Optimal Two-Sided Market Mechanism Design for Large-Scale Data Sharing and Trading in Massive IoT Networks

Matching Pr otocol and Pricing f or Lar ge-Scale Data Sharing in Massiv e IoT Networks T ao Zhang, Quanyan Zhu Abstract — The de velopment of the Internet of Things (IoT) generates a significant amount of data that contains valuable knowledge for system operations and business opportunities. Since the data is the property of the IoT data owners, the access to the data requir es permission from the data owners, which gives rise to a potential market opportunity for the IoT data sharing and trading to create economic values and market opportunities for both data owners and buy ers. In this work, we leverage optimal mechanism design theory to develop a monopolist matching platform for data trading over massive IoT networks. The proposed mechanism is composed of a pair of matching and payment rules for each side of the market. W e analyze the incentive compatibility of the market and characterize the optimal mechanism with a class of cut-off matching rules for both welfare-maximization and re venue-maximization mechanisms and study three matching behaviors including complete-matched, bottom-eliminated, and top-reser ved. I . I N T RO D U C T I O N IoT system generates tons of data that has a significant amount of valuable knowledge hidden. Data collected for fulfilling the indi vidual tasks are often either deleted or stored and locked down in independent data silos. On the one hand, the data owners generally lack knowledge and techniques to conduct kno wledge discov ery from their data and can only gain access to their own data which has little v alue with respect to kno wledge disco very [1]. On the other hand, the data consumers hav e abilities to discover the hidden knowledge accurately and efficiently , but they may not hav e a sufficient amount of targeted data or the rights to access to the data due to priv acy concerns. This asymmetric relationship between the data owners and the data consumers leads to a potential market opportunity for the IoT data sharing. In this work, we focus on massi ve IoT networks with a sufficiently large number of data owners and data consumers and dev elop a two-sided matching market model with a monopolist trading platform for the IoT data sharing. This model captures the different roles of data owners and data consumers in dealing with data and conceptualizes the data owners as the sellers and the data consumers as the buy- ers. Sellers possess IoT data generated during the primary usages of different IoT products to sell and hav e different preferences for the payoffs from participation. On the other hand, buyers offer rew ards to the sellers for sharing their data and hav e dif ferent requirements on the data, including, T ao Zhang and Quanyan Zhu are with Department of Electrical and Computer Engineering, New Y ork University , 370 Jay Street, Brooklyn, 11201, NY , USA. { tz636, qz494 } @nyu.edu for e xample, its features (e.g., category , history , quality , and quantity of the data), the users’ engagements with the IoT system (e.g., the usage frequency of IoT products), and the users’ personal information (e.g., life styles and financial status). W e consider that there is a monopolist intermediary matching platform (platform) that takes advantage of this market opportunity to of fer a matching mechanism that enables and encourages interactions between two sides of participants and matches sellers and buyers based on their priv ate information that summarizes their preferences. The matching mechanism consists of a matching rule and a payment rule. The matching rule matches the owners and the buyers by specifying the datasets that can provide the highest utility for the buyers and determines the rew ards the owners can receiv e from the buyers. As a result, the matching rules produce welfare for owners and buyers. The payment rule specifies the payments from each side of the market once a match is established that generate rev enues to the platform. The proposed model considers the interactions between a large number of owners and buyers and study the matching mechanisms for revenue and welfare maximization [2], respecti vely . W e describe a class of threshold matching rules and show that under certain assumptions both the welfare-maximizing and the rev enue-maximizing matching mechanisms ha ve optimal cut- off matching rules. W e study different matching behaviors and characterize different matching patterns based on the joint marginal effect of matching two indi vidual participants on the platform’ s mechanism design goal. An numerical example is provided to illustrate the theoretical analysis of this work. This work is related to matching mechanism design for matching two-sided market (e.g., one-to-one matching [3] and many-to-many matching [4]). There is a rich literature on the economic analysis and pricing schemes of the market model for data collection in IoT networks [5]–[7] based on a variety of approaches, including smart data pricing scheme [8] such as sealed-bit auctions [9]. Also, utility maximization-based pricing schemes have also been stud- ied [10], [11]. For example, in [12], an optimal dynamic spectrum reservation contract has been designed for mission- critical IoT systems, in which an advance payment is made at the time of reservation and a rebate is made if the reservation is released. The IoT applications are incentivized to reveal the true application type and it has been shown that the incentive compatible mechanism leads to an efficient utilization of the spectrum as well as a greater revenueability of the IoT network operator . I I . I O T D AT A M A R K E T A. Market Model 1) Information structure: There are two sides in the IoT data market: the seller side ( S ) consisting of participants that are data owners, and the buyer side ( B ) consisting of participants who are the data buyers. W e use K to denote one side K ∈ { S , B } and ¯ K to denote its opposite ¯ K ∈ { S , B }\{ K } . W e consider a massiv e IoT networks, in which there is a sufficiently large number of participants on both sides of the market, and each participant has no market po wer . As a result, we model the population of each side of the market as a unit-mass continuum of participants over [ 0 , 1 ] . Each par- ticipant from side K has a type λ K ∈ Λ K = [ λ K , ¯ λ K ] ∈ R . W e assume that λ K is drawn independently from a continuous distribution F K ( λ K ) with density f K ( λ K ) . Each seller’ s type λ S ∈ λ S summarizes the feature of her data (including, e.g., quality , cate gories, or cost performance). On the other hand, each buyer’ s type λ B ∈ λ B summarizes her requirements of data (e.g., quality , quantity) and the offers she can provide to the sellers. The type of each participant from one side also reflects the attracti veness of the participant seen by those in the other side of the market. W e assume that for any two types λ K h ∈ λ K and λ K ` ∈ λ K , participants of λ K h is more attractiv e than those of λ K ` if λ K ` < λ K h , e.g., buyers prefer to be matched to sellers of higher cost performance and sellers prefer to be matched to buyers of better of fers. Informally , let C ( λ S , λ B ) represent the contrib ution generated by matching one seller of λ S to one buyer of λ B to the platform’ s goal. The following assumption specifies a monotone property of C ( λ S , λ B ) . Assumption 1. Let λ K h , λ K ` ∈ Λ K be any two types fr om side K ∈ { S , B } , with λ K ` ≤ λ K h . Then, for a ∈ { h , ` } and ¯ a ∈ { h , ` }\{ a } , C ( λ S ` , λ B ` ) ≤ C ( λ S a , λ B ¯ a ) ≤ C ( λ S h , λ B h ) . Assumption 1 claims that the matching mechanism satis- fies a monotone property , i.e., matching a seller of higher cost performance to a buyer of better offer generates higher contribution than matching participants of lower cost perfor - mance or lo wer offer . Howe ver , each participant’ s type is her own priv ate infor- mation and the matching mechanism requires each partici- pant to report her type to the platform through a message . Due to this information asymmetry , each participant can game the system to her own adv antage by strategically reporting her type to the platform. Let Ω K denote the set of messages a participant from K can choose. Define participant’ s r eporting strate gy α K : Λ K 7→ Ω K , such that a participant reports a message m K = α K ( λ K ) when her true type is λ K . W e consider the direct mechanism, in which participants directly rev eal their type (truthfully or falsified), i.e, Ω K = Λ K . W e say the participant’ s reporting strategy is truthful if α K ( λ K ) = λ K , for all λ K ∈ Λ K , K ∈ { S , B } . 2) Mechanism rules: The data market is driven by the platform (he), who operates as a data market maker that provides a matching mechanism consisting of a matching rule that matches participants from one side to those from the other side and a payment rule that charges participants from each matched participant by a proper price. Let σ K : Λ K 7→ P ( Λ ¯ K ) denote the matching rule for side K ∈ { S , B } , where P ( Λ ¯ K ) denotes the power set of Λ ¯ K , such that σ K ( ˆ λ K ) giv es the set of types on the side ¯ K that a participant of reported type ˆ λ K ∈ Λ K is matched to. Let Ξ K ⊆ Λ K denote any subset of Λ K . This point-to-set mapping captures the fact that a database may be sold to multiple b uyers of different types and a buyer may buy multiple databases from sellers of different types. Let φ K : Λ K 7→ R + denote the payment rule , such that φ K ( ˆ λ K ) specifies a payment that the participant reporting ˆ λ K needs to pay to the platform for being matched. Let σ σ σ = { σ S , σ B } and φ φ φ = { φ S , φ B } . The matching rule σ σ σ is feasible if it satisfies a reciprocal relationship: if a participant of λ K is matched to a participant of λ ¯ K from the opposite side, then the participant of λ ¯ K must also be matched to the participant of λ K , i.e., for K ∈ { S , B } , ¯ K ∈ { S , B }\{ K } , λ K ∈ σ ¯ K ( λ ¯ K ) , if and only if, λ ¯ K ∈ σ K ( λ K ) . (1) The reciprocal relationship (1) holds when λ K and λ ¯ K are the reports sent by the matched participants. 