Consensus reaching with heterogeneous user preferences, private input and privacy-preservation output

le Cadre, Hélène; Bedo, Jean-Sébastien Article Consensus reaching with heterogeneous user preferences, private input and privacy-preservation output Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: le Cadre, Hélène; Bedo, Jean-Sébastien (2020) : Consensus reaching with heterogeneous user preferences, private input and privacy-preservation output, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 7, pp. 1-17, https://doi.org/10.1016/j.orp.2019.100138 This Version is available at: https://hdl.handle.net/10419/246409 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-nc-nd/4.0/ Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp Consensus reaching with heterogeneous user preferences, private input and privacy-preservation output ☆,☆☆ Hélène Le Cadre a,⁎ , Jean-Sébastien Bedo b a VITO/EnergyVille, Thor Scientific Park, Genk 3600, Belgium b Orange, avenue du Bourget, Evere, Belgium ARTICLE INFO Keywords: Matching markets Preferences Nash equilibrium Privacy Consensus ADMM ABSTRACT This paper deals with a generic problem of matching agents with underlying preferences while guaranteeing a certain level of privacy is met. As a general framework, we consider consumers and prosumers who trade energy on a platform. Consumers buy energy to the platform to maximize their usage benefit while minimizing the cost paid to the platform. Prosumers, who have the possibility to generate energy, self-consume part of it to maximize their usage benefit and sell the rest to the platform to maximize their revenue. Inspired by a variant of the Hotelling model, product differentiation is introduced and consumers can specify preferences regarding locality and green origin of their supply. The consumers and prosumers problems being coupled through a matching probability, we provide analytical characterization of the resulting Nash equilibrium, and conditions for existence and uniqueness. Assuming supply shortages occur on the platform, we reformulate the local market clearing problem as a consensus problem that we solve using Consensus Alternating Direction Method of Multipliers (C-ADMM), enabling minimal information exchanges between prosumers and consumers. C-ADMM complexity is recalled and strategyproofness is analysed. The algorithm is then run on a case study made of 300 prosumers from New South Wales in Australia, equipped with solar panels. We consider privacy-preservation output against a centralized benchmark approach, and evaluate C-ADMM computational time under three scenarios with an increasing number of agents. Regarding economic analysis, we observe that it is more profitable for prosumers than for consumers to be flexible within a local energy community, and that belonging to a local energy community incentivizes them to reduce their demands by comparison with their initial targets. Furthermore, the expectation to make a substantial profit is a main driver for prosumers’ engagement within a community; whereas for consumers, the green origin of the supply is determinant. 1. Introduction 1.1. From centralized to decentralized electricity markets The increasing amount of Distributed Energy Resources (DERs), which have recently been integrated in power systems, the development of new storage technologies, and the more proactive role of consumers (prosumers) have transformed the classical centralized power system operation (mostly based on centralized unit commitment) by introducing more uncertainty and decentralization in the decisions. Following this trend, electricity markets are starting to restructure, from a centralized market design in which all the operations were managed by a global (central) market operator, modeled as a classical constrained optimization problem, to more decentralized designs involving local energy communities which can trade energy by the intermediate of the global market operator [22,39] or, in a peer-to-peer setting [15,23,25,29,36,42]. Coordinating local Renewable Energy Sources (RES)-based generators to satisfy the demand of local energy communities, could provide significant value to the power systems, by decreasing the need for investments in conventional generations and https://doi.org/10.1016/j.orp.2019.100138 Received 2 July 2019; Received in revised form 10 December 2019; Accepted 23 December 2019 ☆ A preliminary version of the paper was presented at the 9th EAI International Conference on Game Theory for Networks on April 25–26, 2019 [25]. Part of the Introduction section originally appeared in the conference paper, however it has been significantly enriched in the current version of the paper: we introduce consumers’ preferences relying on a variant of the Hotelling model. This implies different analytical results for the noncooperative game equilibrium model of the two-sided market. Complexity, strategyproofness and computational time of C-ADMM are also additions to the current version of the paper. Finally, the case study relies on an extensive real database from Australia. On the contrary, in [25], we used only synthetic data. ☆☆ The authors would like to acknowledge the two anonymous referees for their detailed reviews and comments. ⁎ Corresponding author. E-mail addresses: [email protected] (H. Le Cadre), [email protected] (J.-S. Bedo). Operations Research Perspectives 7 (2020) 100138 Available online 26 December 2019 2214-7160/ © 2019 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/). T transmission networks. In practice, the radial structure of the distribution grid calls for hierarchical market designs, involving transmission and distribution network operators [24]. But, various degrees of coordination can be envisaged, from full coordination organized by a global market operator (transmission network operator), to bilateral contract networks [30], to fully decentralized market designs allowing peer-to-peer energy trading between the prosumers in a distributed fashion [23,29], or within and between communities/coalitions of prosumers [38,40]. The end of the feed-in-tariff also calls for new market mechanisms to avoid the wasting of prosumers’ energy surpluses while guaranteeing a significant investment in RES-based technologies to reach the ambitious renewable production target in the energy mix fixed by the EU. In the energy sector, peer-to-peer energy trading is a novel paradigm of power system operation, where prosumers providing their own energy from DERs such as solar panels, wind turbines, combined heat and power (CHP), gas boilers, storage technologies, demand response mechanisms, etc., exchange energy/capacity with one another. Zhang et al. provide in [44] an exhaustive list of projects and trails all around the world, which build on new innovative approaches for peer-to-peer energy trading. A large part of these projects rely on digital platforms which match RES-based generators and consumers according to their preferences and locality aspects (such as Piclo in the UK [47], Trans- Active Grid in Brooklyn, US [48], Vandebron in the Netherlands [49], etc.). In the same vein, cloud-based virtual market places to deal with excess generation within microgrids are developed by PeerEnergyCloud [5] and Smart Watts [14] in Germany. Some other projects rely on local community building for investment sharing in batteries, solar PV panels, etc., in exchange of bill reduction or to obtain a certain level of autonomy with respect to the global grid (such as Yeloha and Mosaic in the US [27], SonnenCommunity in Germany which has recently been bought by Shell, etc.). 