents of a cooperative game. The value of a subset S⊂V is the Metcalfe value (10) of the subgraph G(S) induced by S . This is the graph consisting of vertices S and all edges of G , both ends of which lie in S . This is implicitly an aggregate model that clubs together all the minor participants of a CSS with the major participant, modelling the scenario where each minor participant has already been bound to its major participant, perhaps by means of an agreement or due to some external constraints such as geography that limits the choice of the minor participant. As a result, in the following, φu(ν)denotes the share of the crowd-sourced system associated with major participant u , and not just the share of the major participant uin CSS(u)by itself (which is discussed in Remark 7). The following result characterizes the Shapley value of each CSS. Theorem 4. Consider a graph G= (V , E) and a cooperative game with set of agents N=V and value function ν(S) = ∑ v∈S n2 v+∑ (u,w)∈S×S∩E nunw. The Shapley value of each vertex u is given by φu(ν)(≡φCSS(u)(ν)) = n2 u+∑ w∈N(u) nunw. (11) Proof. See Appendix A.2. For example, in the oligopoly shown in Figure 1, we can compute the Shapley values as Games 2023,14, 57 8 of 18 φA(ν) = nA(nA+nB+nC), φB(ν) = nB(nA+nB+nC+nD), φC(ν) = nC(nA+nB+nC), φD(ν) = nD(nB+nD). nA nB nC nD Figure 1. Graph of four crowd-sourced systems with four bilateral agreements and their Shapley allocations (Theorem 4). Remark 6. In the special case where all major participants have an equal number of minor participants within their system, i.e., nv=n for all v ∈V, we obtain φu(ν) = n2(1+deg(u)). (12) Remark 7. If we consider V={g , u1 , . . . , un} , that there is an edge between g and ui for each i= 1, . . . , n , and that nui= 1and nug= 1, we obtain a model that is very close to (3) with k= 2. The only difference here is that the coalitions that do not contain g have a non-zero value that scales linearly with its size. Unsurprisingly, we obtain φg(ν) = 1+deg(g) = 1+n, φui(ν) = 2=⇒ν(N) = 3n+1, =⇒lim n→∞ φg(ν) ν(N)=1 3, lim n→∞ ∑n i=1φui(ν) ν(N)=2 3, exactly the same as in Theorem 1. 4.2. Shapley Value of an Oligopoly: Fine-Grain Model Theorem 4and the Remarks 6and 7derive the Shapely value of each crowd-sourced system ( CSS ) associated with each vertex in the oligopoly game of graph G . What remains is to identify the Shapley fair share of each participant in each CSS associated with each major player v∈G . To achieve this, we need to define a new value function on each CSS which takes its size and sizes of its neighboring CSSs explicitly into account. To this end, we consider a more fine-grained model here. We consider the set Uv of minor participants in the CSS associated with major participant v∈V as separate agents, making the total set of agents of the cooperative game N=∪v∈VUv∪V. A subset S⊂N can be partitioned as Sv=S∩Uv for each v∈V and SV=S∩V . The value of the coalition is computed based on two simple principles: each v∈SV contributes |Sv|2 and each e= (v , w)∈G(SV) contributes |Sv||Sw| to the value of S . That is, ξ(S) = ρ ∑ v∈SV |Sv|2+∑ (u,w)∈SV×SV∩E |Su||Sw| . (13) Games 2023,14, 57 15 of 18 It follows that lim n→∞ φui(ν) ν(N)=2 3fi−f2 i 6, lim n→∞ φg(ν) ν(N)=1−2 3∑ i fi+∑ i f2 i 6=1 3+∑ i f2 i 6. Appendix A.2. Proofs from Section 4 Proof of Theorem 4.The marginal utility of uto coalition Sis ν(S∪ {u})−ν(S) =n2 u+∑ (u,w)∈E,w∈S nunw+∑ (w,u)∈E,w∈S nunw =n2 u+∑ (u,w)∈E,w∈S 2nunw. We can then compute φu(ν) = ∑ S⊂V−{u} |S|!