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On the Efficiency of Sharing Economy Networks
Leonidas Georgiadis
Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Greece, leonid@auth.gr
George Iosifidis
Department of Computer Science and Statistics, Trinity College Dublin, Ireland, george.iosifidis@tcd.ie
Leandros Tassiulas
Department of Electrical Engineering and Institute for Network Science, Yale University, USA, leandros.tassiulas@yale.edu
We consider a sharing economy network where agents embedded in a graph share their resources. This is a fundamental model that abstracts numerous emerging applications of collaborative consumption systems. The agents generate a random amount of spare resource that they can exchange with their onehop neighbors, seeking to maximize the amount of desirable resource items they receive in the long run. We study this system from three different perspectives: a) the central designer who seeks the resource allocation that achieves the most fair endowments after the exchange; b) the game theoretic where the nodes seek to form sharing coalitions within teams, attempting to maximize the benefit of their team only; c) the market where the nodes are engaged in trade with their neighbors trying to improve their own benefit. It is shown that there is a unique family of sharing allocations that are at the same time most fair, stable with respect to continuous coalition formation among the nodes and achieving equilibrium in the market perspective. A dynamic sharing policy is given then where each node observes the sharing rates of its neighbors and allocates its resource accordingly. That policy is shown to achieve long term sharing ratios that are within the family of equilibrium allocations of the static problem. The equilibrium allocations have interesting properties that highlight the dependence of the sharing ratios of each node to the structure of the topology graph and the effect of the isolation of a node on the benefit may extract from his neighbors.
Collaborative consumption (Felson and Spaeth 1978), or the sharing economy, is an emerging economic trend that promotes novel models of sharing, bartering, or renting resources and services, which is opposed to traditional ownershipbased models (Botsman and Rogers 2010, Sundararajan 2016). These solutions are attracting increasing interest due to the global recession that has changed the consumer behavior, the pressing environmental concerns, and the penetration of Internet that facilitates such activities (New York Times 2013, Economist 2013, Guardian 2014). The success of sharing economy is best manifested by the fact that it already encompasses a very diverse set of models. In some cases the payments are implemented with legal tender currency while in others the sharing activities are supported by bespoken credit systems; some applications have geographicallyrestricted scope while others operate in a worldwide scale; in many models the users’ collaboration is mediated by third parties, as in the transportation network Uber, while in other cases the collaboration takes place directly among users who are colocated or have common interests. All these consumptionasaservice schemes offer sustainable and lowcost solutions to daunting consumption problems, and can boost the economy at a local or larger scale (PwC 2016).
The fundamental goal in sharing economy is to leverage the potential of ubiquitous connectivity and enable the exchange of resources among the users by exploiting the complementarity of their resource availability and demands. In decentralized implementations, this can be achieved as follows: whenever a user has some idle resource, she offers it to other users who at that time have excess needs, and benefits from the resources they offer to her in the future. Such solutions can also address problems where users have different preferences for the different resources, and need to exchange them in order each one to acquire the most valuable for her needs (Sonmez and Unver 2011). The common denominator in these scenarios is that users are both resource consumers and producers (or, prosumers), and they are free to decide how their resources will be allocated to others. Moreover, their collaboration opportunities are constrained by various network graphs. For example, ride sharing or food sharing is constrained by the geographic proximity of the participants, renewable energy sharing relies on the grid network, commodity exchanges are conditioned on the matching of the users’ needs or social ties, and so on. We will use hereafter the term sharing economy networks to describe these models where users (or, agents) embedded in a network graph exchange their resources over time, aiming to maximize their individual benefits.
Despite the proliferation and huge potential of these sharing economy networks, very important questions about their salient features and performance remain unanswered. For example, to date it is not understood if these sharing systems admit sharing equilibriums nor if there are meaningful dynamic sharing policies that can lead to these equilibriums. Also, it is not known how efficient these equilibriums are in terms of social welfare, and how robust to strategic behaviors of users who act independently or within coordinated groups. Besides, we cannot assess today how the network affects the overall performance of the sharing system or to what extent the network position of a user shapes the resources she will receive from the network.
Motivated by these important questions, we consider a general model for sharing economy networks and follow a systematic approach to articulate and analyze the following issues:

Definition and Properties of a Fair Sharing Policy. This is one of the most critical issues in sharing economy networks. Ideally, from a system design point of view, each user should receive resource commensurate to its contribution. This is necessary to establish the sense of fairness in the participants. However, this is not always possible because of the underlying graph that prescribes, for each user, the subset of users she can collaborate with. Additionally, there may be multiple feasible sharing solutions that differ on the amount of resource each user receives. We would prefer to select among them a fair outcome that balances the exchanges as much as possible. In the context of sharing economy networks, such a fair allocation is also considered efficient as it minimizes the wasted resources and maximizes the social welfare. The existence and the characterization of the properties of centrally designed fair and efficient policies (e.g., their dependency on the underlying network graph) is an important and currently open question.

Existence and Fairness of Sharing Equilibriums. Most often these systems are not controlled by a central entity that can exogenously impose such a fair solution. Instead, each user is free to decide her strategy and therefore allocate her idle resource to the neighbors from which she receives more service in return. Such interactions give rise to barterstyle competitive sharing markets where the users exchange their resources in a greedy fashion. The main question here is if these myopic strategies lead to an equilibrium allocation where each user cannot unilaterally improve her benefits, and whether these equilibriums are affected by the network constraints. In more advanced settings, groups of users might be able to form coalitions and exclude nonmembers from sharing. For example, in a WiFi sharing community such as FON (FON 2017) a subset of users may decide to serve only each other, expecting to increase their own benefits. Such strategies are very likely to deteriorate the system’s performance, e.g., resulting in isolated users, and it is important to explore if there are coalitional equilibriums that partition the network. Finally, a naturally arising question is how efficient these competitive or coalition equilibriums are, i.e., whether they are related to the above centrally designed fair sharing policy.