3) Utility and P ayoff: In the data market, the sellers are motiv ated to participate by the reward they may obtain by sharing their data to the buyers, while the buyers are motiv ated by the benefits of utilizing the data. Basically , the buyers acknowledge the (potential) value of data and recognize the importance of providing the rew ard to motiv ate the sellers to share their data. Let γ B : Λ B 7→ Γ ⊂ R + denote the reward, such that r = γ B ( λ B ) is the re ward offered by the buyer of type λ B to each of her matched sellers. Let M K : R + × Λ K 7→ R + denote the monetary e v aluation of the rew ard by participants from K , K ∈ { S , B } . Hence, M S ( r , λ S ) is the monetary v alue of the reward r percei ved by the seller of λ S . On the other hand, the shared data produces value to the buyers from utilizing the data. Define the data value as γ S : Λ S 7→ R + , such that M B ( γ S ( λ S ) , λ B ) is the data value provided by seller of λ S that is recognized by the buyer of λ B . W e assume that M K and γ K are continuous and dif ferentiable and they are fixed and known by all participants and the platform. Additionally , we assume that γ B ( λ B ) ≤ M S ( γ B ( λ B ) , λ S ) , i.e., the cost of offering r = γ B ( λ B ) by type- λ B buyer is less than the monetary v alue of r as seen as a re ward by the type- λ S seller . This assumption is based on the setting that the re ward may not be a direct monetary transfer b ut through the terms of services, products, or v ouchers from the buyers’ business, in which the b uyers’ cost is in general lo wer than the tagged price. Giv en σ K , define the utility function as u K ( · , ·| σ K , α K ) : Λ K × P ( Λ ¯ K ) 7→ R + , such that u K ( λ K , Ξ ¯ K | σ K , α K ) specifies the utility for a participant from K when her true type is λ K and she adopts reporting strategy α K . Let R S ( λ S , λ B | σ S , α S ) = M S ( γ B ( λ B ) , λ S ) and R B ( λ B , λ S | σ B , α B ) = M B ( γ S ( λ S ) , λ B ) − γ B ( α B ( λ B )) . Then, for K ∈ { S , B } , ¯ K ∈ { S , B }\{ K } , u K ( λ K , Ξ ¯ K | σ K , α K ) = Z Ξ ¯ K R K ( λ K , x | σ K , α K ) f ¯ K ( x ) d x . (2) W e hav e the follo wing assumption of u K . Assumption 2. The utility u K ( λ K , σ K ( ˆ λ K ) | σ K , α K ) is a continuous in λ K and non-decr easing in report ˆ λ K . It is straightforward to verify that the utility u K satisfying Assumption 2 coincides with the monotonicity of C in Assumption 1. Define payoff function , J K ( ·| σ K , φ K , α K i ) : Λ K × R + × P ( Λ ¯ K ) 7→ R , of a participant of λ K from K , when she adopts reporting strategy α K , J K ( λ K , p K , Ξ ¯ K   σ K , φ K , α K i ) = u K  λ K , Ξ ¯ K   σ S  − p K . (3) T o simplify the notations, we remov e the rules and strategy σ K , φ K , and α K in the left-hand side of J K , unless otherwise stated. I I I . M A T C H I N G M E C H A N I S M In the letter , we focus on the anonymous mechanism de- sign. Specifically , the set of participants from side ¯ K ∈ { S , B } is matched to a participant of λ K from K ∈ { S , B }\{ ¯ K } and the associated payments depend only on the reported type ˆ λ K = σ K ( λ K ) . Let σ K i ( λ K i ) ∈ Λ K and σ K j ( λ K j ) ∈ Λ K be the types of two participants i and j , respectiv ely . If σ K i ( λ K i ) = σ K j ( λ K j ) ,then both participants i and j recei ve the same matched set Ξ ¯ K and payment p K . This is because the matching rule σ K and payment rule φ K , respectiv ely , specify a match and a payment that depend only on her reported type and are independent of other participants from the same side K . As a result, the identity of each participant can be fully characterized by her reported type. Due to the revelation principle [13], our model considers the direct mechanism and requires the matching mechanism to incenti vize all the participants from both sides to re- port their types truthfully . This is established by imposing the incentive compatibility (IC) constraint to the platform’ s mechanism design problem. Definition 1 ( Individual Compatibility ) . The matching mechanism < σ σ σ , φ φ φ > is incentive compatible if J K  λ K , φ K  λ K  , σ K  λ K   ≥ J K  λ K , φ K  ˆ λ K  , σ K  ˆ λ K   , (4) for any λ K , ˆ λ K ∈ Λ K . In incentive compatible matching mechanism, the truthful reporting strategy is the (weakly) dominant strategy for each participant. The follo wing