1.2. Two-Sided matching market literature and the rise of the sharing economy In many papers in the energy market literature and more generally, in classical commodity markets, the market clearing price determines whether a prosumer (or more generally, an agent bidding in the market) is a consumer or a generator. On the contrary, the problem we consider in this paper describes a “two-sided matching market”. The term “two-sided” refers to the fact that agents in such a market belong, from the outset, to one of two disjoint sets [35] – e.g., consumers without generation facility whose demand is supplied by a platform connected to the grid and prosumers with generation facilities who have the possibility to consume part or all of their self-generations without buying anything from the grid to meet their demands. Note that from one time period to another, the roles of the agents might change but we will consider it as fixed for the time period over which the market clearing occurs. Typically, at night, all the agents are consumers, since the solar panels do not produce anything. In our model, the matching process itself is not considered, we consider instead in the utility functions of the prosumers the probability that they are matched to a consumer. As we impose no condition on the number of consumers to which a prosumer having energy in surplus is matched, our matching model is one-to-many. The theoretical part of the two-sided market literature started in 1962 with the seminal work of Gale and Shapley on college admission which allows complex heterogeneous preferences and (possibly) limitations on how parties may split the surplus of a relationship; and the stability of marriages, which assumes simple preferences, with men and women being ranked from the best to the worst and transferable utility functions [2,35]. The family of models has since then been extended by considering stability issues and internal structure of the set of stable outcomes, while proposing computational algorithms for labor market for physicians in the US looking for a position after the medical school; and auction markets where coalitions of agents can collude to influence the outcome [35]. The rise of the sharing economy, understood as an umbrella concept that encompasses several information and communication technology (ICT) developments, among others collaborative consumption (endorsing sharing the consumption of goods and services through online platforms [16]) has been triggering new research questions regarding efficiency in matching, pricing strategies, equilibrium analysis, etc. [13]. Boysen et al. highlight the better performance reached by optimization-based matchings of supply and demand, compared to traditionally used list-based approaches, and detail the resemblance of the matching task in the sharing economy with other problem settings from a structural point of view, such as machine scheduling [20], interval scheduling, jobs assignment, etc. They propose a classification of static and deterministic matching problems and provide complexity analysis through the identification of appropriate polynomial time algorithms, well-known to the operations research community, or NP-hardness proofs [4]. Juding by the recent contributions in the sharing economy literature, platform design is an active area of research [2,9,12,13]. Three needs are identified for platform deployment: a first requirement is to help buyers and sellers find each other, taking into account preference heterogeneity. This requires to find a trade-off between lowentry cost and information retrieval from big, heterogeneous, and dynamic information flows. Buyers and sellers search can be performed in a centralized fashion (Amazon, Uber), or it might allow for effective decentralized search (Airbnb, eBay), or even fully distributed search (OpenBazaar, Arcade City). A second need is to set prices that balance demand and supply, and ensure that prices are set competitively in a decentralized fashion. A third requirement is to maintain trust in the market, relying on reputation, feedback mechanisms and loyalty programs. Sometimes, supply might be insufficient and subsidies should be designed to encourage sharing on the platform [12]. Fang et al. give an example of such subsidies design through loyalty programs in the sharing economy [13]. 1.3. Some definitions of privacy Various privacy models have been developed in the data science and machine learning literature. We review some of them below. In the context of privacy of databases, popular approaches include k-anon- ymity and (epsilon-delta) differential privacy, a detailed review of both is presented in [26] and summarized below. For databases, the first definition of privacy comes with the idea of k-anonymity, which is a property of protecting released data from reidentification. It can be applied when private data – such as energy load profiles – need to be shared for public usage with the constraint that individual subjects of the data cannot be re-identified from the released data, so as to protect their privacy – e.g., in that context, their name, address, telephone number, etc. In other words, all the records in the released database should remain unlinkable to the consumers. A first possibility is to remove the sensitive information. However, quasiidentifier attributes such as age, gender, race, zip code, that can be found from external databases could be used to infer the identity of the consumers [26].k-anonymity requires that in the released data, each record can be mapped to at least krecords in the original data, e.g., each record from the released data will have at least k1 identical records in the same released data. It has been proven that under kanonymity, external data cannot be used to infer private input. Intuitively, this is because each record in released data will have at least k1 same records. Differential privacy has been proven to be more robust than kanonymity against attacks [26]. The intuition underlying the notion of differential privacy is that an agent’s privacy cannot be compromised by a statistical release if their data are not in the database. Therefore with differential privacy, the goal is to give each individual roughly the same privacy that would result from having their data removed. That is, H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 2 the statistical (such as query) functions run on the database should not overly depend on the data of any one individual. In practice, the idea is to add noise to the database. Of course, how much any individual contributes to the result of a database depends in part on how many people’s data are involved in the query. Using additive random vectors to increase privacy is common practice. In differential privacy, because it provides certain privacy guarantees, Laplace noise is usually used [8,26,32]. However, when maximal privacy with minimal distortion is desired, Laplace noise is generally not the optimal solution. The fundamental question to determine the noise distribution achieving maximal privacy for a given allowable distortion level is investigated in an information-theoretic framework in [32]. Similar framework was considered in [23], to analyse a peer-to-peer market involving strategic agents who are not willing to disclose their private information, which is assumed to be known by the other agents up to a certain level of noise caused by the bias introduced voluntarily (in a differential privacy context) to protect input information or involuntarily (when trying to learn the other agents’ private information). In this paper, we will focus on another notion of privacy that comes from the literature on security games [18]. In such games, agents are typically reluctant to share sensitive – even secret – information with other agents, in part because of the potential for leaks. The problem is to coordinate the resource allocation between multiple agents so that social welfare efficiency is reached and minimum amount of sensitive information is shared within the agents. In this paper, the goal is to coordinate prosumers and consumers with heterogeneous preferences so that social welfare efficiency is reached while operational constraints are met, and minimum information is exchanged between the agents. In Game theory, the notion of private information is often linked to the theory of Bayesian games and mechanism design. An asymmetric game where agents have private information – contained in so called “types” – is said to be strategyproof if it is a weakly-dominant strategy for every agent to reveal his/her private information [41], i.e., you are best or at least not worse by being truthful, regardless of what the others do. Goal in such games is to design mechanisms that can take the form of payment functions/penalties guaranteeing social welfare efficiency while inducing the agents to be truthful. In this paper, we will assess the strategyproof property of the energy trading algorithm. 