Dynamics of Sharing Interactions. In such decentralized systems, the issue of how the users can reach the sharing equilibriums is equally important to the existence of the latter. Therefore, it is crucial to understand whether there are meaningful and simpletoimplement dynamic allocation rules that can enable users to exchange their randomly created resources in a fashion that is fair and efficient in the longterm. For the sharing market setting, such dynamic policies must also be incentivecompatible, i.e., aligned with the users’ efforts to maximize their own benefits accrued from the sharing network. Besides, another desirable feature of these sharing policies is to rely on the minimum possible information regarding the structure of the network and the resources or decisions of the users. All these requirements sum up to the following key question: when the users devise their dynamic (over time) sharing decisions in a selfinterested and myopic fashion, having information about the respective actions only of their onehop neighbors, can the sharing network reach an equilibrium that is also fair and robust to group strategic deviations?
The problem of efficient cooperation and resource sharing, even when network constraints are involved, has been studied in several contexts. In communication and computing systems for example, many architectures rely on pooling of resources that belong to different businesses (e.g., Internet Service Providers) or even to different endusers. Examples of the latter cases include bandwidth management in peertopeer file sharing systems (Aperjis et al. 2011, Wu and Zhang, 2010), WiFi Internet sharing communities (Iosifidis et al. 2014, Efstathiou et al. 2010), online content distribution schemes (Misra et al. 2010), and so on. Similar ideas have been explored in other contexts, e.g., for facilitating renewable energy sharing in smart grid networks (Gatzikis et al 2014), or for enabling operating costs’ reduction in networks of service providers or manufacturers (Anily and Haviv 2009, Manea 2016, Falkenhausen and Harks 2013). However, prior works in those areas do not address the questions outlined above, namely the impact of the graph constraints and the analysis of the competitive interactions of the users over time. This latter element gives rise to a new type of bartering markets which are related to the general equilibrium theory and the seminal work of Arrow and Debreu (Arrow and Debreu 1954). Here we study a different setting which, not only has graph constraints similarly to other networked economies (Kakade et al. 2004b, a), but also does not presume the existence of a monetary instrument. Besides our focal point is the dynamics of agents’ decisions, an aspect that remains underinvestigated even in general competitive market models who rely on fullinformation tatonnement processes (MasColell et al. 1995).
Finally, the emergence of sharing economy has motivated early studies focusing on specific applications such as vehicle pooling (Santia et al 2014) or sharing of mobile data plans (Camilo et al 2014). Nevertheless, these studies do not answer the above fundamental questions arising in sharing economy. In the sequel we provide a comprehensive overview of the literature highlighting the recent studies about sharing economy, as well as related resource sharing models that have been proposed in pertinent areas.
In order to shed light on these questions, we employ a general model that captures many basic instances of the emerging sharing economy networks. We consider a set of users, where each one generates over time a random amount of idle resource that she does not need and therefore can allocate to her neighbors. For instance a user may have excess bandwidth in her connection plan, that may either being used to forward some neighbor’s traffic or get wasted. We assume that each user has unsaturated demand for the resources of others. This model captures situations where the users have complementary resource availability and demand generation, or different types of resources and different preferences for them. Neighborhood relationships are described by a bidirectional or directional connected graph which does not change. The sharing ratio (or, simply ratio) of total received over allocated long term average resource characterizes the performance of each user, as it quantifies the accrued benefits over her contributions. We consider that the resources can be shared only among onehop neighbors, and the shared resources are directly consumed by their recipients and cannot be distributed further in the network. This assumption captures practical distribution constraints that arise in many sharing systems.
From a system point of view, a central designer would prefer to have a vector of sharing ratios where each coordinate that corresponds to a user (or, a node in the graph), has value equal to one. Often this will not be possible due to the graph exchange constraints and asymmetries in nodes’ resource availability. For example, in a microgrid energy sharing network, some renewables might create very large amounts of energy which cannot be matched by the neighboring devices. For these cases, the lexicographically maximum (lexoptimal), or maxmin, sharing vector is a meaningful performance criterion as it is Pareto optimal and balances the shared resources as much as possible (Nace and Pioro 2008).
In the absence of a network controller however, we assume that each node makes greedy myopic allocation decisions so as to maximize the aggregate resource it receives in return from the community. The interactions of the nodes give rise to a competitive market, which however differ from previous similar models (Arrow and Debreu 1954, Wu and Zhang, 2010) due to the existence of the graph and the absence of sidepayments (money) among the nodes. We introduce the concept of sharing equilibrium that is appropriate for this setting, characterize the equilibrium allocations, and study its relation to the centrally designed maxmin fair policy.
Accordingly, we assume that subset of nodes can coordinate and form coalitions exchanging resources only with each other. A coalitional graphconstrained game with nontransferable utility (NTU) is identified in the above setup. We focus on the existence and properties of stable equilibrium allocations. Given a certain global allocation, if there is a subset of nodes that when they reallocate their own resources among themselves manage to improve the sharing ratio of at least one node in the subset, then they have an incentive to deviate from the global allocation. Therefore, when an allocation is in equilibrium, it should be strongly stable and no such subset should exist.
We study the above frameworks, that differ on the assumptions about the system control and the users behavior, and find a surprising connection among them. In particular:
(i) It is proved that there is a unique sharing equilibrium ratio vector that is a solution for the competitive market, and lies in the core of the NTU graphconstrained coalitional game, being also strongly stable. This is the maxmin fair ratio vector. This result reveals that a centrally designed meaningful fair solution is robust to nodes’ selfish strategies even if they are allowed to coordinate and form strategic groups seeking to improve their payoff. This finding has many implications for the applicability of such fair policies to sharing economy systems.
(ii) It is shown that the equilibrium exhibits rich structure and a number of interesting properties. For example, in the equilibrium allocation there is exchange of resources only among the nodes with the lowest sharing ratios and the nodes with the highest ratios, the nodes with the second lowest ratios with the set of the second highest ratios, and so on. We also study how the sharing ratios are affected by the graph properties, such as the node degree. This latter aspect is particularly important from a network design point of view as it reveals, among others, the impact a link removal or addition has on the equilibrium. Our findings hold for any graph, and therefore they can help a controller to predict or even dictate the sharing equilibrium.