definition shows a first- order condition for incentiv e compatibility (ICFOC) based on en velope theorem. Definition 2 ( ICFOC ) . A matching mechanism < σ σ σ , φ φ φ > satisfies ICFOC if J K  λ K , φ K  λ K  , σ K  λ K   is a contin- uous and differ entiable function of λ K with the derivative given almost everywher e by ∂ J K  x , φ K ( x ) , σ K ( x )  ∂ x    x = λ K = ∂ ∂ x u K ( x , σ K ( λ K ) | σ K )    x = λ K . (5) The ICFOC condition in Definition 2 implies that the payoff of each matched participant can be characterized by the matching rule σ K , K ∈ { S , B } . Define D K ( λ K ) = ∂ ∂ x u K ( x , σ K ( λ K ) | σ K )    x = λ K , (6) and Q K ( λ K ) = Z λ K ˜ λ K D K ( x ) d x , (7) where ˜ λ K ∈ Λ K is some arbitrarily fixed type. Here, Q K ( λ K ) can be interpreted as the adv antage (an economic rent) a participant of λ K has over some type ˜ λ K due to the platform’ s not knowing that her true type is λ K . For e xample, if J K is an increasing function of λ K , then Q K ( λ K ) with ˜ λ K = λ K is the maximum advantage a participant of λ K holds. In this example, it is straightforw ard to see that participants of ¯ λ K hav e no incenti ve to misreport their true type, while participants of λ K hav e incenti ve to misreport λ K as ¯ λ K because she has no advantage by truthful reporting compared to any type λ K > λ K . Hence, due to the ICFOC, guaranteeing the IC constraint requires to craft the payment rule such that advantages (or disadvantages) that different types hold in the utilities can be balanced. The following proposition shows the necessary and sufficient conditions for matching mechanisms being incenti ve compatible. Proposition 1. Suppose Assumption 2 holds. A matching mechanism < σ σ σ , φ φ φ > is incentive compatible if and only if the payment rule is constructed as follows, for all K ∈ { S , B } , φ K ( λ K ) = u K ( λ K , σ K ( λ K )) − Z λ K λ K D K ( x ) d x − J K ( λ K , φ K ( λ K ) , σ K ( λ K )) , (8) wher e D K is given in (6). The construction of φ K in (8) can be interpreted as the utility of truthfully reporting λ K , advantage of type λ K ov er λ K , and the payoff of λ K . Proposition 1 also requires that the payoff of the lowest type λ K is designed by choosing a constant. Given the IC constraint, we rewrite the payoff in (3) as J K ( λ K | σ K , φ K ) . Another important constraint in our matching mechanism design is the individual rationality (IR) constraint. In ad- dition to the IC constraint, the IR constraint requires that the payoff of each participant of λ K is non-negativ e, i.e., J K ( λ K | σ K , φ K ) ≥ 0, for all λ K ∈ Λ K , K ∈ { S , B } . A. Revenue and W elfar e Maximization The expected social welfare generated by the matching mechanism is gi ven as Z W ( σ σ σ , φ φ φ ) = ∑ K ∈{ S , B } Z Λ K u K ( x , σ K ( x ) | σ K ) f K ( x ) d x , (9) and the expected re venue obtained by the platform is gi ven as Z R ( σ σ σ , φ φ φ ) = ∑ K ∈{ S , B } Z Λ K φ K ( x ) f K ( x ) d x . (10) By substituting the construction of φ K in (8) into (10), we hav e Z R ( σ σ σ , φ φ φ ) = ∑ K ∈{ S , B } Z Λ K h u K ( x , σ K ( x ) | σ K ) − Z x λ K D K ( r ) d r − J K ( λ K | σ K , φ K ) i f K ( x ) d x . (11) Corollary 1. Suppose Assumption 2 holds. In the incentive compatible matching mechanism < σ σ σ , φ φ φ > , the followings hold. (i) The mechanism is individually rational if and only if, for K ∈ { S , B } , J K ( λ K | σ K , φ K ) ≥ 0 . (12) (ii) In the matching mechanism that maximizes the social welfar e Z W ( σ σ σ , φ φ φ ) , J K ( λ K | σ K , φ K ) = 0 . B. Cut-off Rule In this section, we describe a class of matching mechanism based on a cut-off rule. Let τ K ( ·| δ K ) : Λ K 7→ Λ ¯ K , for K ∈ { S , B } , be the cut-off function with the thr eshold δ K ∈ Λ K , such that the matching rule is defined as follo ws: σ K ( λ K ) =[ τ K ( λ K | δ K ) , ¯ λ K ] 1 { λ K ≥ δ K } + / 0 1 { λ K < δ K } , (13) where 1 {·} is the indicator function. The matching rule in (13) can be interpreted as, for e xample, the seller with overall data quality characterized by λ S that is at least δ S is matched to buyers whose data requirements (and the associated reward) characterized by λ B is at least τ S ( λ S | δ S ) ; otherwise the seller is not matched. Similar interpretation can be made for the buyers’ side. The following definition sho ws the conditions of the cut- off based matching rules such that the reciprocal relationship in (1) is satisfied. Definition 3 ( Cut-Off Matching Rule ) . The cutoff rule in (13) is a feasible matc hing rule if (i) τ K ( λ K | δ K ) = inf { λ ¯ K : τ ¯ K ( λ ¯ K | δ ¯ K ) ≤ λ K } ; (ii) δ K = inf { λ K : τ ¯ K ( λ ¯ K | δ ¯ K ) ≤ λ K , λ ¯ K ∈ Λ ¯ K } ; (iii) τ K ( λ K | δ K ) is non-incr easing in λ K . In Definition 3, the conditions (i) and (ii) specify the reciprocal relationships of the cut-of f functions and the thresholds