1.4. Classifying the information Our goal is to determine the optimal demands of the consumers, self-usage quantities and quantities shared by the prosumers on the platform. The information involved can thus be classified into two categories – static and dynamic – depending on whether it is available from the outset or evolving dynamically. •Static information is the information that is private to the agents (consumers and prosumers) and available to them from the outset. For the prosumers, it is their own self-usage benefits and associated parameters (target self-consumption, calibration parameters), and their cost functions. For the consumers, it is their own usage benefits and associated parameters, their preferences regarding the green and local origin of the supply, and their cost functions. Throughout the text, it will be called the private input of the agents. •Dynamic information is the information obtained as output of the market clearing, i.e., the optimal self-usage and shared quantities of the prosumers and the demand of the consumers. Throughout the text, it will be called the output of the market clearing problem. Goal is to keep it private to the agents. To compute the optimal decisions of the agents, some information is shared iteratively between the agents such as the local market clearing price updates. 1.5. Adversial attack and trust From an ICT perspective, a fully decentralized electricity market design provides a robust framework since if one node in a local market is attacked or in case of failures, the communication network architecture should remain in place and information could find other paths to circulate from one point to another, avoiding malicious nodes/corrupted paths [36]. However, among the peers, some nodes might perform data injection attacks to alter the estimation of the system real state, enabling them to manipulate the market clearing price to obtain economic benefits. As such, security, detection of malicious behaviors and robustness against adversial attacks remain major issues for peer- to-peer electricity markets to emerge. Security and trust enforcement among the peers requires blockchain technology. A blockchain is a continuously growing list of records, called blocks, which are linked and secured using cryptography. By design, blockchains are inherently resistant to modification of the data [36]. The validation of new blocks relies on a distributed consensus algorithm and miner node selection which is specific to the blockchain protocol in place. Most current protocols are heavily energy greedy (see Bitcoin). In [25], an innovative miner selection rule based on a fixedshare exponentially-weighted average density function is analysed. It is far less energy greedy than classical Proof-of-Work methods, and integrates the peers’ past performance contrary to Proof-of-Steak methods used in Ethereum which relies for a large part of it on random miner selection. On top of blockchain technology, smart contracts are autonomous computer systems, written in code, that manage executions in the form of rules between parties on the blockchain. For example, the reaching of a consensus between nodes, specific events (like adversial attacks) can be detected online, and the execution of the smart contract is automatically triggered [37]. To avoid any influence of a malicious node, consensus algorithms are employed [21,31]. The core idea behind various distributed decision applications is the ability of individual agents to reach agreement globally via local interactions [39]. Several algorithms for consensus can be found in the literature and have attracted much attention in the last decades in the broader framework of sensor management and data fusion: they differentiate on the basis of the amount of communication and computation they use, on their scalability with respect to the number of nodes, on their (online) adaptability, and, finally, they can be deterministic or randomized [10]. In this paper, we will focus on Consensus ADMM (C-ADMM) [3], which can play the role of a smart contract: once coordination among the agents is reached – meaning that the local decisions of the agents give rise to a Pareto efficient solution under minimal information exchange among the nodes – buying and selling offers are matched on the virtual trading platform. Note that adversial attack, miner selection process, and more generally blockchain design, will not be considered in this paper. 1.6. Contributions We decompose our contributions to the two-sided matching markets literature, according to three main tracks. 1) We first formulate the two-sided market matching problem as a noncooperative networking game [1] that we reformulate as a Mixed Complementarity Problem (MCP), and analyse its solutions in terms of existence and uniqueness relying on appropriate solution concepts [1]. We also determine conditions under which supply shortages occur on the platform, highlighting the needs for the design of subsidies and loyalty programs. 2) Second, we compute analytically the centralized market clearing solution in case the local Market Operator determines the optimal demand, self-usage and shared quantities on the platform that maximize the social welfare of the local energy community. 3) Though a closed form result can be inferred for the centralized market clearing, it does not allow privacy preservation. To allow output H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 3 privacy-preservation, we introduce a distributed (algorithmic) approach. Our algorithm, C-ADMM, is an application of the classical Alternating Direction Method of Multipliers (ADMM)[3] to the market clearing problem that we reformulate as an optimal exchange problem and solved as a consensus problem. We analyse formally the algorithmic properties of C-ADMM regarding its strategyproofness, the level of privacy preservation, and its complexity. Computational properties as a function of the number of agents involved is quantified in a case study. The remainder of the paper is organized as follows. In Section 2, we formally introduce our two-sided market platform model. The agents, their utility functions, as well as appropriate solution concepts to analyse the platform outcome are detailed. In Section 3, the platform market clearing problem is formulated as an optimal exchange problem, output privacy-preservation is formally defined. The different approaches we use to solve the market clearing problem, either centralized or distributed, are described in Section 4. Finally, a case study providing computational results for C-ADMM and economic guidelines regarding the emergence of local energy communities is introduced in Section 4. 