(iii) Finally, we propose a distributed stochastic algorithm that can be used by the nodes in order to make sharing decisions over time. The algorithms is simple and with minimal information requirements as it allocates the resource generated at each time instance at a node to its neighbor having the highest exchange ratio at that point. This strategy is intuitive as well, since it maximizes the current sharing benefits for the users. Interestingly, it is proved that this dynamic algorithm leads to the above fair and robust sharing equilibrium points.
The rest of this paper is organized as follows. In Section id1 we present the model and the problem statement ; in Section id1 we introduce a policy that solves the problem for all three frameworks; Section id1 presents extensive numerical results and Section id1 surveys the related literature in different areas. We conclude in Section id1 where we also discuss our model assumptions. All the proofs can be found in the Appendix of the paper.
We use capital letters to denote sequences of random variables, e.g., or . Time averages of sequences are denoted with the same letter and a bar on top, e.g.,
Let denote a connected undirected graph with a set of nodes and a set of links. We denote by the set of neighbors of node , that is . We consider a system that evolves over time and we assume a slotted time operation, where slot is the time interval The “beginning” and “end” of slot are respectively the times and . The dynamics of the nodes interactions can be described as follows. At the beginning of time slot , node generates resource , where are i.i.d with mean ; to avoid complications in the discussion we will assume that are bounded, i.e., there is a real number such that The longterm average amount of produced resource by node , , will be referred to as “endowment” of node . This resource is distributed to the neighbors of according to a policy which is formally defined below.
Definition 1
A policy is a set of rules according to which the distribution of resources among the nodes in is effected over time. More specifically, a policy determines the amount or resource node gives to node at time based on the generated and allocated resources up to time We denote the class of all policies by
Under a policy , at time node gives to node amount of the resource it generates up to and since the node cannot give more than it generates, it holds for any ,
(1) 
The average amount of resource node gives to its neighbors by time is
The amount of resource node receives from its neighbors at time is and the average amount of resource received by time is, . We denote the long term average resource that user receives under policy as . Note that in general is a random variable. However, as we will see in the next section, in order to obtain policies that satisfy the objectives of interest in this work, it suffices to restrict attention to policies for which exists and has a constant and finite value.
The set of feasible long term average received resource vectors that can be achieved by policies in is denoted by . That is,
(2) 
Node is interested in maximizing its longterm average received resource . Clearly, the objectives of nodes are conflicting, as neighbors have to compete for the same resources and therefore the key issue is to decide how to allocate the resources produced by the nodes. There are two basic approaches to address this issue. Namely, one could formulate this problem as a centrally defined fairallocation problem and take into account the resource contribution of each node to the community in the longterm, so as to decide how much resource to return to it. In a different context, each node is interested in maximizing it own received resource, and this gives rise to competitive interactions and hence creates a sharing economy market. In that case, the amount of resource each node receives in the long run depends on the attained equilibrium, if any exist. Additionally, it is possible in some settings that users can coordinate with each other and form sharing groups, or coalitions, aspiring to improve their benefits by excluding nonmembers. Our goal is to analyze the longterm average performance of the nodes’ dynamic interactions in the three frameworks described above. This is formalized in the next subsection.
In this section, we consider three different problem formulations whose objectives are based on longterm averages of the quantities of interest.
In this setting, we consider a centralized policy designed to allocate resources to nodes in proportion to their contribution to the community. Ideally, in such a setting each nodes distributes in the longrun all its endowment and it would be desirable to allocate to every node resource equal to its contribution, i.e., . However, due to the resource sharing constraints imposed by the graph and the different resource endowments of the nodes, such policies will not be feasible in general. Given this, the designer would prefer to ensure the “most balanced” longterm allocation. A suitable method to achieve this goal is to employ the lexicographic optimal (or, lexoptimal) criterion, which has been extensively used for resource allocation and load balancing, for example, in communication networks (Georgiadis et al. 2002), (Nace and Pioro 2008), (Radunovic and Le Boudec 2007). This multiobjective optimization method first increases as much as possible the allocated resource to the node with the smaller sharing ratio, . Next, if there are many choices, it attempts to increase the resource allocated to the node with the second smaller sharing ratio, and so on. The resulting longterm average allocation is maxmin fair, thus as balanced as possible. Next we provide the necessary definitions.
Definition 2
Lexicographical order. Let and be dimensional vectors, and and the dimensional vectors that are created by sorting the components of and respectively, in nondecreasing order. We say that is lexicographically larger than , denoted by , if the first nonzero component of the vector is positive. The notation means that either or, .
Within this framework we are interested determining policies that induce lexicographically optimal sharing ratio vector, i.e., defining , we are interested in determining a policy such that . In the following, a vector whose sharing ratio vector is lexicographically optimal will also be called simply “lexicographically optimal” or “lexoptimal”.
Assume now that each node is an independent decision maker, interested in maximizing the longterm average resource it receives. An approach in this setup is to define exchange ratios for the node resources that have the following property: each node receives resources from its neighbors in such a manner that the node maximizes its received resource subject only to the constraint that the cost of received resource does not exceed its wealth determined by the exchange ratio and the size of its endowment (the constraints regrading the size of the endowments of the neighbors of the node are not taken into account in this optimization). The solution concept for this setup is effectively the competitive (or, Walrasian) equilibrium (Arrow and Debreu 1954), (MasColell et al. 1995), which has been also applied in communication networks (Aperjis et al. 2011), and extended to graphical economies (which exhibit localities) (Kakade et al. 2004a, b). However, for the problem under consideration, we avoid explicit exchange ratios, and introduce a closely related equilibrium concept:
Definition 3
Sharing Equilibrium. A sharing equilibrium is determined by a vector of sharing ratios with the following properties. a) If at time node gives resource to node , node expects in return (either at time or in the future) resource^{}^{}endnote: This return resource may be obtained either from node or from any other neighbor of node . This can be interpeted as follows. If node provides resource to node it gets resource credit . Node can receive this amount of resource from any of its neighbors either at time or at some time in the future. , and b) node gives the resources it generates to its neighbors in such a manner that it maximizes its received resource in the long run, under the constraint that it does not exceed the amount it is entitled by the specified exhange rate and its endowment, that is, . It is easy to see that the resulting policy should be such that,

