I Introduction
Ia Background and Motivation
WiFi technology is playing an increasingly important role in today’s wireless communications. According to Cisco [2], more than of global mobile data traffic will be carried via WiFi in 2021. However, each WiFi access point (AP) often has a limited coverage, generally tens of meters [3]. Hence, to provide a citywide or nationwide WiFi coverage, the network operator needs to deploy a huge number of WiFi APs, which is usually very expensive. As an example, KT, LG, and SK Telecom in Korea have invested 44 million dollars to provide WiFi access at 10,000 hotspots in Seoul [4].
On the other hand, inhome WiFi network is becoming more and more popular during the past several years. According to [5], of global households had deployed WiFi APs by 2014, and the percentage is expected to grow to by 2019. This motivates us to study a new type of WiFi network that relies on aggregating (crowdsourcing) the large number of existing home WiFi APs already deployed by individual users, instead of deploying new WiFi APs. Such a novel WiFi network based on crowdsourcing is often called Crowdsourced Wireless Community Network [6].
Specifically, a crowdsourced wireless community network enables a set of individual users, who own private home WiFi APs, to form a community and share their home WiFi APs with each other [6]. These AP owners (APOs) are often called community members. By crowdsourcing the massive home WiFi APs, it is possible to achieve a large (e.g., citywide or nationwide) WiFi coverage with a small deployment cost. A successful commercial example of such a network is FON [7], the world largest WiFi community network with more than 20 million members globally. In practice, however, individual APOs may not know each other personally, hence lack the proper incentive to share WiFi APs with each other [10, 9, 11, 12, 8]. Therefore, one of the important issues in such a community network is to provide sufficient economic incentive for APOs, such that they are willing to join the community and share their WiFi APs with each other.^{1}^{1}1Another equally important issue is to guarantee the security and privacy for community members, which can be achieved through proper hardware and software solutions (as did in [7]), and is beyond the scope of this work.
In this work, we focus on studying the incentive issue in crowdsourced wireless community networks. In particular, we consider the network scenario under incomplete information, where each APO has certain private information (e.g., his mobility pattern and his provided WiFi access quality) not known by the operator and other APOs. We aim to design an incentive mechanism that (i) encourages individual APOs to join the community network and share their home WiFi APs properly, and (ii) offers a considerable revenue for the network operator to manage such a community network.
IB Model and Problem Formulation
We consider such a crowdsourced wireless community network, where a set of individual APOs form a community and share their WiFi APs with each other. The community network is open not only to the community members (i.e., APOs) but also to those users (called Aliens) who do not own WiFi APs. Figure 1 illustrates an example of a community network with 3 APOs and 2 Aliens, where APO 1 stays at home and accesses his own AP, APO 2 travels to APO 3’s home location and accesses AP 3, APO 3 travels to a location without any member WiFi coverage, and Aliens access APs 1 and 2, respectively.
Inspired by the success of FON [7] and the existing studies [10, 9, 8], we consider two different sharing schemes for APOs (to share their home WiFi APs), corresponding to two different membership types: Linus and Bill. Specifically,

As a Linus, an APO can get free WiFi access at other APs in the community, and meanwhile he has to share his own AP without any monetary compensation;

As a Bill, an APO needs to pay for accessing other APs in the community, and meanwhile he can earn some money from sharing his own AP with others.
Note that Aliens always need to pay for accessing any AP in the community (as they do not contribute APs). Thus, from the network operator’s perspective, she will get all of the money collected at Linus APs, while only part of the money collected at Bill APs. Such a dual membership scheme has achieved a great success in practice (e.g., FON [7]), as it captures two major motivations for APOs to join the community: (i) getting free WiFi access and (ii) earning money.
In such a community network with dual membership, it is natural to answer the following questions:

Membership Selection: which membership (i.e., Bill or Linus) will each APO select?

WiFi Pricing: how to set the WiFi access price (for Bills and Aliens) on each AP?