(|V|−|S| − 1)! |V|!(ν(S∪ {u})−ν(S)) =∑ S⊂V−{u} |S|!(|V|−|S| − 1)! |V|!n2 u+ ∑ S⊂V−{u} |S|!(|V|−|S| − 1)! |V|!∑ (u,w)∈E,w∈S 2nunw =n2 u+∑ S⊂V−{u} |S|!(|V|−|S| − 1)! |V|!∑ (u,w)∈E,w∈S 2nunw. The second term above can be reduced using Fubini’s theorem as we did before to obtain φu(ν) = n2 u+∑ (u,w)∈E ∑ w∈S⊂V−{u} |S|!(|V|−|S| − 1)! |V|!2nunw =n2 u+∑ (u,w)∈E 2nunw∑ w∈S⊂V−{u} |S|!(|V|−|S| − 1)! |V|! =n2 u+∑ (u,w)∈E 2nunw |V|−1 ∑ s=1 s!(|V| − s−1)! |V|!|V| − 2 s−1 =n2 u+∑ (u,w)∈E 2nunw |V|−1 ∑ s=1 s |V|(|V| − 1) =n2 u+∑ w∈N(u) nunw. Proof of Theorem 5.The value function ξcan be split as ξ(S) = ∑ v∈V ξu(S) + ∑ (v,w)∈E ξu,w(S)(A1) Games 2023,14, 57 16 of 18 where ξv(S) = (|Sv|2if v∈S 0 otherwise, and ξv,w(S) = (2|Sv||Sw|if v,w∈S 0 otherwise. Each of these value functions can be considered as the value function of a cooperative game in its own right. By Linearity of Shapley value, φa(ξ) = ∑ v∈V φa(ξv) + ∑ (v,w)∈E φa(ξv,w)(A2) for any player a∈N. Notice that for any v∈V, if a/∈ {v} ∪ Uv, then ξv(S∪ {a})−ξv(S) = 0, for all S⊂N, which means a is a null player in the game ξv . This implies φa(ξv) = 0 by the null player axiom of the Shapley value. The game ξv on the remaining set {v} ∪ Uv is just the single major player game (3), and the result of Theorem 1gives us φv(ξv) = n2 v 3+O(nv),φuv(ξv) = 2nv 3−O(1). Similarly, for any a/∈ {v,w} ∪ Uv∪Uw, ξv,w(S∪ {a})−ξv,w(S) = 0, for all S⊂N, which implies φa(ξv,w) = 0 by the null axiom of the Shapley value. All that remains to be computed is the Shapley value of players {v , w} ∪ Uv∪Uw with the value function ξv,w . Since we showed that all other players in this game are dummy, we can find the Shapley value of players {v , w} ∪ Uv∪Uw as their Shapley value in the simplified game whose set of players is ˜ N={v , w} ∪ Uv∪Uw and value function ξv,w . To achieve this, we again split the value function ξv,was ξv,w(S) = ∑ uv∈Uv ∑ uw∈Uw ξuv,uw v,w(S) where ξuv,uw v,w(S) = (2 if S={v,w,uv,uw} 0 otherwise, and compute the Shapley value of the players for each of these value functions separately. We observe that any player ¯ uv∈Uv− {uv} or ¯ uw∈Uw− {uw} is a null player of the value function ξuv,uw v,w, which means φa(ξuv,uw v,w) = 0, if Uv3a6=uvor Uw3a6=uw. This means we can again reduce the set of players to {v , w , uv , uw} and value function to ξuv,uw v,w , and notice that this value function is symmetric with respect to all four players {v,w,uv,uw}which implies φa(ξuv,uw v,w) = (1 2if a∈ {v,w,uv,uw} 0 otherwise. Games 2023,14, 57 17 of 18 We can now compute the original Shapley values as φa(ξv,w) = ∑ uv∈Uv ∑ uw∈Uw φa(ξuv,uw v,w). When a=vor a=w, φa(ξv,w) = ∑ uv∈Uv ∑ uw∈Uw 1 2=|Uv||Uw| 2=nvnw 2. When a∈Uv, φa(ξv,w) = ∑ uv∈Uv ∑ uw∈Uw 1 2δuv,a=|Uw| 2=nw 2 and when a∈Uw, φa(ξv,w) = ∑ uv∈Uv ∑ uw∈Uw 1 2δuw,a=|Uv| 2=nv 2. Putting everything together in (A2) , we obtain the results of Equations (11) and (15) . References 1. Waze Company. Available online: http://waze.com/company (accessed on 13 July 2023). 2. Helium. People-Powered Network. Available online: http://helium.com (accessed on 13 July 2023). 3. Voshmgir, S. Token Economy: How the Web3 Reinvents the Internet; Token Kitchen: Berlin, Germany, 2019. 4. Lanier, J. Who Owns the Future? Simon and Shuster: New York, NY, USA, 2013. 5. Jia, R.; Dao, D.; Wang, B.; Hubis, F.A.; Hynes, N.; Gürel, N.M.; Li, B.; Zhang, C.; Song, D.; Spanos, C.J. Towards efficient data valuation based on the shapley value. In Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, Naha, Okinawa, 16–18 April 2019; pp. 1167–1176. 