215 13th International Sympoium on Modeling and Optimization in Moile, Ad Hoc, and Wirele Network (WiOpt) Moile Data Trading: A Behavioral Economic Perpective Junlin Yu, Man Hon Cheung, Jianwei Huang, and H. Vincent Poor Atract Motivated y the recently launched 2CM data trading platform of China Moile Hong Kong, we tudy the optimal uer moile data trading prolem under the future demand uncertainty. We conider a rokerage-aed market, where eller and uyer propoe their elling and uying price and quantitie to the trading platform, repectively. The platform act a a roker, which facilitate the trade y matching the upply and demand. To undertand uer realitic trading ehavior, we ue propect theory (PT) from ehavioral economic in the modeling, which lead to a challenging non-convex optimization prolem. Neverthele, we are ale to detere the unique optimal olution in cloed-form, y utilizing the unimodal tructure of the ojective function. When comparing with the enchmark expected utility theory (EUT), we how that a PT uer with a low reference point i more willing to uy moile data. Moreover, when the proaility of high demand i low, comparing with an EUT uer, a PT uer i more willing to uy moile data due to the proaility ditortion. I. INTRODUCTION With the increaing computation and communication capailitie of moile device, gloal moile data traffic ha een growing tremendouly in the pat few year 1], 2]. In order to alleviate the tenion etween moile data demand and network capacity, moile ervice provider have een experimenting with everal innovative pricing cheme, uch a uage-aed pricing, hared data plan, and ponored data pricing 3] 5]. However, the aove cheme do not fully take advantage of the heterogeneou demand acro moile uer. Recently, China Moile Hong Kong (CMHK) launched the firt 4G data trading platform in the world, called the 2nd exchange Market (2CM), which allow it uer to trade their 4G moile data directly with each other. In thi platform, a eller can lit hi deirale elling price (within a predefined range), together with the amount of data to e old (up to hi monthly ervice plan quota). If there i a uyer who i willing to uy (part of) the data at the lited price, the platform will clear the tranaction and tranfer the correponding data to the uyer monthly quota limit. In the current form of 2CM, only a eller can lit hi price and elling quantity, ut not a uyer. Thi mean that a uyer need to frequently check the platform to ee whether he i J. Yu, M. H. Cheung, and J. Huang are with the Department of Information Engineering, the Chinee Univerity of Hong Kong, Hong Kong, China; Email: {yj112, mhcheung, jwhuang}@ie.cuhk.edu.hk. H. V. Poor i with the Department of Electrical Engineering, Princeton Univerity, Princeton, NJ, USA; Email: poor@princeton.edu. Thi work i upported in part y the General Reearch Fund (Project Numer CUHK 412713 and 1422814) etalihed under the Univerity Grant Committee of the Hong Kong Special Aditrative Region, China, and in part y the U. S. Army Reearch Office under MURI Grant W911NF-11-1-36. willing to uy according to the current (lowet) elling price. Thi motivate u to propoe a new market mechanim that i aed on the widely ued Walraian auction in the tock market 6], 7], in which oth eller and uyer can umit their elling and uying price and quantitie to the platform. The platform clear the market whenever the uying price of a uyer i no maller than the elling price of a eller. In thi paper, we focu on a ingle uer trading deciion under the future demand uncertainty, given the price and quantitie of other eller and uyer. More pecifically, we focu on the following quetion: (i) Should a uer chooe to e a eller or a uyer? (ii) How much hould he ell or uy? One way of anwering the aove quetion i to compute the imum expected utilitie for eing a eller and a uyer, conidering hi future demand uncertainty and the atifaction lo for exceeding the monthly data quota. Then y comparing thee utilitie, the uer can decide whether to e a eller or a uyer. Such an approach relie on the expected utility theory (EUT), which ha een widely ued in tudying deciion prolem under uncertainty 8]. However, utantial empirical evidence ugget that prediction aed on EUT can ignificantly deviate from real world oervation, due to the complicated