Revenue Division: how to divide the collected revenue at each Bill AP between the operator and the Bill?^{2}^{2}2Note that the operator will get all revenue collected at a Linus AP.
Similar as in [10, 9, 8], we consider such an operation scenario where APOs make their membership selection decisions and the network operator makes the WiFi pricing and revenue division decision. We formulate the interactions between the network operator and APOs as a twostage decision process: in Stage I, the network operator decides the WiFi pricing and revenue division strategy; in Stage II, each APO chooses the membership, given the network operator’s strategy.
IC Solution and Contributions
Our focus is to design the pricing and revenue division strategy for the network operator that provides sufficient incentive for APOs (to join the community and choose the proper memberships) and meanwhile maximizes her own profit. The problem is challenging due to several reasons. First, each APO is associated with certain private information (e.g., his mobility pattern and the quality of his provided WiFi access), which cannot be observed by the network operator. Thus, it is difficult for the network operator to predict the APO’s behavior without the complete information. To this end, some incentive mechanisms are necessary for eliciting the private information of APOs. Second, APOs’ decisions are coupled with each other, as one APO’s membership choice will affect other APOs’ payoffs, hence affect their membership choices. Thus, the comprehensive analysis of APOs’ equilibrium choices is challenging, even if the operator’s pricing and revenue division strategy are given [8, 9, 12].
Contract theory [13] is a promising theoretic tool for dealing with problems with private information, and has been widely adopted in wireless communication networks [14, 17, 18, 19, 15, 16, 20, 21]. Therefore, we propose a contract theoretic framework, in which (i) the network operator offers a contract to APOs, which contains a set of contract items, each consisting of a WiFi access price and a subscription fee for Bills; and (ii) each APO chooses the best membership, and the best contract item if choosing to be Bill. The key difference between the contract in this work and the existing contracts in [14, 17, 18, 19, 15, 16, 20, 21] is that in our contract, each APO’s best choice depends not only on his private information (as in existing contracts), but also on other APOs’ choices. This greatly complicates the contract design, as the operator needs to analyze the equilibrium choices of all APOs, rather than the best choice of each single APO. In this work, we will first analyze the feasibility of contract and then design the optimal contract systematically.
In summary, the key contributions are summarized below.

Novel Model: To our best knowledge, this is the first work that applies the novel contractbased incentive mechanism and derives the optimal contract to address the incentive issue in crowdsourced wireless community networks.

Novel Theoretical Results: The proposed contract mechanism is different from the existing contract mechanisms in the literature due to the coupling of users’ choices. We perform a comprehensive analysis for the users’ equilibrium choices, based on which we analyze the feasibility and optimality of contract systematically.

Practical Insights: Our analysis helps us to understand how different users choose their WiFi sharing schemes, which facilitates the network operator to better optimize her profit in different scenarios. Specifically, we find that (i) a user who provides a higher quality WiFi access is more willing to be a Bill, and choose a larger price and a higher subscription fee regardless of the mobility pattern; (ii) a user who travels less is more likely to choose to be a Bill to earn money by sharing his WiFi with others; and (iii) in a network with a larger average WiFi quality, the operator can gain more profit by setting lower prices and subscription fees to all users.
The rest of the paper is organized as follows. In Section II, we review the related works. In Section III, we present the system model. In Sections IVVI, we provide the contract formulation, analyze the contract feasibility, and design the optimal contract for the homogeneous mobility scenario. We provide simulation results in Section VII and conclude in Section VIII. Due to space limit, we put the detailed proofs and the analysis for the heterogeneous mobility scenario in the online technical report [29].
Ii Literature Review
Several recent works have studied the economics and incentive issues in crowdsourced wireless community networks, including the user behavior analysis [8, 10, 9], the pricing scheme design [11], and the competition analysis [12]. However, all of the above works focused on the scenario of complete information, where the network operator is assumed to know the complete information of all APOs. In practice, however, it is often difficult for the network operator to obtain some APO information, e.g., the APO’s provided WiFi access quality and daily mobility pattern. In our work, we consider the more practical incomplete information scenario.
Contract theory [13] is a promising theoretic tool for dealing with problems under incomplete information and has been widelyused in supply chain.^{3}^{3}3Another important tool for dealing with problems with incomplete information is auction theory [22], which has also been widelyused in wireless networks [23, 25, 24]. Auction is more useful for the scenario of allocating limited resources among a set of users with incomplete (asymmetric) information, while contract is more useful for the scenario of motivating different types of users (with private information) behave in a desired way. Recently, contract theory has also been introduced to analyze various wireless network problems, such as spectrum trading/reservation [14, 16, 15], dynamic spectrum sharing [17, 18], D2D communications [19], energy harvesting [20], and smallcell caching [21].^{4}^{4}4A comprehensive survey of the applications of contract theory in wireless networks can be found in [HanZhu5]. The key difference between our contract and the existing contracts in [14, 17, 18, 19, 15, 16, 20, 21] is that in our contract, each APO’s best choice depends not only on his private information but also on other APOs’ choices. Moreover, in our contract, the private information of each APO is multidimensional (i.e., his provided WiFi access quality and his daily mobility pattern), which further complicates the contract design.
Iii System Model
We consider a set of individual APOs, each owning a private home WiFi AP, who form a community and share their WiFi APs with each other, leading to a crowdsourced wireless community network. The community network is managed (coordinated) by a network operator, who is responsible for the necessary security, privacy, and incentive issues in such a WiFi network crowdsourcing. The community network is open not only to APOs but also to a set of users (called Aliens) who do not own WiFi APs, where is the ratio between Aliens and APOs. Aliens have to pay for accessing the community network (as they do not contribute APs) through, for example, purchasing WiFi passes from the operator.
To ensure the necessary security and privacy, each AP’s channel is divided into two parts: a private channel (for the APO’s own use) and a public channel (for other traveling users’ access). This can be achieved by either specialized hardwares (e.g., customized AP routers as in FON [26]) or specific software built in the AP. Under such a channel division, a traveling user’s access to an AP (on the public channel) will not affect the communication of the owner of that particular AP (on the private channel), but will affect other traveling users who access the same AP simultaneously.
1) WiFi Access Quality: The WiFi access quality provided by different APOs for traveling users (on their public channels) may be different, due to the different wireless characteristics (such as WiFi standard, backhaul, public channel bandwidth, channel fading, and network congestion) and location popularities. For example, the WiFi access in a popular location may be more valuable than that in a unpopular location, and hence the APO may choose to install an AP with a carrier grade WiFi standard that provides a better quality than a regular home WiFi. The WiFi access quality provided by an APO will affect traveling users’ lengths of connection on this APO’s WiFi, hence affect the APO’s potential revenue from sharing his AP (see Section IVA2 for more details).
2) APO Type: To facilitate the later analysis, we consider a discrete APO (quality) model with a discrete set of
possible WiFi access qualities. Thus, we can classify all APOs into
types, according to the WiFi access qualities that they provide for traveling users (on their public channels).^{5}^{5}5Note that such a discrete quality model can well approximate the continuous quality model by choosing a sufficiently large . Namely, the same type APOs will provide the same WiFi access quality. Let denote the WiFi access quality provided by a type APO, and denote the number of type APOs. Without loss of generality, we assume thatIt is important to note that the exact type (quality) of each APO is his private information, and cannot be observed by the network operator or other APOs. Similar as in many existing works [14, 17, 18, 19, 15, 16, 20, 21], we assume that the distribution information regarding APO type (i.e., and , where ) is public information and known by the network operator and all APOs. Such distribution information of APO type can be obtained through, for example, historical information or longterm learning.
3) APO Membership: As in [8, 7, 10, 9], an APO can share his WiFi AP in two different ways, corresponding to two different membership types: Linus and Bill. Specifically,