each node distributes all its endowment to its neighbors in the long run, i.e.,
(3) 
each node distributes its generated resource at all times to the neighbors that have the smallest sharing ratio. Moreover, since each node attempts to maximize its received resource without taking into account the avaibable endowments of its neighbors, the optimization should result in received resource rate vector that satisfies: .
In this context, we are interested in determining whether equilibrium sharing ratios and associated policies exist. Moreover, we are interested in determining policies that operate without a priori knowledge of the equilibrium rates (provided that they exist), but adjust sharing ratios over time in such a manner that they eventually converge to the equilibrium ones; in addition, the longterm received resources are those that are obtained by employing policies that know a priori the equilibrium sharing ratios.
Before providing the details of this framework, let us introduce some additional notation. We denote by the subgraph of induced by a nonempty set of nodes , i.e., the graph with node set , and link set . We denote by the set of policies that operate on graph and by the set of all longterm received resource vectors that can be obtained by employing policies in
Note that the Graph may not be connected. However, the definition of policy in Section 2.1 still holds an hence the set of policies is well defined. Also, all the stated results for connected graphs hold for each of the connected components of .
In this setting we assume that subsets of nodes can coordinate to form coalitions and deviate from the proposed fair solution if this will ensure higher resources for some of them. In game theoretic terms, this behavior leads to a coalitional (or, cooperative) game (Myerson 1997) played by the nodes. Specifically, we call any nonempty subset of nodes a coalition when they allocate their resources only among each other. That is, there is no resource exchange among nodes in and nodes in its complement set . Hence, the feasible longterm resource vectors that nodes in get, are the dimensional vectors in . We refer to the set as the grand coalition. This coalitional game is one with nontransferable utilities, as resources cannot be split arbitrary among the nodes, due to the exchange constraints imposed by the graph. Our goal is to study the existence and the properties of selfenforcing longterm allocations. This property is formally captured by the notion of stability for the grand coalition.
Definition 4
Coalitional Stability. A grand coalition along with a policy that induces longterm received resource vector is called strongly stable if for any nonempty node set , there is no policy that induces an (dimensional) vector such that for all , and for at least one node . The allocation is called weakly stable if for any nonempty node set , there is no policy that induces a vector such that for all .
Note that strong stability implies weak stability but not the other way around. In particular, the concept of weak stability for the grand coalition is directly related to the concept of the core. In this coalitional framework, we ask the question: Is there a policy that renders the grand coalition stable?
In this section we describe a simple policy that achieves the objectives of all three frameworks defined in Section LABEL:subsec:OperatingFrameworks. According to each node maintains a ratio which may be interpreted as resource sharing ratio (or simply “ratio”) at time At time every node gives its generated resource to the node that has the smallest sharing ratio among its outgoing neighbors. Specifically, the policy operates according to Algorithm id1. Note that the only a priori information required for the operation of the policy is the set of endowments of the nodes. However, as will be discussed in Section id1, the policy can also operate by replacing with the time average .
Theorem 1
The following hold.