between sides S and B and the condition (iii) specifies the monotonicity of the cut-of f function, such that the matching rule defined in (13) is feasible, i.e. (1) is satisfied. Lemma 1. Suppose Assumption 2 holds. Then, matching rules of the welfar e-maximizing and the re venue-maximizing mechanisms ar e cut-off rules. Hence, the platform’ s mechanism design problem is to maximize the social welfare or his revenue by determining the cut-of f functions τ τ τ = { τ S , τ B } with thresholds δ δ δ = { δ S , δ B } and the payment rule φ φ φ , i.e., for Y ∈ { W , R } , max τ τ τ , δ δ δ , φ φ φ Z Y ( { τ τ τ , δ δ δ } , φ φ φ ) s.t. IC, IR, and (i)-(iii) in Definition 3 . (14) I V . C H A R AC T E R I Z AT I O N T H E M AT C H I N G M E C H A N I S M From Assumption 2, it is straightforward to see that both sellers and buyers prefer a match to opponents of higher types, i.e., τ K ( λ K | δ K ) ≤ ¯ λ ¯ K , for all λ K ∈ Λ K , K ∈ { S , B } , and ¯ K ∈ { S , B }\{ K } , giv en that IC and IR constraints are satisfied. Their preferences over the opponents of lower types, ho wever , may exhibit inconsistency between two sides. Suppose a seller of λ S ∈ Λ S is matched to a buyer of λ B . Then the seller receives a rew ard M S ( γ B ( γ B ) , λ S ) and the buyer receiv es a rew ard M B ( γ S ( λ S ) , λ B ) − γ B ( λ B ) . Since γ B ( λ B ) is non-negativ e and the same data can be copied multiple times without additional cost, the seller also prefers to be matched to buyers of a wide range of types. The buyers, on the other hand, may recei ves ne gati ve re ward by being matched to lower types. As a result, buyers of some types may not prefer to be matched to sellers of (suf ficiently) low types. W e define the following matching patterns when the platform adopts the cut-off matching rule defined in Definition 3. Definition 4 ( Matching Patterns ) . W e consider the follow- ing match patterns, for K ∈ { S , B } , ¯ K ∈ { S , B }\{ K } . (i) Complete-matched on side K : δ K = λ K . (ii) Bottom-eliminated on side K : δ K > λ K . (iii) T op-r eserved on side K : τ ¯ K ( δ ¯ K | δ ¯ K ) < ¯ λ K . Here, complete-matched on side K means all participants on side K are matched to (some or all of) participants on side ¯ K . If the market is bottom-eliminated on side K , then the participants of types in [ λ K , δ K ) are not matched to any participants. If the market is top-reserved on side K , then there is a group of types in the top subset [ τ ¯ K ( δ ¯ K | δ ¯ K ) that are matched to all the participants on side ¯ K whose types are in [ δ ¯ K , ¯ λ ¯ K ] , i.e., those are not bottom-eliminated on side ¯ K . Based on the cut-off rule, we can rewrite the utility function in (2) as follo ws, with a slight ab use of notation: u K ( λ K , τ K ( λ K | δ K ) | σ K ) = Z ¯ λ ¯ K τ K ( λ K | δ K ) R K ( λ K , x | σ K ) f ¯ K ( x ) d x . (15) The platform’ s rev enue in (11) can be rewritten, by setting J K ( λ K | σ K , φ K ) = 0 and integral by parts as follo ws: Z R ( σ σ σ , φ φ φ ) = ∑ K ∈{ S , B } Z Λ K h u K ( x , τ K ( x | δ K ) | σ K ) − D K ( x ) 1 − F K ( x ) f K ( x ) i f K ( x ) d x . (16) Then, the virtual surpluses of the welfare and the revenue maximization of the participant of λ K , respectiv ely , are as follows: for K ∈ { S , B } , U K W ( λ K , τ K ( λ K | δ K )) = u K ( λ K , τ K ( λ K | δ K ) | σ K ) , (17) and U K R ( λ K , τ K ( λ K | δ K )) = u K ( λ K , τ K ( λ K | δ K ) | σ K ) − D K ( λ K ) 1 − F K ( λ K ) f K ( λ K ) . (18) Suppose a participant has the type λ K ∈ [ τ ¯ K ( λ ¯ K ) , ¯ λ K ] , i.e., she is matched to participants from ¯ K of type λ ¯ K and other participants whose types are abo ve λ ¯ K . Hence, the marginal effect of this participant on the utility of the participants from ¯ K of λ ¯ K can be modeled as ω ¯ K ( λ K ) = γ K ( λ K ) f K ( λ K ) . (19) The direct marginal ef fect of this participant (not from influencing the matched participants from ¯ K through the matching rule) on the social welf are defined in (10) is proportional to the following, for some arbitrarily fixed non- zero λ ¯ K ∆ ∈ Λ ¯ K , θ K W ( λ K , λ ¯ K ∆ ) = − ∂ ∂ x U K W ( λ K , x ) f K ( λ K )    x = λ ¯ K ∆ . (20) Similarly , the direct marginal effect of participant i on the platform’ s rev enue defined in (9) is proportional to the following, for some arbitrary fixed λ ¯ K ∆ ∈ Λ ¯ K , θ K R ( λ K , λ ¯ K ∆ ) = − ∂ ∂ x U K R ( λ K , x ) f K ( λ K )    x = λ ¯ K ∆ . (21) W e hav e the follo wing assumption. Assumption 3. F or K ∈ { S , B } , Y ∈ { W , R } , θ K Y ( λ K , λ ¯ K ∆ ) ω ¯ K ( λ K ) is a strictly increasing function of λ K , for any fixed non-zer o λ ¯ K ∆ ∈ Λ ¯ K . In Assumption 3, the term θ K Y ( λ K , λ ¯ K ∆ ) ω ¯ K ( λ K ) is the ratio of a participant of λ K ’ s direct marginal contribution to Z Y , Y ∈ { W , R } , and her marginal contribution to each of her matched