As a Linus, an APO can get free WiFi access at other APs in the community, and meanwhile he has to share his own AP without any monetary compensation.

As a Bill, an APO needs to pay for accessing other APs in the community, and meanwhile he can earn some money from sharing his own AP with others.
Moreover, Aliens always need to pay for accessing any AP in the community, as they do not contribute APs. Thus, from the network operator’s perspective, she will get all of the money collected at Linus APs (paid by Bills and Aliens), while only part of the money collected at Bill APs (paid by other Bills and Aliens). In this work, the network operator will design the WiFi access pricing and revenue sharing (with Bills), and APOs will choose the proper memberships.
Let ($/min) denote the WiFi access price that the network operator designs for AP , for charging Bills (other than APO ) and Aliens who access AP . Then,

When an APO chooses to be a Bill, he needs to pay for accessing every other AP at a unit price (per unit WiFi connection time) and meanwhile he can charge other users (i.e., Bills and Aliens) for accessing his AP at a unit price .

When an APO chooses to be a Linus, he neither pays for accessing other APs, nor charges other users for accessing his AP. In this case, the network operator will instead charge other users (i.e., Bills and Aliens) for accessing AP .
The revenue division between the network operator and each Bill (APO) is via a subscription fee that the operator charges the Bill. Let ($) denote the subscription fee that the network operator charges Bill . Note that can be negative, which is essentially a bonus from the operator to incentivize APO to be a Bill. Moreover, Linus does not need to pay any subscription fee. For clarity, we summarize the properties of different users (i.e., Bill, Linus, and Alien) in Table I.
Type 