pychological apect of human deciionmaking. Reearcher in ehavioral economic howed that propect theory (PT) provide a pychologically more accurate decription of the deciion making under uncertainty, and can explain quite a few human ehavior that eem to e illogical under EUT 9]. PT how that a deciion maker evaluate an outcome ignificantly differently from what people have commonly aumed in EUT in everal apect: (1) Impact of reference point: A PT deciion maker evaluation i aed on the relative gain or loe comparing to a reference point, intead of the aolute value of the outcome. (2) Aymmetric value function: A PT deciion maker tend to e rik avere when conidering gain and rik eeking when conidering loe. Furthermore, the PT deciion maker i lo avere, in the ene that he trongly prefer avoiding loe to achieving gain. (3) Proaility ditortion: A PT deciion maker tend to overweigh low proaility event and underweigh high proaility event. A PT ha een hown to e more accurate than EUT in predicting human ehavior 9] 11], it ha een uccefully applied to have a etter undertanding of financial market 12] and laor market 13]. In thi paper, we aim to undertand a uer realitic trading ehavior in a moile data market, conidering the uer future demand uncertainty. Specifically, we formulate the prolem a a two-tage optimization prolem, where the uer will decide 978-3-918-8274-6/15/$31. 215 IEEE 363
215 13th International Sympoium on Modeling and Optimization in Moile, Ad Hoc, and Wirele Network (WiOpt) whether to e a eller or a uyer in Stage I, and hi elling price and quantity (a a eller) or uying price and quantity (a a uyer) in Stage II. We will dicu the practical inight y comparing the analyi under PT and EUT (which i a pecial cae of PT with properly choen ytem parameter). The reearch of uing ehavioral economic (and PT in particular) to undertand uer deciion in networking i at it infancy tage. Li et al. in 14], 15] conidered a linear value function with the proaility ditortion, and compared the equilirium trategie of a two-uer random acce game under EUT and PT. Xiao et al. in 16] and Wang et al. in 17] conidered a linear value function with the proaility ditortion, and characterized the unique Nah Equilirium of an energy exchange game among microgrid under PT. Yu et al. in 18] conidered the general S-haped value function in tudying a econdary wirele operator pectrum invetment prolem. To the et of our knowledge, thi paper i the firt work that capture all three characteritic of PT when modeling and analyzing a wirele networking prolem. A a reult, we are ale to gain a more thorough undertanding of the uer optimal deciion and derive more inight. Our key contriution are ummarized a follow: Behavioral economic modeling of uncertainty: Weue propect theory to model the uer trading ehavior under future demand uncertainty. We conider all three key characteritic of PT, and derive key inight that characterize the optimal elling and uying deciion. Non-convex optimization: Depite the non-convexity of the uer deciion prolem, we are ale to otain a cloed-form characterization of the unique optimal olution. We further evaluate how different ehavioral characteritic (i.e., reference point, proaility ditortion, and S-haped valuation) affect thi optimal deciion. Engineering inight: Comparing with the reult under EUT, we how that a PT uer with a low reference point i more willing to uy moile data and le willing to ell moile data. Moreover, a PT uer i more willing to uy moile data when the proaility of high demand i low, mainly due to the proaility ditortion. The ret of thi paper i organized a follow. In Section II, we decrie the data uage trading platform and formulate the uer utility function under oth EUT and PT. In Section III, we compute the unique optimal uer deciion. In Section IV, we numerically evaluate the enitivity of the uer optimal deciion with repect to everal model parameter. We conclude the paper in Section V. II. SYSTEM MODEL We conider a moile data trading market with a large numer of uer. Each uer make the trading deciion in two tage. In Stage I, he decide whether to ell or to uy in the market, or not participate in the market. In Stage II, he decide the price and quantity a a eller or a a uyer, depending on hi choice in Stage I. For implicity, we aume that the uer make the trading deciion only once in a illing cycle, although different uer may make deciion at different Tale I: A naphot of the data trading platform. Buyer Market Seller Market Price (per GB) Availale (GB) Price (per GB) Availale (GB) $16 6 $2 5 $13 2 $21 45 $11 3 $24 2............ time 1. Since the numer of uer in the market i large, a ingle uer choice will have a negligile impact on the market. More pecifically, in Stage I, a uer make a deciion a A = {, }, where and correpond to eing a eller and a uyer, repectively. In Stage II, a eller detere hi offer {q,π }, which mean that he i willing to ell q (GB) of data at a unit price of π (dollar per GB). A uyer detere hi id {q,π }, which mean that he i willing to uy a total of q of data at a unit price of π. Tale I how an example of the market, which include the price and quantitie of uer who have made their deciion. In thi example, the highet uying price from the uyer market (π = $16) i lower than the lowet elling price from the eller market (π = $2). Thi i ecaue thoe elling offer with price le than $16 have already een cleared y the market, and o are thoe uying requet with price higher than $2. Under a large network aumption, it i reaonale to aume that the quantity aociated with the imum uyer price and the quantity aociated with the imum eller price are oth large enough. Thi mean that for a ingle uer who want to complete the trade immediately, he only need to conider the imum uyer price π and imum eller price π, and ignore all other price 2.In particular, if a uer chooe to e a eller in Stage I, hi elling price in Stage II will e π = π, o that he can ell the data immediately with the imum price that ome exiting uyer can accept. Similarly, if a uer chooe to e a uyer in Stage I, he will et hi uying price in Stage II a π = π, o that he can uy the data immediately with the imum price that ome exiting eller can offer. Hence we will ignore the uer pricing deciion in the ret of the paper. The key iue that the uer need to conider i the future data demand uncertainty. If hi total monthly data conumption d exceed hi monthly data quota Q, he will incur a atifaction lo. For implicity, we conider a linear atifaction lo function, where {, y, L(y) = κy, y <, (1) y = Q d. (2) When y i negative, it mean that the quota i exceeded. The linear coefficient κ repreent the uage-aed pricing impoed 1 In the future work, we will conider the multi-period cenario, where each uer can make multiple equential trading deciion in a ingle illing cycle. 2 If a uer doe not need to complete the trade immediately, he may chooe to lit a elling price higher than π or lit a uying price lower than π. We will conider thi more general cae in our future work. 364
215 13th International Sympoium on Modeling and Optimization in Moile, Ad Hoc, and Wirele Network (WiOpt) 2 1 v(x) 1 Lo 1 Gain Sujective Proaility.8.6.4.2 α =.4 α =.6 α =.8 α = 1 2 λ = 1, β = 1 λ = 1.5, β =.3 λ = 2, β =.3 λ = 2, β =.6.2.4.6.8 1 Ojective Proaility p Figure 2: The proaility ditortion function in PT. 3 2 1 1 2 x Figure 1: The S-haped aymmetrical value function v(x) in PT. y the operator 3. By elling or uying data in the market, a uer can change hi monthly data quota (for the current month only), and hence will change the expected value of the atifaction lo. Next we derive the uer expected utilitie of eing a eller and a uyer, under oth EUT and PT. A. Utilitie under Expected Utility Theory (EUT) We aume that the uer total monthly data conumption ha I poile value {d i : i =1,..., I}, with the correponding proailitie {p i : i =1,..., I} uch that I i=1 p i =1.Ifauer uy q GB of data, hi expected utility i where π U(, q )= elling price π I i=1 p i π q + L(Q + q d i ) ], (3) q i the cot for uying the data at the imum,andl(q + q d i ) i the atifaction lo if the total data conumption i d i. If a uer ell q GB of data in the market, then hi expected utility i U(, q )= I i=1 p i π q + L(Q q d i )], (4) where π q i the revenue for elling the data at the imum uying price π. B. Utilitie under Propect Theory (PT) Here, we conider the three feature of PT, namely S-haped value function v(x), proaility ditortion function, and reference point R p 9], 2]. A higher reference point R p indicate that the uer ha a high expectation on the utility, and he conider an outcome a a lo if it i le than hi expectation. A lower reference point R p indicate that the uer ha a low expectation, and he conider an outcome a a gain if it i aove hi expectation. 