2. Model description We consider a set of Nnodes. Each node can be either a prosumer, P, having the possibility to generate and consume (part) of her own energy while selling the excess by the intermediate of a sharing platform operated by a local Market Operator (MO), or a consumer-only, C, without generation facility. We denote by , the prosumer set, and by , the consumer-only set. Furthermore, we have the relationships: = and = . Local energy demand and supply balance is guaranteed by the local MO, who can sell excess production or buy shortage to the power grid. Our inspiration for the prosumer-consumer interaction model comes for the literature of two-sided markets [2,12,13], though the structure of electricity markets and asymmetry of prosumer role, who can benefit from consumption of self-production (therefore, behaving as consumers) and excess production selling by the intermediate of the sharing platform (therefore, behaving as producers), makes extensions of this literature tricky. The consumer-prosumer platform framework is visualized in Fig. 1. Note that we assume a uniform platform price, p, t at every time period t(e.g., there is no discrimination between the nodes). Futhermore, there exist lower and upper bounds on the platform price, such that p p p t at every time period t, with 0 ≤ p and < < +p0 . We do not consider contract for the supply provision, though this can be an interesting instrument for risk hedging in case uncertainty associated with the RES-based generation supply is considered. In our model, the consumers just pay the prosumers for the quantity of energy supplied. The payments are performed at each time step t, relying on the unit price pt defined uniformly by the local market operator. Remark 2.1. In practice, the prosumers’ households are equipped with smart meters that provide near real-time data and granular information. Energy surpluses are then traded online on the electronic trading platform (in the form of tokens) by the intermediate of a smart contract [36]. In case of excess production (resp. supply shortages), the excess (resp. missing) quantity is sold (resp. bought) to (resp. from) the power grid. In this paper, we assume that storage can be performed at interface transmission grid nodes [24], assuming that the suppliers in these nodes invest in some forms of storage such as hydro-electric dams, or prosumers collectively invest in global storage technologies (see SonnenCommunity, in Germany). The cost of battery acquisition at the prosumers’ level being still quite high, we do not consider individual storage technology at the residential level. Remark 2.2. In the formulation of our optimization problems, we do not describe the techno-economical constraints of the generating technologies that are captured through complex bids in the energy market literature. Such a setting deeply complexifies the noncooperative game analysis as it introduces non-convexities in the agents’ optimization problems. However, this can be an interesting direction for future work. 2.1. Modeling consumers For each consumer C, we denote the usage benefit obtained from consuming a quantity yt C of energy, by U y( ) Ct C . We assume that U C (.) is only known to the consumer and is not public knowledge. We make the assumption that U C (.) is continuous and strictly concave and non-negative on + . Following the approach in [23] and to fix the idea, we assume that consumer Cusage benefit is a quadratic function of the consumer demand y, t C leading to the following definition: = +U y y y( ) ( ) ˜, Ct CC t C t CC2 (1) where ,˜ C C are positive parameters, and yt C is the target demand of consumer Cat time period t. For the usage benefit to remain non-ne- gative on the interval of definition of yt C (e.g., the interval [0; κ C ]), we impose conditions on the parameters such that U C (0) ≥ 0 and U C (κ C ) ≥ 0, leading to y t, 0. C t C ˜ ˜ C C C C Note that the maximum usage benefit is reached in =U y( ) ˜ Ct CC and in case =U(0) 0, C i.e., zero demand implies zero usage benefit, we have the following relationship between the consumer target demand and usage benefit parameters: = CU y y ( ) ( ) CtC tC2 . We refine the consumer model by introducing horizontal product differentiation. In [23], the preferences were captured through (product) differentiation prices. These prices can model taxes to encourage/ refrain the development of certain technologies (micro-CHPs, storage, solar panels) in some nodes. They can also capture agents’ preferences to pay regarding certain characteristics of trades (RES-based generation, location of the prosumer, transport distance, size of the prosumer, etc.). In this paper, we capture the agents’ preferences relying on a variant of the discrete choice model introduced by Hotelling [17] for horizontal product differentiation with quadratic distance [7]. In the Hotelling model, consumers’ preferences are located by points on the same unit segment. The extremities of the same line are used to represent the two alternatives. We assume that each consumer has the choice between two alternatives: “buying 100% green certified energy” 1 located in 0 or “buying energy without any guarantee of origin” located in 1. Beyond this, these two energy supplies are seen as perfect substitutes by the consumers. For <, C 1 2 consumer Chas strict Fig. 1. Example of a sharing platform involving consumers-only on one side and prosumers on the other side. 1 In practice, it is very difficult to determine where the electrons that make the supply come from. A blockchain technology on top of a digital peer-to-peer energy trading platform can help trace back the origin of the supply and provide certificates of green origin [31,36]. In our paper, the supply is 100% green if, and only if, the demand is exclusively covered by the prosumers’ RES generations. H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 4 preference for a 100% green energy supply; whereas for >, C 1 2 consumer Cwould rather be supplied by a mix without any guarantee of origin. For =, C 1 2 consumer Cis indifferent between the two alternatives. Remark 2.3. In this paper, the platform provides demand and supply matching for consumers and prosumers providing RES-based generation. This means that as long as the demand does not exceed the local supply, the consumers have the guarantee to be supplied by green energy only. We introduce γ> 0 as the coefficient (interpreted as unit transport price in [17]) which determines the importance of the distance (between the consumer’s preferences and the two alternatives), by comparison with the energy price on the platform. We define θ C ∈[0; 1], as the preference of consumer Cregarding these two alternatives. The utility consumer Cobtains from energy consumption y, t C y( ), Ct C is given by the usage benefit U C (.) minus the cost to buy energy on the platform operated by the local MO, pt times the consumption y, t C minus the cost associated with the distance between his preference and the two alternatives for the origin of his supply. Formally, we have: =y U y p y y C U y p y y C ( ) ( ) if is supplied by 100% green energy ( ) (1 ) if is supplied by a mix without any guarantee of origin . Ct C Ct C tt C t C C Ct C tt C t C C 2 2 (2) Consumers determine their product choice (e.g., their demand) based on the difference between the usage benefit of consuming yt C and the supply cost, and discrepancy between the green supply feature and their own desire. By varying the values of γand (θ C ) C , we model different markets where market clearing price has different effect on the supply origin and where consumers can have varying sensitivities to the green origin of their supply. Close to our work, Fang and Huang characterize the effect of brand in market competition, relying on a variant of the Hotelling model [11]. To determine in which class each consumer falls, we assume without loss of generality that the consumers are ordered according to their preferences such that < … +C, {1, ,card( ) 1}, C C 1 and that the consumers are served by the local MO one after the other, in the increasing order of their θs. In practice, this means that the platform supply is used to fulfill the demand of the consumers with the smallest θs (e.g., the ones with the highest sensitivity to the RES-origin of their supply) until supply shortage occurs and the local MO is forced to buy the missing quantities to the grid to fulfill the demand of the remaining consumers with the highest θs. We denote C ¯ the index of the first consumer that is not