6. Li, H.; Vincent, N.; Chancellor, S.; Hecht, B. The Dimensions of Data Labor: A Road Map for Researchers, Activists, and Policymakers to Empower Data Producers. In Proceedings of the 2023 ACM Conference on Fairness, Accountability, and Transparency (FAccT ’23), Chicago, IL, USA, 12–15 June 2023; Association for Computing Machinery: New York, NY, USA, 2023; pp. 1151–1161. [CrossRef] 7. Ma, R.T.; Chiu, D.M.; Lui, J.C.; Misra, V. Internet Economics: The use of Shapley value for ISP settlement. IEEE/ACM Trans. Netw. 2010,18, 775–787. [CrossRef] 8. Ma, R.T.; Chiu, D.M.; Lui, J.C.; Misra, V.; Rubenstein, D. On cooperative settlement between content, transit, and eyeball internet service providers. IEEE/ACM Trans. Netw. 2010,19, 802–815. [CrossRef] 9. Cheung, Y.; Chiu, D.M.; Huang, J. Can bilateral ISP peering lead to network-wide cooperative settlement. In Proceedings of the 2008 IEEE 17th International Conference on Computer Communications and Networks, St. Thomas, VI, USA, 3–7 August 2008; pp. 1–6. 10. Misra, V.; Ioannidis, S.; Chaintreau, A.; Massoulié, L. Incentivizing peer-assisted services: A fluid Shapley value approach. ACM SIGMETRICS Perform. Eval. Rev. 2010,38, 215–226. [CrossRef] 11. Web 3.0 Foundation. Available online: https://web3.foundation/ (accessed on 13 July 2023). 12. 4th Industrial Revolution. Available online: https://en.wikipedia.org/wiki/Fourth_Industrial_Revolution (accessed on 13 July 2023). 13. Mu´ck, J.; McAdam, P.; Growiec, J. Will the “True” Labor Share Stand Up? An Applied Survey on Labor Share Measures. J. Econ. Surv. 2018,32, 961–984. [CrossRef] 14. Estimating the U.S. Labor Share. Available online: https://www.bls.gov/opub/mlr/2017/article/estimating-the-us-labor-share. htm (accessed on 13 July 2023). 15. Shapley, L.S. Notes on the n-Person Game—II: The Value of an n-Person Game; RAND Corporation: Santa Monica, CA, USA, 1951. 16. Roth, A.E. Introduction to the Shapley value. In The Shapley Value; Cambridge University Press: Cambridge, UK; 1988; pp. 1–27. 17. Metcalfe, B. Metcalfe’s law after 40 years of ethernet. Computer 2013,46, 26–31. [CrossRef] 18. Peterson, T. Metcalfe’s Law as a Model for Bitcoin’s Value. Altern. Invest. Anal. Rev. Q 2018,2, 9–18. [CrossRef] 19. Zhang, X.Z.; Liu, J.J.; Xu, Z.W. Tencent and Facebook data validate Metcalfe’s law. J. Comput. Sci. Technol. 2015 ,30, 246–251. [CrossRef] 20. Alexis, M. How Much Does Spotify Pay Per Stream? 2022. Available online: https://twostorymelody.com/spotify-pay-per- stream/ (accessed on 13 July 2023). Games 2023,14, 57 18 of 18 21. Mulroy, C. Spotify Pays Artists (Sort of), but Not per Stream. 2022. Available online: https://www.usatoday.com/story/life/20 22/10/22/how-much-per-spotify-stream/8094437001/ (accessed on 13 July 2023). 22. Kormoczi, R. How Much Can an Artist Earn with Pandora Radio? 2020. Available online: https://timesinternational.net/howmuch-can-an-artist-earn-with-pandora-radio/ (accessed on 13 July 2023). 23. Bergen, M.; Shaw, L. Youtube Has Paid out $30 Billion to Creators as the Competition for Online Content Intensifies? 2021. Available online: https://fortune.com/2021/08/23/youtube-30-billion-ad-sales-online-creators/ (accessed on 13 July 2023). 24. How Much Does YouTube Pay Video Creators? 2021. Available online: https://screencast-o-matic.com/blog/how-much-does- youtube-pay-video-creators (accessed on 13 July 2023). Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.