3 We have aumed a two-part pricing tariff, where the uer pay a fixed fee for the data conumption up to a monthly quota, and a linear uage-aed cot for any extra data conumption. Such a pricing model i widely ued y major operator 19]. For example, for a 4G CMHK uer, κ = $6 with a monthly data quota of 1 GB. A we will ee next, whether an outcome i conidered a a lo and gain will ignificantly affect the uer ujective valuation of the outcome. Figure 1 illutrate the value function v(x), which map an outcome x to the uer ujective valuation v(x). Notice that all the outcome are meaured relatively to the reference point, which i normalized to x =. Behavioral tudie how that the function v(x) i S-haped, which i concave in the gain region (i.e., x>, the outcome i larger than the reference point) and convex in the lo region (i.e., x<, the outcome i maller than the reference point). Moreover, the impact of lo i larger than the gain, i.e., v( x) >v(x) for any x>. A commonly ued value function in the PT literature i 9] { x β, x, v(x) = λ( x) β (5), x <, where <β 1 and λ 1. Hereβ i the rik averion parameter, where a maller β mean that the value function i more concave in the gain region, hence the uer i more rik avere in gain. Similarly, a maller β mean that the value function i more convex in the lo region, hence the uer i more rik eeking in loe. The valuation of the lo region i further characterized y the lo penalty parameter λ, where a larger λ indicate that the uer i more lo avere. Figure 2 illutrate the proaility ditortion function, which capture human pychological over-weighting of low proaility event and under-weighting of high proaility event. A commonly ued proaility ditortion function in the PT literature i 2] =exp( ( ln p) α ), <α 1, (6) where p i the real proaility of an outcome, and i the correponding ujective proaility. Here α i the proaility ditortion parameter, which reveal how a peron ujective evaluation ditort the ojective proaility. A maller α mean a larger ditortion. Conidering the aove three feature in PT, a uyer expected utility i U(, q )= I i=1 w(p i )v ( π q + L(Q + q d i ) R p ), (7) 365
215 13th International Sympoium on Modeling and Optimization in Moile, Ad Hoc, and Wirele Network (WiOpt) Tale II: Buyer optimal uying quantity in Stage II under EUT Condition Optimal Buying Quantity q π <κp q = d h Q π κp q = Tale III: Buyer optimal uying quantity in Stage II under PT with R p = Condition Optimal Buying Quantity q ] 1 π β <κ q +w(1 p) = d h Q ] 1 π β κ q +w(1 p) = and a eller expected utility i I U(, q )= w(p i )v (π q + L(Q q d i ) R p ). i=1 (8) We note that the utility function under EUT ((3) and (4)) are pecial cae of thoe under PT ((7) and (8)), with the parameter choice of λ = β = α =1and R p = 4. In the next ection, we will tudy the uer optimal trading deciion in a large market. III. SOLVING THE TWO-STAGE OPTIMIZATION PROBLEM In thi ection, we ue ackward induction to olve the twotage equential optimization prolem. Firt, we derive the uer optimal elling or uying amount in Stage II. Then, we conider whether the uer chooe to e a eller or a uyer in Stage I, y comparing hi imum achievale utilitie under oth cae. To implify the preentation and etter illutrate the inight, we aume I =2for the ret of the paper. More pecifically, there are two poile realization of a uer monthly data conumption (or imply called demand): d h and d l, with d h > Q>d l > 5. The proaility of oerving a high demand d h i p, and the proaility of oerving low demand d l i 1 p. The analyi and inight can e generalized to the cae of I>2 with lightly more complicated algeraic manipulation. We further focu on two choice of reference point. The firt choice i R p =, which reflect the uer expectation of oerving the low demand and hence having no exceive demand. The econd choice i R p = κ(q d h ) <, which reflect the uer expectation of oerving the high demand and paying for the correponding exceive demand (without trading). Hence the ame outcome i more likely to e conidered a a gain under R p = κ(q d h ) than under R