fully served by the platform. Formally, it can be defined as = > = = C C y s y s ¯: max{{ | { } { }}; 0} i C t i Pt P i C t i Pt P 1 1 1 . Note that, by convention, in case of excess of supply compared to the platform demand, =C ¯0 and all the consumers are served by the platform. Each consumer Cdetermines his demand yt C so as to maximize the sum of his utility function (2), under non-negativity and maximum capacity of consumption, κ C , constraints: C y( ) max ( ), y Ct C t C (3) s t y. . , ( ) t CCt C (4) y0 . ( ˜) t C t C (5) Note that the dual variables associated with constraints (4) and (5) are denoted by Greek letters between brackets at the right of the constraints. We will follow the same convention throughout this article. We prove in the proposition below that there always exists a unique solution to the consumer utility maximization problem. Proposition 1. There exists a unique solution to the consumer optimization problem C ( ) . Proof of Proposition 1. We start by computing the Lagrangian function associated with the consumer’s optimization problem C ( ) : = +y y y y( , , ˜) ( ) ( ) ˜. Ct C t C t C Ct C t C t CCt C t C (6) Then, we distinguish between the two classes of consumers introduced in (2). •Consumer Cis served by the platform: The consumer Lagrangian function takes the form =y( , , ˜) Ct C t C t C + + +y y p y y y y( ) ˜( ) ˜ C t C t CC tt C t C C t C t CCt C t C 22 . Derivating the consumer’s utility function (2) with respect to yt C a first time, we obtain =y y p2 ( ) y y C t C t C tC ( ) 2 CtC tC ; and a second time, we get = <2 0 y y C ( ) ( ) CtC tC 2 2 . We conclude that Π C (.) is strictly concave in y, t C meaning that KKT conditions are necessary and sufficient conditions to find the optimum solution of C ( ) . Derivating the Lagrangian function with respect to y, t C the stationary condition implies that at the optimum = + + y y p * ˜ 2 . t C t CtC t C t C C 2 (7) Primal feasibility constraints impose that y* t CC and y0* t C . From dual feasibilty constraints we get: 0 t C and ˜0 t C . Finally, with complementarity slackness conditions, we have the relationships: =y(*) 0 t C t CC and =y ˜*0 t C t C . •Consumer Cis served by the grid: The consumer Lagrangian function takes the form =y( , , ˜) Ct C t C t C + + + +y y p y y y( ) ˜(1 2 ) C t C t CC tt C t C Ct CC 22 y y( ) ˜ t C t CCt C t C . Derivating the consumer’s utility function (2) with respect to yt C a first time, we obtain =y y p2 ( ) (1 ) y y C t C t C tC ( ) 2 CtC tC ; and a second time, we get = <2 0 y y C ( ) ( ) CtC tC 2 2 . We conclude that Π C (.) is strictly concave in yt C . Derivating the Lagrangian function with respect to y, t C the stationary condition implies that at the optimum = + + y y p *(1 ) ˜ 2. t C t CtCt C t C C 2 (8) Primal and dual feasibility constraints as well as complementarity slackness conditions remain the same as in the case consumer Cis served by the platform. □ In the following proposition, we aim at finding a link between the consumer total demand and statistical measures (such as empirical mean and variance) of the consumers’ preference sample. To that purpose, we define = ^[ ]: 1 card( ) C C and = ^( ): 1 card( ) ^[ ] C C 22 as the empirical mean and (biased) empirical variance of the consumers’ preference sample. We also introduce the conditional empirical mean of the consumers’ preference sample as = + ^[ | : 1 card( ) C ¯1 C C C C ¯ ¯ . Proposition 2. Assuming that = C and <p y C2 , , t C t C at H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 5 the optimum, the sum of the consumers’ demands can be expressed as a closed form expression in the empirical mean, variance, and conditional empirical mean of the consumers’ preference sample. Proof of Proposition 2. From Proposition 1, we infer that in case consumer Cis served by the platform, < <y0* t CC is equivalent to < C y p2Ctt . Whereas in case consumer Cis served by the grid, < <y0* t CC is equivalent to >1 C y p2CtC t . From these two relationships, we infer that < + <p y C C2 , ¯ tCC t C2 and <p y C C2 (1 ) , ¯ t C t CC2 . Since θ C ∈[0; 1], a sufficient condition to have < <y0* t CC in case consumer Cis served either by the platform or the grid, is to check <p y2 t C t C . Using the definitions of the empirical mean, variance, conditional empirical mean and (7), (8), the sum of the consumers’ demands at the optimum can be analytically expressed as follows = + + y y p *card( ) 2 card( ) 2 ^( ) ^[ ] card( ) C ¯1^[ | 1 2. C t C C t C t 2 C ¯ (9) □ Remark 2.4. As a corollary of Proposition 2, it is worth noting that the sum of the consumers’ demands is decreasing in the platform price pt . It is also linearly decreasing in the variance (which can be interpreted as the spread) of the consumers’ preference sample. This means that the more heterogeneous the consumers’ preferences are, the smaller the total demand is. For the sake of simplicity, in the following, we set =V y: 2 t CC t C C 2 and =W y: 2 (1 ) t CC t CC2 . Proposition 3. At the optimum, the consumer demand can be expressed as a stepwise linear decreasing function in the platform price, pt : = < y p V p C C W p C C 1 1 ( ) 2if ¯, 2if ¯. t C t t C t Cp V t C t Cp W [0; [ [0; [ ttC ttC (10) Proof of Proposition 3. The analytical expression of the optimal demand of consumer Cis given in Equation (7) in case <C C ¯ and in Equation (8) in case C C ¯ . Complementarity constraints are detailed in Proposition 1 proof. In case <C C ¯, the consumer demand can take three values: =y t CC (then = ˜0 t C ), =y0 t C (then =0 t C ), or = + y y ]0; [ t C t CpC 2 tC C 2 . It is straightforward to prove that having yt C equal the last value is equivalent to < <y p y2 ( ) 2 C t CCt C t C C 2 2 . Then, we obtain the following expression for the demand of consumer Cat the optimum: = +y1 1 t CCp V V p p V V22] 2 ; [ ttC C C t Ct CttC C C tC . Since <V2 0 t CC C if <yt CC and >p p 0, t the expression of the consumer demand can be simplified to give the expression in the Proposition statement. In case C C ¯, similar reasoning applies, replacing Vt C by Wt C . □ Substituting the expression of the consumer demand derived in Proposition 3 in the consumer’s utility (2) assuming <C C ¯, we obtain: = + + + + = p V p y p V p Vp V y p V ( ) (2)˜1 2( ) 1 2( ) 2if [0; [, ( ) ˜( 0) if , by definition of and ˜. Ct CtC t CtCC Ct CCt Ct tC CC t t C CtCCtt C C C 2 2 2 2 2 (11) In case C C ¯, we obtain a similar expression replacing Vt C by Wt C . We note that p( ) Ct is strictly convex in p V[0; [, tt C indeed = > C0, p p ( ) ( ) 1 2 Ct tC 2 2 . This means that the platform price that maximizes the consumers’ utility is reached in one corner of the interval p p V[ ; min{ ¯; min }] Ct C . Same holds with Wt C . Remark 2.5. From Proposition 1 proof and Proposition 3, if <p y C2 , , t C t C all the consumers get an equitable access to the market platform, i.e., no consumer is denied access to the platform because of a too high market clearing price. 