Policy is Lexicographically optimal.

Under the node sharing ratios and longterm received resources converge to the equilibrium sharing ratios and equilibrium received resources.

Policy is coalitionally stable.
Lemma 1
Under any policy it holds for any
(4)  
(5) 

The longterm average of received resources exist, i.e, .

All endowments generated by the nodes are eventually consumed, i.e.,
(6)  
(7) 
Lemma 2
is submodular i.e., it holds for every
(8) 
Lemma 3
It holds:
Lemma 4
If is a polymatoid with base , then for any there exist an such that . Hence the lexicographically optimal vector in lies in
Theorem 2
Let be a polymatroid. A vector in , is lexicographically optimal if and only if the following hold.
(9)  
(10) 
where, The lexicographically optimal vector exists and is unique.
Lemma 5
Let then: a) If then . b) If , then and .
(11) 
Theorem 3
A vector , is lexicographically optimal if and only if the following hold. If then . If then

is an independent set in graph , for .

, for .

, for .

, for .

If is odd, then
Theorem 4
Let be a lexicographically optimal vector. The ratios are equilibrium sharing ratios for the competitive framework.
Theorem 5
A policy that achieves the lexicographically optimal vector is strongly stable.
(12)  
(13) 
(14) 
(15)  
(16) 

All nodes in give always their generated resource to .

All nodes in give always their generated resource to .
(17) 
(18) 

Policy is Lexicographically optimal.

Under the node sharing ratios and longterm received resources converge to the equilibrium sharing ratios and equilibrium received resources.

Policy is coalitionally stable.
Theorem 6
On a probability space equipped with a sequence of fields let be nonnegative and measurable random variables such that
where
(19) 
Then, exists, is finite and a.s.
Lemma 6
Let It hodls
(20) 
For all it holds,
(21) 