p =. A. Stage II 1) Buyer Prolem: To olve the Stage II prolem, we firt conider the uyer prolem, where the uyer decide 4 In fact, a long a λ = β = α =1, chooing a non-zero value of R p will jut induce a contant hift of the EUT utilitie in (3) and (4), without affecting the optimal deciion under EUT. 5 The analyi for the cae where oth d h and d l are higher (or lower) than the monthly quota Q i relatively trivial, and hence i omitted here due to pace limitation. Tale IV: Buyer optimal uying quantity in Stage II under PT with R p = κ(q d h ) Condition Optimal Buying Quantity q π κ < q w(1 p)+ = d h Q π κ w(1 p)+ & β =1 q = π κ w(1 p)+ & <β<1 q = κ(q d h ) (κ π ) β ] 1 β 1 +π w(1 p)π the uying quantity at the lowet eller price π. The correponding optimization prolem i u() = U(, q ), (9) q where U(, q ) i the utility function in (7) with I =2poile demand. A we have mentioned efore, EUT i a pecial cae of PT under λ = β = α =1and R p =. Theorem 1. The uyer optimal uying quantity under EUT i ummarized in Tale II. The uyer optimal uying quantitie under PT with high reference R p =and low reference R p = κ(q d h ) are ummarized in Tale III and Tale IV, repectively. The proof of Theorem 1 i given in Appendix A. Tale II, III, and IV how how the optimal uying quantity depend on the imum eller price π. In each tale, we oerve a uyer threhold price, elow which the optimal uying amount equal d h Q: π PT1 π EUT = κp (Tale II), (1) ] 1 β = κ (Tale III), (11) +w(1 p) π PT2 κ = (Tale IV). (12) w(1 p)+ In Tale II and III, we oerve that the optimal uying quantity i dicontinuou at the uyer threhold price. Thi i due to the linearity of utility function in the EUT cae and the convexity of utility function in the PT cae with R p =. Detail are given in Appendix A. From Tale II to IV, we have the following oervation. Oervation 1. When the proaility ditortion parameter α =1, a PT uyer with a high reference point R p =(Tale III) ha a maller threhold price than an EUT uyer (Tale II), i.e., π PT1 < π EUT. Thi mean that comparing with an EUT uyer, a PT uyer with a high reference point i le willing to purchae moile data. Oervation 2. When the proaility ditortion parameter α =1, a PT uyer with a low reference point R p = κ(q d h ) (Tale IV) ha the ame threhold price a an EUT uyer (Tale II), i.e., π PT2 = π EUT. However, the optimal uying quantity q of the PT uyer (Tale IV) i higher than the EUT uyer under the ame price π. Thi mean that comparing with an EUT uyer, a PT uyer with a low reference point i more willing to purchae moile data. Notice that uying data reduce the rik that the data conumption exceed the quota. When a uyer ha a high 366
215 13th International Sympoium on Modeling and Optimization in Moile, Ad Hoc, and Wirele Network (WiOpt) Tale V: Seller Optimal Selling quantity in Stage II under PT with R p = ) β 1 > λ(κ π π β w(1 p) 1 λ(κ π π β w(1 p) ) β ) β 1 λ(κ π π β w(1 p) Condition ( 1+ ( 1+ ( 1+ κ(d h Q) (κ π κ(d h Q) (κ π Optimal Selling Quantity q )(Q d l ) ) β 1 q = Q d l ) β 1 )(Q d l ) & β =1 q = ) κ β 1 )(Q d l ) & <β<1 q κ π = ( κ(d h Q) (κ π (d h Q) w(1 p)π β λ(κ π ) β ) 1 β 1 1 Tale VI: Seller Optimal Selling quantity in Stage II under PT with R p = κ(q d h ) Condition Optimal Selling Quantity q λ(κ π )(Q d l )] β <w(1 p){(π κ)q + κd h π d l ] β κ(d h Q)] β } q = Q d l λ(κ π )(Q d l )] β w(1 p){(π κ)q + κd h π d l ] β k(d h Q)] β } q = Tale VII: Seller Optimal Selling quantity in Stage II under EUT Condition Optimal Selling Quantity q π >κp q = Q d l π κp q = expectation (e.g., R p = ), he i more likely to encounter loe than gain under uncertainty. A we have mentioned in Section II, a maller β mean the uyer i more rik eeking in loe and will not uy data. When a uyer ha a low expectation (e.g., R p = κ(q d h ) < ), the uyer i more likely to encounter gain than loe. A we have mentioned in Section II, a maller β (which i the cae for a PT uer comparing with an EUT uer) mean the uyer i more rik avere in gain and will uy an amount equal to d h Q, which will completely eliate the rik that the data conumption exceed the updated quota d h. 2) Seller Prolem: Next we conider the eller prolem, where he need to decide the elling quantity q at the highet uyer price π : u() = U(, q ), (13) q Q where U(, q ) i the utility function in (8) with I =2poile demand. A we have mentioned, EUT i a pecial cae of PT under λ = β = α =1and R p =. Theorem 2. The eller optimal elling quantity q under EUT i ummarized in Tale VII. The eller optimal elling quantity q under PT with high reference R p =and low reference R p = κ(q d h ) are ummarized in Tale V and Tale VI, repectively. The proof of Theorem 2 i given in our online technical report 21]. Tale VII, V, and VI how how the optimal elling quantity depend on the imum uyer price π. In each tale, we oerve a eller threhold price, aove which the optimal elling amount equal Q d l. The eller threhold price π EUT, π PT1, and π PT2 are the unique olution of the following three equation: π EUT = κp (Tale V), (14) λ(κ π PT1 ) β ( ) β 1 κ(d h Q) ( π PT1 ) β 1+ w(1 p) (κ π PT1 =1 (Tale VI), )(Q d l ) (15) w(1 p){( π PT2 κ)q + κd h π PT2 d l ] β κ(d h Q)] β } = λ(κ π PT2 )(Q d l )] β (Tale VII). (16) In Tale V and VII, we oerve that the optimal elling quantity q i dicontinuou at the eller threhold price. Thi i due to the linearity of utility function in the EUT cae and the unimodality of utility function in the PT cae with R p = κ(q d h ). Detail are given in our technical report 21]. From Tale V-VII, we have the following oervation. Oervation 3. When the proaility ditortion parameter α =1, a PT eller with a high reference point R p =(Tale VI) ha a maller threhold price than an EUT eller (Tale V), i.e., π PT1 < π EUT. Thi mean that comparing with an EUT eller, a PT eller with a high reference point i more willing to ell moile data. Oervation 4. When the proaility ditortion parameter α =1, a PT eller with a low reference point R p = κ(q d h ) (Tale VII) ha a larger threhold price than an EUT eller (Tale V), i.e., π PT2 > π EUT. Thi mean that comparing with an EUT eller, a PT eller with a low reference point i le willing to ell moile data. Contrary to uying data, elling data increae the rik that the data conumption exceed the quota. When a eller ha a high expectation (e.g., R p =), he i more likely to encounter loe than gain under uncertainty. A we have mentioned in Section II, a maller β mean the eller i more rik eeking in loe and will ell an amount equal to Q d l.whena eller ha a low expectation (e.g., R p = κ(q d h ) < ), the eller i more likely to encounter gain than loe. A we have mentioned in Section II, a maller β mean the eller i more rik avere in gain and will not ell data. B. Stage I In Stage I, the uer decide whether to e a eller or a uyer, y comparing the imum utilitie that he can achieve in oth cae (aed on the calculation in Stage II). He need to olve the following optimization prolem: a {,} u(a), (17) where u() and u() are defined in (9) and (13), repectively. In the cae of EUT, we can compute the cloed-form optimal olution of prolem (17). 367
215 13th International Sympoium on Modeling and Optimization in Moile, Ad Hoc, and Wirele Network (WiOpt) Threhold Price 35 25 β =.6 β =.8 β = 1 Threhold Price 5 4 3 2 p =.2 p =.5 p =.8 1 15 1 2 3 Lo Penalty Parameter λ Figure 3: Buyer threhold price π PT1 λ with different β. veru lo penalty parameter Tale VIII: uer Optimal Deciion in Stage I under EUT Condition The uer Optimal Deciion (a,qa ) κp > π (, d h Q) π κp π (, ) or (, ) π >κp (, Q d l ) Theorem 3. The uer optimal deciion in Stage I under EUT i ummarized in Tale VIII. The proof of Theorem 3 i given in our technical report 21]. The reult in Tale VIII depend on the proaility p of high demand a well a the market price π and π. When p i high (κp > π ), the uer chooe to e a uyer and uy an amount equal to d h Q. When p i medium (π κp π ), the uer will not participate in the market, ince neither elling nor uying will ring a higher utility. When p i mall (π >κp), the uer chooe to e a eller, and ell an amount equal to Q d l. In the cae of PT, we compare the correponding U(, q ) and U(, q) with q and q otained in Tale III, IV, V, and VI, to find the optimal Stage I olution of the prolem. Different from the EUT cae, the optimal elling or uying quantity may not e equal to the difference etween monthly quota and high/low demand. We will further illutrate the reult in the next ection. IV. NUMERICAL RESULTS In thi ection, we illutrate the impact of the PT model parameter (λ, β, andα), market parameter (π and π ), and