2.2. Modeling prosumers Prosumers have two ways to derive benefits from their production: using it themselves or selling it through the sharing platform by the intermediate of the local MO. We let xt P be prosumer Pself-usage quantity and st P be the quantity of energy that prosumer Pshares through the platform. When prosumers consume their own energy production, they experience benefit from the consumption, like consumers-only. But, unlike consumers-only, they do not have to pay the local MO for their consumption, though their consumption may lead to production costs that can be interpreted as usage (in case of micro-CHP activation for example) or maintenance cost, or government taxes, etc. We denote the benefit from self-usage by U x( ) Pt P and the production cost incurred by +c x s( ) Pt P t P . As in the case of the consumers-only, we assume that U P (.) is continuous and strictly concave and non-negative on + . In the same spirit as the consumer model, we assume that prosumer Pusage benefit is a quadratic function of the prosumer selfconsumption x, t P leading to the following definition: = +U x x x( ) ( ) ˜, Pt PP t P t PP2 (12) where ,˜ P P are non-negative parameters, and xt P is the target selfconsumption of prosumer Pat time period t. For the self-consumption benefit to remain non-negative on the interval of definition of xt P (e.g., the interval [0; κ P ]), we impose conditions on the parameters such that U P (0) ≥ 0 and U P (κ P ) ≥ 0, leading to x t, 0. P t P ˜ ˜ P P P P Similarly to the consumers, the maximum usage benefit is reached in =U x( ) ˜ Pt PP and in case =U(0) 0, P we have the additional relationship = PU x x ( ) ( ) Pt P t P2 . When the prosumers share their excess production through the platform, they receive a revenue and incur costs. The revenue they receive from sharing depends on how many other prosumers are also sharing their excess production. We introduce the probability μ(y t ,s t ) that a prosumer is matched to a consumer as follows: =y sµy s ( , ): min ; 1 . tt Ct C Pt P (13) Naturally, μ(y t ,s t ) < 1 if, and only if, <y s , Ct C Pt P i.e., there is an excess of supply compared to the actual demand on the platform. And, =y sµ( , ) 1 tt in case the consumer total demand is larger than the prosumers supply, therefore requiring that the local MO buys the missing quantity to the grid. In the following, for the sake of simplicity, we will write: μ t ≔μ(y t ,s t ). The utility function of a prosumer is the sum of the benefit she derives from the consumption of her self-production plus the expected H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 6 revenue she derives from the selling of her excess production conditionally to her matching with a consumer minus her production costs, leading to the following mathematical expression: = + +y sx U x p µ s c x s( , , ) ( ) ( ), Pt P tt p t P ttt PPt P t P (14) where c P (.) is prosumer Pcost function. Assuming that prosumer Pcost function is quadratic in her selfusage production, we set = + +c x c x c x c x( ) , P P P P221 0 with c P2 , c P1 ,c P0 non-negative parameters. Each prosumer Pdetermines her self-usage quantity xt P and the quantity to share on the platform st P that maximize her utility function (14), under non-negativity of her self-usage and shared quantities in (17), and maximum capacity of generation, κ P , in (16), by solving the following optimization problem: P y sx( ) max ( , , ), x s Pt P tt , t P t P (15) +s t x s. . , ( ) t P t PP t P (16) x s0 , . ( ˜,˜) t P t P t P t PS (17) Observing the form of P ( ), we can already distinguish between two cases: • =µ1 t implying that the prosumers’ optimization problems are decoupled from one another, and from the consumer’ ones. As a result, solving P ( ) is equivalent to solve an optimization problem in x s, t P t P . •μ t < 1 implying that the prosumers’ optimization problems are coupled through their utility functions Π P (.). The consumers’ demand y t also impact the prosumers’ utilities but since it is optimized by the consumers independently of the prosumers’ reactions, we will consider it as a fixed parameter. Proposition 4. In case =µ1, t provided >c P, , P P2 there exists a unique optimum solution to the prosumer optimization problem P ( ) . In case μ t < 1, there exists a Nash equilibrium solution to the noncooperative game involving the prosumers and consumers C P ( ) ( ) . Proof of Proposition 4. We start by computing the Lagrangian function associated with the prosumer’s optimization problem P ( ) : = + +y s y sx s x s x s x s ( , , , , , ˜,˜) ( , , , ) ( )˜ ˜ . Pt P t P tt P t P t P t PS Pt P t P tt P t P t P t P P t P t P t PS t P (18) Then, we distinguish between two cases depending whether supply shortage occurs on the platform. • =µ1, t the prosumers’ optimization problems are decoupled: Derivating the prosumer’s utility function (14) with respect to xt P a first time, we obtain = x s x ( , ) Pt P t P t P +x x c x s c2 ( ) 2 ( ) P t P t P Pt P t P P2 1 ; and a second time, we get = <2 0 x s x P ( , ) ( ) Pt P t P t P 2 2 . Similarly, derivating the prosumer’s utility function with respect to st P a first time, we get = +p c x s c2 ( ) x s stPt P t PP ( , ) 2 2 PtPtP tP ; and a second time, we have = <c2 0 x s sP ( , ) ( ) 2 Pt P t P t P 2 2 . Then, cross-derivatives give = = <c2 0 x s x s x s s x P ( , ) ( , ) 2 Pt P t P t P t P Pt P t P t P t P 2 2 . The Hessian matrix associated to the two-variable utility function Π P (.) admits as determinant c c4 ( ), P P P2 2 we conclude that the determinant is positive if, and only if, η P >c P2 . Under this assumption, since the first minor ( 2P ) is negative, we conclude that Π p (.) is concave with respect to x s, t P t P . Furthermore, the Hessian matrix being definite negative in any point of the space of definition, we get the stronger result that Π P (.) is strictly convex in x s, t P t P . As a result, the optimization problem P ( ) admits a unique solution. To determine the analytical expression of the optimum, we compute the stationary conditions which give = + x x p * ˜ 2, t P t Ptt PS P (19) = + + + + +s x cpc c c *1 2(1 1 )2 1 2(1 1 )˜. t P t P PPt Pt P P P Pt PS 2 1 2 2 (20) Primal feasibility constraints impose that +x s **, t P t PP x s0*,* t P t P . From dual feasibility constraints we get: 0, ˜0, ˜0 t P t P t PS . Finally, the complementarity slackness conditions give the following relationships: + = = =x s x s(* * ) 0, ˜*0, ˜*0 t P t P t PP t P t P t PS t P . •μ t < 1, the prosumers’ optimization problems are coupled: To simplify the notations, we set =Y Y: tCt C and =S s: tPt P . Derivating the prosumer’s utility function (14) twice with respect to xt P and s, t P we obtain = + <c2( ) 0 s yx x P P ( , , ) ( ) 2 Pt Ptt t P 2 2 and = <p c2 1 2 0 s yx st Y S s SP ( , , ) ( ) 2 PtPtt tP t t tP t 2 2 respectively, while the cross-derivatives give = = <c2 0 s y s yx x s x s x P ( , , ) ( , , ) 2 Pt Ptt t P t P Pt Ptt t P t P 2 2 . The determinant of the Hessian matrix associated with Π P (.) being positive, e.g., + + >c p c4( ) (1 ) 4 0, PPt Y S s SPP 2 2 t t tP and the first minor +c2( ) P P3 being negative, we conclude that Π P (.) is concave in x s, t P t P . We now want to prove that in the most general setting, there is no guarantee on the uniqueness of the Nash equilibrium [34]. To that purpose, we introduce the Jacobian block matrix of the pseudogradient of the non negative weighted sum of the two prosumers P≠P′ utility functions with weights equal to 1 defined as x x s x x x s s x s s x s s x x x s x x s s x s s s x s ( ) ( ) ( ) ( ) P t P P t P t P P t P t P P t P t P P t P t P P t P P t P t P P t P t P P t P t P P t P t P P t P P t P t P P t P t P P t P t P P t P t P P t P 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = +c c c p c p s p p s p p s p 2( ) 2 0 0 2 2 (1 ) 2 0 2 0 0 0 0 0 2 0 2 . PP P Pt Y S s SPt Y St P t Y S t Y St P t Y St Y St P t Y S 2 2 2 2 t t tP t t t t t t t t t t t t t 2 3 2 3 2 3 2 The determinant of the sum of the Jacobian block matrix and its transpose being null, we cannot conclude that the sum of the Jacobian block matrix and its transpose is negative definite. Therefore, there might exist multiple Nash equilibria solutions of the non-co- operative game C P ( ) ( ) . The stationarity conditions indicate that a Nash equilibrium should check the following relationships + + + + =c x x c s c2( ) 2 2 ˜0, P Pt PP t P Pt P Pt P t P 2 2 1 (21) + + + + =pY S s Sc x s c P( 1) 2 ( ) ˜0, . t t t t P t Pt P t PPt P t PS 2 1 (22) The primal and dual feasibility constraints as well as complementarity slackness conditions are the same as the ones introduced in the case =µ1 t and should hold for any P . □ In the proposition below, we give explicit conditions on the prosumers’ optimization problem parameters and constraints to guarantee H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 7 uniqueness of the Nash equilibrium. Proposition 5. In case μ t < 1, assuming that >card ( ) 1, =c c , P2 2 =P¯, , P and s t > 0, x