participants from ¯ K . Assumption 3 requires that the direct marginal contribution of a participant of λ K to the social welfare or the platform’ s rev enue changes faster than her marginal contribution to any of her matched opponents from ¯ K through the matching rule. Define, for K ∈ { S , B } , Y ∈ { W , R } , η K Y ( λ K , λ ¯ K ) = θ K Y ( λ K , λ ¯ K ) + θ ¯ K Y ( λ ¯ K , λ K ) . (22) Here, η K Y ( λ K , λ ¯ K ) is the joint marginal ef fect of matching λ K to λ ¯ K to the social welfare ( Y = W ) or platform’ s rev enue ( Y = R ) if they are matched. Here, η K Y coincides with the contribution C in Assumption 1. The following proposition shows the uniqueness of the cut-off functions. Proposition 2. Suppose Assumptions 2 and 3 hold. If η K Y ( λ K , λ ¯ K ) < 0 , for Y ∈ { W , R } , K ∈ { S , B } , then the cut-off function τ K specifies a unique matched lowest type for each type on side K , i.e., τ K ( λ K | δ K ) 6 = τ K ( ˜ λ K | δ K ) , for any two differ ent λ K , ˜ λ K ∈ [ δ K , ¯ λ K ] . Pr oof. See Appendix A. The follo wing proposition shows the conditions under which the market e xhibits certain matching patterns. Proposition 3. Suppose Assumptions 2 and 3 hold. Then, the followings ar e ture . (i) If η K Y ( λ K , λ ¯ K ) ≥ 0 , then the market is complete-matched on both sides of the market, for Y ∈ { W , R } , K ∈ { S , B } . (ii) If η K Y ( λ K , λ ¯ K ) < 0 and η K Y ( ¯ λ K , λ ¯ K ) > 0 , then the market is top-reserved on side K and complete-matched on side ¯ K . Additionally τ K ( λ K | δ K ) = δ ¯ K for all λ K ∈ [ τ ¯ K | δ ¯ K , ¯ λ K ] . (iii) If η K Y ( λ K , λ ¯ K ) < 0 and η K Y ( ¯ λ K , λ ¯ K ) < 0 , then the market is not top-res erved on side K and bottom-eliminated on side ¯ K . (iv) If η K Y ( λ K , λ ¯ K ) < 0 and η K Y ( ¯ λ K , λ ¯ K ) = 0 , then the market is not top-reserved on side K and complete-matched on side ¯ K . Pr oof. See Appendix B. W ithout loss of generality , we refer to K as the seller and ¯ K as the buyer and focus on the social welfare maxi- mization, i.e., Y = W . In Proposition 3, η S W ( λ S , λ B ) is the joint marginal effect of matching the sellers’ lowest type λ S to the buyers lowest type λ B . It η S W ( λ S , λ B ) ≥ 0, then η S W ( λ S , λ B ) ≥ 0 for all λ S ∈ Λ S and λ B ∈ Λ B due to the monotonicity in Assumption 2. If η S W ( λ S , λ B ) < 0, then the lowest types should not be matched because they contrib ute negati vely to the social welfare. Here, η S W ( ¯ λ S , λ B ) measures the joint marginal effect of matching a seller of the highest type ¯ λ S to a buyer of the lowest type λ B . T ogether with η S W ( λ S , λ B ) < 0, η S W ( ¯ λ S , λ B ) > 0 means that buyers of the lowest type can generate positive contrib ution to the social welfare, which implies that buyers of all kinds of types are matched to sellers and τ B ( λ B | δ B ) < ¯ λ S , i.e., top-reserved, due to the non-increasing property of τ B . On the other hand, η S W ( λ S , λ B ) < 0 and η S W ( ¯ λ S , λ B ) < 0 imply that the buyers of the lowest type should be eliminated from being matched. This exhibits the bottom-eliminated on the buyer side, i.e., δ B > λ B . Hence, τ B ( δ B | δ B ) < τ B ( λ B | δ B ) . Since λ B is excluded (and not matched to ¯ λ S ), τ B ( δ B | δ B ) = ¯ λ S , i.e., the buyers of lowest matched type is matched only to the sellers of the highest type. From the reciprocal relationship, measuring η B W on λ S , λ B , and ¯ λ S can complete the matching patterns on both sides. Corollary 2 directly follo ws Proposi- tions 2 and 3. Corollary 2. Suppose Assumptions 2 and 3 hold. Then, the followings hold. (i) If η K Y ( λ K , λ ¯ K ) < 0 , then η K Y ( λ K , τ K ( λ K | δ K )) = 0 . (ii) If η K Y ( λ K , λ ¯ K ) < 0 and η K Y ( ¯ λ K , λ ¯ K ) < 0 , then τ ¯ K ( δ ¯ K | δ ¯ K ) = ¯ λ K . (iii) If η K Y ( λ K , λ ¯ K ) < 0 and η K Y ( ¯ λ K , λ ¯ K ) = 0 , then τ ¯ K ( λ ¯ K | λ ¯ K ) = ¯ λ K . (a) (b) (c) (d) Fig. 1: (a): W elfare-maximizing matching rule. δ K = λ K and τ K ( λ K | δ K ) = λ ¯ K . i.e., each data seller is matched to all data buyers. (b): Rev enue-maximizing matching rule. Each data seller of λ S is matched to all data buyers of λ B ∈ [ τ S ( λ S | 40 9 ) , 10 ] , where τ S ( λ S | 7 2 ) = 10 λ S − 5 4 λ S − 11 . (c): W elfare vs. δ K . When δ K = 1, the welfare reaches its maximum, i.e., τ ¯ K ( δ ¯ K | δ ¯ K ) = λ K . (d): Marginal Rev enue vs. λ S . The platform has a non-negati ve marginal revenue for λ S ≥ 7 2 . τ S ( λ S | 7 2 ) = 10 λ S − 5 4 λ S − 11 is optimal and specifies a unique cut-of f for each λ S ∈ [ 7 2 , 10 ] . A. Numerical Analysis In this section, we present a numerical example for the matching to illustrate the optimal matching rules for the social welfare and the platform’ s re venue maximization. Fig. 1 sho ws the numerical results. Consider the case when Λ K = [ 1 , 10 ] , γ S ( λ S ) = λ S , γ B ( λ B ) = 1 2 λ B , M K ( r , λ K ) = C K λ K r , with C S = 1 and C B = 1 2 , for K ∈ { S , B } . Let R S ( λ S , λ B | σ S ) = 1 2 λ S λ B and R B ( λ B , λ S | σ B ) = 1 2 λ B ( λ S − 1 2 ) . Suppose the type λ K is uniformly distributed over Λ K , for K ∈ { S , B } , i.e., F K ( λ K ) = λ K − 1 9 , and f K ( λ K ) = 1 9 . Then, ω B ( λ S ) = 1 9 λ S , ω S ( λ B ) = 1 18 λ B , θ S W ( λ S , λ B ∆ ) = 1 18 λ B ∆ λ S , θ B W ( λ B , λ S ∆ ) = 1 18 λ B ( λ S ∆ − 1 2 ) , θ S R ( λ S , λ B ∆ ) = 1 9 λ B ∆ ( λ S − 5 ) , θ B R ( λ B , λ S ∆ ) = 1 9 ( λ B − 5 )( λ S − 1 2 ) . It is straightforward to verify that Assumption 3 holds on both sides of the market for both the social welfare and the rev enue maximization. For K ∈ { S , B } , η K W ( λ K , λ ¯ K ) = 1 36 λ B ( 4 λ S − 1 ) , η K R ( λ K , λ ¯ K ) = 1 9 ( 2 λ B λ S − 5 λ S − 11 2 λ B + 5 2 ) . Since η K W ( λ K , λ ¯ K ) > 0 for all λ K ∈ Λ K and λ ¯ K ∈ Λ ¯ K , social welfare maximizing matching rule matches all sellers to all buyers and reciprocally all buyers to all sellers. F or re venue maximization, since η K R ( 1 , 1 ) < 0, for K ∈ { S , B } , Proposition 2 implies that the matching rule sets a unique τ K ( λ K | δ K ) for each λ K ∈ [ δ K , ¯ λ K ] . On the seller side, η S R ( 10 , 1 ) = − 33 9 < 0. One the buyer side, η B R ( 10 , 1 ) = − 75 2 < 0. Hence, Proposition 3 implies that the market is bottom-eliminated and not top- reserved on both sides. From Corollary 2, we have τ S ( λ S | δ S ) = 10 λ S − 5 4 λ S − 11 , with δ S = 7 2 , and τ B ( λ B | δ B ) = 11 λ B − 5 4 λ B − 10 , with δ B = 95 29 . V . C O N C L U S I O N In this paper , we hav e proposed a tw o-sided matching mar - ket framework for IoT data trading as a sustainable pricing model that incentivizes both the data sellers and the data buyers. A monopolist platform has been introduced to match the data owners and the data buyers based on their priv ate information. W e hav e established a quantitative framework to model the IoT data trading market for the welfare and the rev enue maximization. This work has characterized a class of feasible cut-off matching rules. 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Pr oof of Pr oposition 2 Expanding θ K W and θ K R as follo ws: θ K W ( λ K , λ ¯ K ) = − ∂ ∂ x U K W ( λ K , x ) f K ( λ K )    x = λ ¯ K = R K ( λ K , λ ¯ K | σ K ) f ¯ K ( λ ¯ K ) f K ( λ K ) , and θ K R ( λ K , λ ¯ K ∆ ) = − ∂ ∂ x U K R ( λ K , x ) f K ( λ K )    x = λ ¯ K ∆ = R K ( λ K , λ ¯ K | σ K ) f ¯ K ( λ ¯ K ) f K ( λ K ) + ∂ ∂ x R K ( x , λ ¯ K | σ K ) f ¯ K ( λ ¯ K )( 1 − F K ( λ K )) . Define, ρ K W ( λ K , λ ¯ K ) = R K ( λ K , λ ¯ K | σ K ) f K ( λ ¯ K ) , and ρ K R ( λ K , λ ¯ K ) = R K ( λ K , λ ¯ K | σ K ) f K ( λ ¯ K ) + ∂ ∂ x R K ( x , λ ¯ K | σ K )( 1 − F K ( λ K )) , such that θ K Y ( λ K , λ ¯ K ) = ρ K Y ( λ K , λ ¯ K ) f ¯ K ( λ ¯ K ) . If ρ K Y ( λ K , λ ¯ K ) < 0, for all K ∈ { S , B } , Y ∈ { W , R } , then η K Y ( λ K , λ ¯ K ) < 0. Let τ K ( | δ K ) be the matching rule that sat- isfies the uniqueness property in Proposition 2. Let ˆ τ K ( | ˆ δ K ) with threshold ˆ δ K be the cut-off function that does not specify unique matched cut-off types for different types. Hence, participants from side K are excluded from matching if λ K < ˆ δ K . Otherwise, participants from side K are matched to participants from side ¯ K whose types are above ˆ δ ¯ K . Let g K Y = inf { λ K ∈ Λ K : ρ K Y ≥ 0 } . Suppose the platform adopts the unique matching rule τ K that matches each participant of type λ K ≥ ˆ δ K to the same set as the non-unique matching rule ˆ τ K and matches each participant of type λ K ∈ [ g K Y , ˆ δ K ] to the set [ ˜ λ ¯ K , ¯ λ ¯ K ] , where ˜ λ ¯ K = max { g K Y , ˆ δ K } . Then, the platform can increase his payoff (utilities or revenues) by switching ˆ τ K to τ K . Now , suppose ˆ δ K < g K Y for K ∈ { S , B } . Let τ K Ω be defined as follo ws: τ K Ω ( λ K )      [ ˆ δ ¯ K , ¯ λ ¯ K ] if, [ g K Y , ¯ v K ] [ g ¯ K Y , ¯ λ K ] if, [ ˆ δ K , g K Y ] / 0 if, [ λ K , ˆ δ K ] . By adopting τ K Ω , the platform can improve his payoff than using the original ˆ τ K with ¯ δ K because all types that lead to negati ve ρ K Y are eliminated, for K ∈ { S , B } , Y ∈ { S , B } . Next, suppose ˆ δ = g K Y , for K ∈ { S , B } and ˆ δ ¯ K ≤ g ¯ K Y , for ¯ K ∈ { S , B }\{ K } , Y ∈ { W , R } . Let ˜ δ K + ε = g K Y with sufficiently small ε > 0. Define τ K Ω ( λ K )      [ ˆ δ ¯ K , ¯ λ ¯ K ] if, [ g K Y , ¯ v K ] [ g ¯ K Y , ¯ λ K ] if, [ ˜ δ K , g K Y ] / 0 if, [ λ K , ˜ δ K ] . Here, participants of type λ K ∈ [ ˜ δ K , g K Y ] contribute ne gati vely to the platform’ s payof f b ut they are sufficiently small such that they do not offset the positiv e contribution from participants of types in [ g K Y , ¯ v K ] . Hence, the platform can increase his payoff by using τ K Ω to replace the original ˆ τ K with ˆ δ K . Now , consider ρ ¯ K Y ( λ ¯ K , λ K ) < 0 ≤ ρ K Y ( λ K , λ ¯ K ) , η K Y ( λ K , λ ¯ K ) < 0 and the platform adopts a non-unique cut-off function. First, suppose that the market is complete- matched on both sides, i.e., ˆ δ K = λ K , τ ¯ K ( λ ¯ K ) = λ K . Since η K Y ( λ K , λ ¯ K ) < 0, the platform’ s effort of increasing τ K ( λ K , ) abov e λ ¯ K while keeping the cut-off type unchanged for all other types on side K contributes to his payoff positi vely . Hence, the platform can improve his payoff by using the unique cut-of f by increasing τ K at the right neighborhood of λ K while keeping the cut-off type unchanged for all other types on side K . Next, suppose that the original cut-off function eliminates some participants and match each of the participants of types abov e ˆ δ K to the same matched set. As similar to the above analysis, optimal rule has to be in the case when ˆ δ ¯ K < g ¯ K Y and ˆ δ K = λ K , and Z Λ K η K Y ( x , ˆ δ ¯ K ) d x = 0 which requires that the total effect of an increase of the size of the matched set on ¯ K is zero. From Assumption 3, we hav e there exists a λ K ∆ ∈ Λ K such that R ¯ λ K λ K ∆ η K Y ( x , ˆ δ ¯ K ) d x > 0. Thus, there exists a δ K ∆ < ˆ ω K such that the platform can improv e his payof f by using the follo wing cut-of f functions, instead of the original cut-of f function: τ K Ω ( λ K ) ( [ δ K ∆ , ¯ λ ¯ K ] if, [ λ K ∆ , ¯ v K ] [ ˆ ω K , ¯ λ K ] if, [ λ K , λ K ∆ ] . Therefore, we can conclude that unique cut-off rule is optimal for the platform. B. Pr oof of Pr oposition 3 Let ρ K Y be defined in Appendix A, such that θ K Y ( λ K , λ ¯ K ) = ρ K Y ( λ K , λ ¯ K ) f ¯ K ( λ ¯ K ) . If ρ K Y ( λ K , λ ¯ K ) ≥ 0, for all K ∈ { S , B } , Y ∈ { W , R } , then η K Y ( λ K , λ ¯ K ) ≥ 0. The monotonicity of the ρ K Y implies that the social welfare ( Y = W ) and the platform’ s rev enue ( Y = R ) are maximized by matching all sellers to all b uyers. Hence, the market is complete-matched on both sides. Next, consider ρ ¯ K Y ( λ ¯ K , λ K ) < 0 ≤ ρ K Y ( λ K , λ ¯ K ) . Suppose η K Y ( λ K , λ ¯ K ) ≥ 0, then the platform uses a matching rule τ K Γ ( λ K ) ≥ v ¯ K for some λ K ∈ Λ K . Let λ K a be any type in Λ K , such that τ K is strictly decreasing at the right neighborhood of λ K a . Hence, the marginal effect of reducing τ K ( λ K ) below λ ¯ K is η K Y ( λ K , λ ¯ K ) . Let [ λ K ` , λ K h ] ⊂ Λ K , in which τ K ( λ K ) = λ ¯ K for all λ K ∈ [ λ K ` , λ K h ] , i.e., τ K specifies a constant cut-off type λ ¯ K , in which the marginal effect of the reduction of τ K ( λ K ) belo w λ ¯ K is R λ K h λ K ` η K Y ( x , λ ¯ K ) d x . Under Assumption 3, we hav e η K Γ ( λ K , λ ¯ K ) > 0 for any λ K ∈ Λ K and λ ¯ K ∈ Λ ¯ K , which implies that a complete-matched pattern for both sides is optimal. Let g K Y = inf { λ K ∈ Λ K : ρ K Y ≥ 0 } . The platform’ s objective functions can be written in terms of U K Y as follo ws: for Y ∈ { W , R } , Z Y ( { τ τ τ , δ δ δ } ) = ∑ K ∈ S , B Z ¯ λ K δ K U K Y ( x , τ K ( x )) f K ( x ) d x Since η K Y ( λ K , λ ¯ K ) < 0, g K Y > λ K . From Appendix A, δ K ∈ [( λ ) K , g K Y ] at the optimum. Moreo ver , δ ¯ K Y ∈ [ λ ¯ K , g K Y ] and τ K ( g K Y ) = g ¯ K Y . Assume that g K Y > λ ¯ K . (If not, then δ ¯ K = λ ¯ K and τ K K ( λ K ) = λ K for all v k ≥ g K Y ). Hence, we can partition Z Y into two independent problem, for K ∈ S , B , Z K Y = Z g K Y δ K U K Y ( x , τ K ( x )) f K ( x ) d x + int ¯ λ ¯ K g ¯ K Y U ¯ K Y ( x , τ ¯ K ( x )) f b arK ( x ) d x . Here, the problem Z K Y can be solved piecewisely . Let λ K ∆ = inf { λ K ∈ [ λ K , g K Y ] : η K Y ( λ K , κ ( λ K )) = 0 } . Then, the follo wing cut-off function maximizes Z K Y : ˜ τ K ( λ K ) = ( ¯ λ ¯ K if, λ K ∈ [ λ K , λ K ∆ ] , κ ( λ K ) if, λ K ∈ ( λ K ∆ , ¯ λ K ] . Next, let cut-of f function associated to the matching rule τ K = ˜ τ K for any λ K ∈ [ λ K ∆ , g K Y ] . Gi ven τ K : [ δ K , g K Y ] 7→ [ δ ¯ K , g ¯ K Y ] , reciprocally , we have τ ¯ K : [ δ ¯ K , g ¯ K Y ] 7→ [ δ K , g K Y ] . It is straightforward to see that matching rule characterized by the above τ K with δ K is unique in the sense that it specifies a unique cut-off type for each matched type on side K . It is easy to see that the market if bottom-eliminated on side K if λ K ∆ > λ K and top-reserved on side ¯ K if λ K ∆ = λ K . From Assumption 3, we hav e η K Y ( λ K , ¯ λ K ) < 0. Next consider that λ K ∆ = λ K , in which there exists a κ ( λ K ) ∈ [ g ¯ K Y , ¯ λ K ] such that η K Y ( λ K , κ ( λ K )) = 0. Suppose κ ( λ K ) < ¯ λ ¯ K . From Assumption 3, η K Y ( λ K , ¯ λ ¯ K ) > 0. Hence, we hav e that if η K Y ( λ K , ¯ λ ¯ K ) > 0, the market is complete- matched on side ¯ K and top-reserved on side K . Finally , consider κ ( λ K ) = ¯ λ ¯ K , in which δ K = λ K and τ K ( λ K ) = ¯ λ ¯ K , i.e., the mark et is not top-reserved on side ¯ K and complete- matched on side K .