demand uncertainty parameter (p) on the uer optimal deciion. Due to pace limitation, we will only conider a high reference point R p =for the PT cae. Comparing with the EUT enchmark, numerical reult illutrate the following inight for a PT uer: (i) Rik eeking under a high reference point: A PT uyer i rik eeking and i le willing to uy moile data. A PT eller i alo rik eeking and i more willing to ell moile data. (ii) Proaility ditortion: When the proaility of high demand i low, a PT uyer i rik avere and i more willing to uy moile data comparing with an EUT uyer. On the other hand, when the proaility of high demand i high, a PT uyer i rik eeking and i le willing to uy moile data comparing with an EUT uyer. Impact of the lo penalty parameter λ and the rik averion parameter β on a uyer threhold price π PT1 in.6.7.8.9 1 Proaility Ditortion Parameter α Figure 4: Buyer threhold price π PT1 parameter α with different p. Optimal Selling Quantity q * 12 1 8 6 4 2 λ = 1 λ = 1.5 λ = 2 1 2 3 4 Maximum Buyer Price π veru proaility ditortion Figure 5: Seller elling quantity q veru imum uyer price π with different λ. (11): Here we aume p =.5 and α =1. Figure 3 illutrate the reult tated in Theorem 1, where π PT1 i increaing in β for a fixed value of λ, and doe not change in λ for a fixed value of β. Note that a higher threhold price mean that the uyer i more willing to uy moile data. Thi i ecaue under a reference point R p =, the uyer will not enounter a gain. In thi cae, a maller β mean that the uer i more rik eeking, and i hence le willing to purchae moile data to reduce the rik that the demand exceed the quota. Meanwhile, notice that λ only affect the lo region in (2). A the uer will never encounter a gain in thi cae, the threhold price i independent of λ. Impact of the proaility ditortion parameter α on a uyer threhold price π PT1 in (11): Figure 4 conider three different proailitie of high demand: high (p =.8), medium (p =.5), and low (p =.2). Here we aume β =.8 and λ =2. We can ee that π PT1 decreae in α when p =.2, i independent of α when p =.5, and increae in α when p =.8. A a maller α mean that the uyer will overweigh the low proaility more, he ecome more rik avere when p i mall. Similarly, ince a maller α mean that the uyer will underweigh the high proaility more, he i more rik eeking when the p i large. Impact of the lo penalty parameter λ and the rik averion parameter β on a eller optimal elling quantity q in Tale VI: Figure 5 illutrate how the eller elling quantity q change with the imum uyer price π and λ. Here we aume that α =1and β =.8. Figure 5 how that a π increae, q increae accordingly until reaching 368
215 13th International Sympoium on Modeling and Optimization in Moile, Ad Hoc, and Wirele Network (WiOpt) Optimal Selling Quantity q * 12 1 8 6 4 2 β =.6 β =.8 β = 1 1 2 3 4 Maximum Buyer Price π Figure 6: Seller elling quantity q veru imum uyer price π with different β. a fixed value. Thi i ecaue a π increae, the eller gain more revenue from the trade, and he want to ell more. However, he will not ell more than Q d l, ecaue hi revenue from the trade cannot cover hi atifaction lo otherwie. Figure 5 alo how that under the ame value of π, q i non-increaing in λ. Thi i ecaue, a λ increae, the eller ecome more lo avere, hence he will ell le in order to avoid a heavy lo when the demand i high. Figure 6 illutrate how the eller elling quantity q change with the imum uyer price π and β. Here we aume that α =1and λ =2. Figure 6 how that under a fixed π, q i non-increaing in β. Thi i ecaue, under a high reference point R p =, the eller will encounter either a mall gain or a large lo. In thi cae, a maller β mean that the uer i more rik eeking, and hence ecome more willing to ell moile data. V. CONCLUSION AND FUTURE WORK In thi paper, we have conidered a moile data trading market that i motivated y the CMHK 2CM platform. We have conidered a large market regime, and analyzed the optimal trading deciion of a ingle uer. We have compared and contrated the uer optimal deciion under propect theory (PT) and expected utility theory (EUT), and have highlighted everal key inight. Comparing with an EUT uer, a PT uer with a low reference point i more willing to uy moile data and le willing to ell moile data. Moreover, when the proaility of high demand i low, a PT uer i more willing to uy moile data comparing with an EUT uer. On the other hand, when the proaility of high demand i high, a PT uer i le willing to uy moile data. Thi tudy demontrated that a more realitic ehavioral modeling aed on PT can hed important inight in undertanding ome eegly illogical human ehavior. In our future work, we will tudy how a uer make multiple equential trading deciion in the ame illing period, with updated etimation of future demand. We are alo conducting a market urvey to evaluate the prediction power of our analyi aed on realitic uer data. APPENDIX A. Proof of Theorem 1 For all three cae, we divide the feaile interval of uying quantity q into two uinterval,,d h Q] and d h Q, ), and analyze the optimal uying quantity q that imize U(, q ) within each uinterval. Such a diviion i aed on the fact that the atifaction lo L(Q + q d h )=when q d h Q, ). 1) Buyer Prolem Under EUT (Tale II): Cae I: q,d h Q]. In thi cae, from (1), the atifaction lo under low demand i L(Q+q d l )=, and the atifaction lo under high demand i L(Q+q d h )=κ(q + q d h ). The expected utility from (3) i U(, q )= (κp π )q + κp(q d h ), (18) which i a linear function in q. It i increaing in q when π <κp, and decreaing in q when π > κp. The optimal uying quantity i then q = d h Q when π <κp,andq =when π >κp. When π = κp, the utility i independent of q. Without lo of generality, we aume that q =when π = κp. Cae II: q d h Q, ). In thi cae, the atifaction lo under oth low demand and high demand equal to, and the utility U(, q )= π q. Since the utility function U(, q ) i linearly decreaing in q,wehave q = d h Q in thi cae. Cog the aove analyi, we have the following reult for the EUT cae in Tale II: When π < κp, q = d h Q, U(, Q d h ) = π (Q d h ). When π κp, q =, U(, ) = κp(q d h). 2) Buyer Prolem Under PT with R p =(Tale III): Cae I: q,d h Q]. In thi cae, from (1), the atifaction lo under low demand i L(Q+q d l )=, and the atifaction lo under high demand i L(Q+q d h )=κ(q + q d h ). The expected utility from (3) i U(, q )= λ(π q κ(q + q d h )) β λ(π q ) β w(1 p). (19) The econd order partial derivative of U(, q ) with repect to q i U 2 (, q ) 2 = λβ(β 1) π 2 (π q ) β 2 w(1 p) q ] +(π κ) 2 (π κ)q κq+κd h ] β 2 >, (2) which implie that U(, q ) i a convex function in q, and the optimal olution mut lie at one of the oundary point 6.Henceq = d h Q if U(, ) <U(, d h Q), and q =if U(, ) U(, d h Q). 6 In the cae β =1and U(, ) = U(, d h Q), we will chooe q = without lo of generality. 369
215 13th International Sympoium on Modeling and Optimization in Moile, Ad Hoc, and Wirele Network (WiOpt) Cae II: q d h Q, ). In thi cae, the atifaction lo under oth low demand and high demand equal to, and the expected utility i U(, q )= λ+w(1 p)](π q ) β. (21) Since the firt order partial derivative U(, q )/ q <, U(, q ) i a decreaing function of q,andq = d h Q in thi cae. Cog the aove analyi, we have the following reult for the PT cae with R p =in Tale III: When π <κ +w(1 p) ] 1 β, q = d h Q. When π κ +w(1 p) ] 1 β, q =. 3) Buyer Prolem Under PT with R p = κ(q d h ) (Tale IV): Cae I: q,d h Q]. In thi cae, the atifaction lo under low demand i L(Q + q d l )=, and the atifaction lo under high demand i L(Q + q d h )= κ(q + q d h ). The expected utility i U(, q )= λ(π q + κ(d h Q)) β w(1 p) λ((κ π )q ) β. (22) The econd order partial derivative of U(, q ) with repect to q i U 2 (, q ) 2 = β(β 1){(κ π ) β q β 2 q +w(1 p)(π ) 2 π q +κ(d h Q)] β 2 }<, (23) o U(, q ) i a trictly concave function of q. A a reult, the optimal olution q atifie the firt order condition, or lie at one of the oundary point. We conider the firt order partial derivative of U(, q ) with repect to q : U(, q ) q +w(1 p)(π = β(κ π ) β q β 1 ) π q + κ(d h Q)] β 1 ]. (24) If β =1, U(, q )/ q i independent of q. When π κ < +w(1 p), U(, q )/ q >, oq = d h Q. When π κ +w(1 p), U(, q )/ q, oq =. If <β<1, olving U(, q )/ q =,wehave κ(q d q = h ) >. If q <d h Q, (κ π ) β w(1 p)π ] 1 β 1 +π then the optimal olution q = q. Otherwie, q = d h Q. Cae II: q d h Q, ). In thi cae, the atifaction loe under oth low demand and high demand equal to, and the expected utility i U(, q )= λ+w(1 p)](π q + κ(d h Q)) β. 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