t > 0, there exists a unique Nash equilibrium solution of the noncooperative game involving the prosumers and the consumers C P ( ) ( ) . Proof of Proposition 5. The detailed proof can be found in Appendix A.1. Proposition 6. In case =µ1, t the prosumer self-usage and shared quantity on the platform can be expressed as stepwise linear decreasing and increasing functions respectively, in the platform price, pt : = + = + + + + + + x p x x p s p x cpc c 1 1 1 1 ( ) 2 2, ( ) 1 2(1 1 )2 . t P tt P p in c Pt P t Pp c c c t P tt P PPt P P p c c c Pc c p [0; ] ] ;2 [ 2 1 2] ;2 [ [2 , ¯] tPtP P PP tP P PP PPP 1 1 2 1 1 2 1 2 1 Proof of Proposition 6. In case =µ1 t and >c P, , P P2 the analytical expression of the optimal self-consumption and shared quantity of prosumer Pare given by Eqs. (19) and (20). Complementarity constraints are detailed in Proposition 4 proof. Each variable can take three values: 0, xt P for xt P (resp. κ P for st P ), or be such that +x s ]0; [ t P t PP . Using the analytical expressions of x s, t P t P mentioned above, it is straightforward to prove that this last condition is equivalent to < < +c p c c2 PtPPP1 2 1 . □ Remark 2.6. To avoid that the prosumers self consume all their production and find incentives to share part of it on the platform, it is realistic to assume p p p t with =p cmax { } P P1 and +p c c P2 , P P P 1 2 . Substituting the expressions of the prosumer self-usage and shared quantity derived in Proposition 6 in case =µ1 t in the prosumer’s utility (14), we obtain: = + + + + ppccpc cccpc c c p x c c c p ( ) ( 2)˜(1 2 2 ) ( 1 2 2 ) (2)1 2(1 1 )( ) . Pt Pt P PP Pt P P P Pt P P P tt PP P P Pt 22 2 1 2 21 2 1 2 0 1 2 2 2 (23) We note that p( ) Pt is strictly convex in p, t indeed = + > 0, p pc ( ) ( ) 3 1 Pt tPP 2 22 P . This means that the platform price that maximizes the prosumers’ utility is reached in one corner of the interval p p[ ; ¯] . 2.3. On the need for an optimal design of subsidies Proposition 7. Assuming that =C, , C and =P¯, , P at >s*0, t there exists a market clearing price upper bound, p¯ , below which supply shortages occur on the platform. Proof of Proposition 7. Starting with the consumers side: from Proposition 2, the condition <p y C2 , tt C implies that the sum of the consumers’ demands at the optimum takes the closed form expression (9). Continuing with the prosumers side: from the assumption >s*0, t the prosumers’ complementarity slackness conditions impose that = ˜0 t PS . Then, from Proposition 4, in case =µ1 t (e.g., consumers’ total demand is larger than the prosumers’ supply), we infer from (20) and (19) that <s x t PP t P is equivalent to < +p c c P2 , tpPP2 1 . So, assuming these relationships hold, the sum of the prosumers’ shared quantities on the platform can be expressed as a closed form expression in the platform price = + +s x cpc c 1 2(1 1 ¯)2. P t P P t P PPt P P P2 1 2 (24) By definition =µ1 t ⇔ y s Ct C Pt P . By substitution of the closed form expressions of the sum of the consumers’ demand and shared quantities obtained in (9) and (24), we infer that =µ1 t if and only if + + + + + + + + p y x ^( ) ^[ ] card( ) C ¯1^[ | . t CtC PtP P cP cP PcP card PcP 1 22 1 2 1 2 card( ) 2 ¯ card( ) 2 ( ) 2 2C ¯1 2 1 2 1 2 card( ) 2 ¯ card( ) 2 (25) So, if the upper bound on the market clearing price is chosen so that <p y¯ min{2 min { } ; (25)}, Ct C then supply shortages always occur on the platform. □ Proposition 7 coincides with the results obtained in [12] for Didi Chuxing, the largest ridesharing platform in China: if the platform market clearing price is not high enough, suppliers might lack incentives to share their production on the platform and consumer shortages might happen. In such cases, optimal design of subsidies might be necessary to give incentives to suppliers (prosumers) to share their supply. The designs of optimal subsidies and loyalty programs are discussed in [12,13], but is out of the scope of our paper. 3. Interpreting the market clearing problem as an optimal exchange problem We will suppose that Proposition 7 holds in the rest of the paper. This seems a reasonable assumption, as many experimental studies led on sharing platforms lead to such an observation [12,13]. This means that we assume that there exist upper and lower bounds p and p¯ on the platform clearing price such that prosumers have incentives to share their productions on the platform but supply shortages occur: p p p t¯, 0. t On the platform, supply and demand balance gives rise to the following equation at every time period t: + =y s q 0, C t C P t P t (26) where q t is the import/export to/from the platform from/to the grid. In case supply shortage occurs, the local MO can import energy from the grid at the wholesale market unit price pt 0 . As proved in Propositions 3 and 6, the optimal demand, self-usage and shared quantities can be expressed as closed form expressions in the platform price. Similarly, the quantity exchanged between the platform and the grid q t can be expressed as a function of the platform price. This last result is summarized in the proposition below: Proposition 8. At the optimum in y x sp p p( ), ( ), ( ), tttttt the quantity exchanged between the platform and the grid, q t (.), can be expressed as a linear increasing function of pt . Proof of Proposition 8. By definition, from the balancing Eqs. (26), =q s p y p( ) ( ) tPt P tCt C t . Substituting the expressions of s p( ), Pt P t y p( ) Ct C t obtained in Propositions 7 (precisely (24)) and 2(precisely (9)) respectively, we obtain the following expression for q t (.) as a function of the platform price pt : H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 8 provided. From this work, one possible new research direction would be to introduce uncertainty through random variables modeling the prosumers’ RES generations, which is unpredictable and only partially controllable. Risk measures could be used to quantify the impact of such variability on the market clearing outcome. From a methodological point of view, the coordination of agents with heterogeneous risk measures still raises open research questions. Another direction would be to include the prosumers’ techno-economical constraints, power grid operational constraints and power flow constraints in the market clearing problem resulting in a large-scale non convex optimization problem. ADMM is known to still be applicable in non convex environments but the scale of the problem and the introduction of the strategic behaviors of the agents still remain an issue. Fig. 5. Prosumers and consumers optimal decision variables. H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 15 CRediT authorship contribution statement Hélène Le Cadre: Conceptualization, Formal analysis, Methodology, Writing - review & editing, Visualization. Jean- Sébastien Bedo: Formal analysis, Writing - review & editing, Data curation, Software, Validation. Declaration of Competing Interest The authors declare that they have no known competing financial interests that could have appeared to influence the work reported in this paper. Appendix A. Appendix A1. Proof of Proposition 5 By substraction of the stationarity condition (22) from (21) at s t > 0, x t > 0, we obtain = = + x s S x p Ys S S : ( , ), 2 ¯ ( 1) 1. t P Pt Pt t P t tt P t t (43) But, from (21), we also get = + +s c s S c xc c (¯1) ( , ) ¯ 2. t P Pt P tt PP 2 2 1 2 (44) Taking the sum of (44) over all P, we obtain = + +S c s S c x c(¯1) ( , ) 1( ¯ 2 ). t P Pt P t P t P P 2 2 1 (45) But, taking the sum over all P in (43), we obtain another expression for s S( , ) PPt P t = +s S x p Y card S ( , ) 2 ¯ 1 ( ) . P PtPt P tPtt t (46) By substitution of (46) in (45) and multiplying both parts of the equality by S t , we obtain a second order polynomial equation in S t + + + + = Scxcx c S c p Y card(¯1) 1( ¯ 2 ) ( ¯1) 2 ¯ 1 ( ) 0. tPtP PtPP a ttt b 2 2 2 1 2 StSt Since the polynomial equation constant coefficient is negative in zero (assuming >card ( ) 1 ) and has a positive coefficient in front of S, t 2 we conclude that it admits a unique positive solution in S t = + S a a b * 4 2 . t SSS 2 ttt By substitution in (43), we get = +x x p Y s S S *2 ¯ (* *1) 1 *. t P t Pttt P t t Then, by substitution of the previous value in (44), we obtain = + + + + ( ) sc c p Y a a b cx * 1 12¯ 4 (1 ¯) . tPc p Y Ptt SSS tP ¯ 2¯ 1 222 ttttt 2 Substituting s* t P and S* t in (43) gives the expression of x* t P in the Nash equilibrium. □ References [1] Altman E, Boulogne T, El-Azouzi R, Jimenez T, Wynter L. A survey on networking games in telecommunications. Comput Oper Res 2006;33:286–311. [2] Azevedo EM, Leshno JD. A supply and demand framework for two-sided matching markets. J Political Econ 2016;124(5). [3] Boyd S, Parikh N, Chu E, Peleato B, Eckstein J. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and trends in machine learning. 2018. [4] Boysen N, Briskorn D, Schwerdfeger S. Matching supply and demand in a sharing economy: Classification, computational complexity, and application. European journal of operational research (EJOR). 2019. [5] Brandherm B, Baus J, Frey J. Peer energy cloud - civil marketplace for trading renewable energies. In 8th international conference on intelligent environment (IE). 2012. p. 1–4. [6] Chen Y. Alternating direction method of multipliers. Large-scale optimization for data science lecture notes. Princeton University; 2018. [7] d’Aspremont C, Gabszewicz J, Thisse JF. On hotelling’s stability in competition. Econometrica 1979;47(5):1145–50. [8] Dwork C. Differential privacy: a survey of results. Theory and applications of models of computations. Berlin, Heidelberg: Springer Berlin Heidelberg; 2008. p. 1–19. [9] Einav L, Farronato C, Levin J. Peer-to-peer markets. Ann Rev Econ 2015;8(1):615–35. [10] Fagnani F, Zampieri S. Randomized consensus algorithms over large scale networks. IEEE J Sel Areas Commun 2018;26(4):634–49. [11] Fang Z, Huang L. Market share analysis with brand effect. 55th conference on decision and control (CDC). 2016. H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 16 [12] Fang Z, Huang L, Wierman A. Prices and subsidies in the sharing economy. 26th international conference on world wide web - WWW ’17. ACM Press; 2017. p. 53–62. [13] Fang Z, Huang L, Wierman A. Loyalty programs in the sharing economy: Optimality and competition. 18th ACM international conference. 2018. [14] (2019). Fraunhofer website. https://www.iis.fraunhofer.de/en/ff/lv/iot-system. html(Online November. [15] Gregoratti D, Matamoros D. Distributed energy trading: the multiple-microgrid case. IEEE Trans Ind. Electron 2015;62(4):2551–9. [16] Hamari J, Sjöklint M, Ukkonen A. The sharing economy: why people participate in collaborative consumption. ASIS& T, Wiley Online Library; 2015. [17] Hotelling H. Stability in competition. Econ J 1929;39(153):41–57. [18] Jiang AX, Procaccia AD, Qian Y, Shah N, Tambe M. Defender (mis)coordination in security games via ADMM. IJCAI’13. ACM Press; 2018. p. 220–6. [19] Khanafer A, Touri B, Başar T. Consensus in the presence of an adversary. 3rd IFAC workshop on distributed estimation and control in networked systems. 2012. [20] Kovalyov M, Kress D, Meiswinkel S, Pesh E. A parallel machine schedule updating game with compensations and clients averse to uncertain loss. Comput Oper Res 2019;103:148–57. [21] Lamport L, Shostak R, Pease M. The byzantine generals problem. ACM Trans Program LangSyst 1982;4(3):382–401. [22] Le Cadre H. On the efficiency of local electricity markets under decentralized and centralized designs: a multi-leader stackelberg game analysis. Cent Eur J Oper Res (CEJOR) 2019;27(4):953–84. [23] Le Cadre H, Jacquot P, Wan C, Alasseur C. Peer-to-peer electricity market analysis: From variational to generalized nash equilibrium. European journal of operational research (EJOR). 2019. in press [24] Le Cadre H, Mezghani I, Papavasiliou A. A game-theoretic analysis of transmissiondistribution system operator coordination. Eur J Oper Res (EJOR), 2019;274(1):317–39. [25] Le Cadre H, Rivero Puente E, Höschle H. Consensus reaching with heterogeneous user preferences. In book: Theory and applications of models of computation. Paris, France: Springer; 2019. p. 151–70. [26] Lu X, Au MH. An Introduction to Various Privacy Models, Mobile Security and Privacy. Syngress; 2017. 235–245 [27] Maamar HR, Boukerche A, Petriu E. MOSAIK - a mobile peer-to-peer networksbased 3d streaming supplying partner protocol. IEEE/ACM 14th international symposium on distributed simulation and real time applications. 2010. [28] Matamoros J. Asynchronous online ADMM for consensus problems. IEEE international conference on accoustics, speech, and signal processing. LA, USA: New Orleans; 2017. [29] Moret F, Pinson P. Energy collectives: a community and fairness based approach to future electricity markets. IEEE transactions on power systems, in press. 2018. Doi.10.1109/TPWRS.2018.2808961 [30] Morstyn T, Teytelboym A, McCulloch MD. Bilateral contract networks for peer-to- peer energy trading. IEEE transactions on smart grid, in press. 2018. Doi.10.1109/ TSG.2017.2786668 [31] Munsing E, Mather J, Moura S. Bolckchains for decentralized optimization of energy resources in microgrid networks. International conference on control technology and applications. IEEE; 2017. Doi.10.1109/CCTA.2017.8062773 [32] Murguia, C., Shames, I., Farokhi, F., Nesic, D., & Poor, H. V. (2019). On privacy of dynamical systems: an optimal probabilistic mapping approach. ArXiv Preprint https://arxiv.org/abs/1910.13559 (Online December). [33] Rasouli M, Teneketzis D. An efficient market design for electricity networks with strategic users possessing local information. IEEE transactions on control of network systems. 2019. [34] Rosen JB. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 1965;33(3). [35] Roth AE, Sotomayor MO. Two-sided matching. Econometric society monographs. Cambridge University Press; 1992. [36] Siano P, De Marco G, Rolán A, Loia V. A survey and evaluation of the potentials of distributed ledger technology for peer-to-peer transactive energy exchanges in local energy markets. IEEE systems journal. 2019. Doi.10.1109/JSYST.2019.2903172 [37] Szabo N. Smart contracts: building blocks for digital markets. Extropy 1996;16. [38] Tiago S, Soares T, Pinson P, Moret F, Baroche T, Sorin E. Peer-to-peer and community-based markets: acomprehensive review. Renew Sustain Energy Rev 2019;104(C):367–78. [39] Tsitsiklis JN. Problems in Decentralized Decision Making and Computation. MIT; 1984. [40] Tushar W, Yuen C, Mohsenian-Rad H, Saha T, Poor V, Wood KL. Transforming energy networks via peer to peer energy trading: potential of game theoretic approaches. IEEE Trans Signal Process 2018;247(935):529–51. [41] Vazirani VV, Nisan N, Roughgarden T, Tardos E. Algorithmic game theory. Cambridge University Press; 2007. [42] Wang Y, Saad W, Han Z, Poor V, Başar B. A game-theoretic approach to energy trading in the smart grid. IEEE Trans Smart Grid 2014;5(935):1439–50. [43] Xu Z, Taylor G, Li H, Figueiredo MAT, Yuan X, Goldstein T. Adaptive consensus ADMM for dsitributed optimization. 34th international conference on machine learning - PMLR. 2017. [44] Zhang C, Wu J, Long C, Cheng M. Review of existing peer-to-peer energy trading projects. Energy Procedia 2016;105:2563–8. [45] Ausgrid, Solar home electricity data, (2019). https://www.ausgrid.com.au (Online May). [46] Australian energy market operator. (2019). https://aemo.com.au/Electricity/ National-Electricity-Market-NEM/Data-dashboard(Online November). [47] Piclo (2019). https://piclo.energy/about (Online November). [48] Transactive grid. LO3 Energy. (2019). https://lo3energy.com/ (Online November). [49] Vandebron. (2019). https://vandebron.nl/ (Online November). [50] EDF commUNITY project. (2019). https://www.regen.co.uk/wp-content/uploads/ Anes-Dallagi-commUNITY.pdf(Online November). H. Le Cadre and J.-S. Bedo Operations Research Perspectives 7 (2020) 100138 17