Linus  No  No  No  
Bill  Yes  Yes  Yes  
Alien  Yes  N.A.  N.A. 
4) APO Mobility: To concentrate on the impact of WiFi access quality, in this work we first consider a simple homogeneous mobility pattern for APOs, where (i) APOs stay at “home” with the same probability , hence travels outside with probability , and (ii) when traveling outside, each APO travels to every other AP with the same probability .^{6}^{6}6Here we use the approximation as is usually large.
Moreover, we assume that each Alien travels to every AP with the same probability
. Our analysis in Sections IV–VI will be based on this homogeneous mobility pattern.In our online technical report [29], we will further consider a more general heterogeneous mobility pattern, where different APOs may have different mobility patterns. That is, different APOs may have different probabilities of staying at home (traveling outside). Note that in this case, an APO’s mobility pattern is also his private information.
Iv Contract Formulation
To deal with the private information, we propose a contractbased operation scheme, where the operator provides a set of contract items for each APO to choose.
Specifically, each contract item is the combination of a WiFi access price and a subscription fee, denoted by . A special contract item is used to indicate the choice of being Linus. Based on the revelation principle [27], the operator needs to design one contract item for each type of APOs, to induce them to truthfully reveal their types. Hence a contract is such a list of contract items, denoted by
(1) 
where denotes the contract item designed for the type APOs. Note that the same type APOs will choose the same contract item, as they have the same parameters, which will be justified later in Section IVB.
A contract is feasible if each APO is willing to choose the contract item designed for his type (i.e., achieving the maximum payoff under the item designed for his type). A contract is optimal if it maximizes the network operator’s profit, taking the APOs’ truthful information disclosure into consideration. We will study the feasibility and optimality of contract in Sections V and VI, respectively.
Iva Operator Profit
The set of APO types,  

The WiFi quality provided by the type APOs  
The number of the type APOs  
The unit WiFi access price at the type APs  
The subscription fee of the type APOs  
The contract item designed for the type APOs  
The contract item chosen by Linus  
The contract provided by the network operator  
The set of APO types choosing Bills under  
The set of APO types choosing Linus under  
The number of Bills,  
The number of Linus,  
The number of Aliens,  
The ratio between Aliens and APOs,  
The probability of an APO staying at home 
We now characterize the network operator’s profit under a given feasible contract , where each APO will choose the contract item designed for his type.
Let denote the set of APO types choosing Linus and denote the total number of Linus. Let denote the set of APO types choosing Bills and denote the total number of Bills. Whenever there is no confusion, we will also write as for notational convenience. Table II lists the key notations in this paper.
IvA1 Profit Achieved from a Bill AP
If a type APO is a Bill who chooses , the operator’s profit achieved from this AP is simply the subscription fee .
IvA2 Profit Achieved from a Linus AP
If a type APO is a Linus who chooses , the network operator will charge other users (Bills and Aliens) accessing this AP directly, at a price per unit WiFi connection time. With the assumption of homogeneous mobility, the expected number of Bills accessing this AP is , and the expected number of Aliens accessing this AP is . We further denote the average WiFi connection time of a Bill or Alien on this (type) AP under the price as . Intuitively, reflects a Bill’s or Alien’s demand for using a type AP, which is nonnegative, monotonically decreasing with the price , and monotonically increasing with quality . We will further discuss the demand function at the end of Section IV. Based on the above analysis, the total profit from this type Linus AP is:
(2) 
where is the price charged by the operator on Linus AP.
IvA3 Total Profit
To summarize, the network operator’s profit achieved from all APs can be computed as follows:
(3) 
where the first term is the total profit from all Linus APs and the second term is the total profit from all Bill APs.
IvB APO Payoff
We now define the payoff of each APO under a feasible contract , where each APO will choose the contract item designed for his type.
IvB1 Payoff of a Linus
If a type APO chooses to be a Linus (i.e., ), he neither pays for using other APs, nor gains from sharing his own AP. Hence, his payment is zero, and his payoff is simply the difference between utility and cost. Let constant denote the utility when staying at home and accessing WiFi through his own AP, constant denote the utility when traveling outside and accessing WiFi through other APs, and constant denote the cost (e.g., energy, bandwidth) of serving other users. Note that , , and do not depend on the APO’s membership choice. Then a type Linus APO’s payoff is:
(4) 
IvB2 Payoff of a Bill
If a type APO chooses to be a Bill (i.e., ), his payment consists of (i) the revenue earned at his own AP, (ii) the payment for accessing other APs, and (iii) the subscription fee to the network operator.
First, the revenue earned at his own AP equals the payment of other Bills and Aliens accessing his AP. The average WiFi connection time of a Bill or Alien on this (type) AP under the price is . The average payment of a Bill or Alien for accessing this AP is
(5) 
which monotonically increases with the WiFi quality , as the demand monotonically increases with . The expected number of paying users (including the other Bills of type, all Bills of other types, and all Aliens) accessing this AP depends on the given contract , and can be computed as follows:
(6) 
Note that does not depend on , i.e., same for all Bills. Thus, the revenue earned at his own AP is .
Second, the expected payment of this (type) APO for accessing other APs includes (i) the payment for accessing the other Bill APs of type, (ii) the payment for accessing all Bill APs of other types, and (iii) the payment for accessing Linus APs, which can be calculated as
(7) 
Similarly, when there is no confusion, we will also write and for and , respectively.
Third, the subscription fee to the network operator is . Hence, a type Bill APO’s total revenue (when choosing under a feasible contract ) is . Based on the definition of , , and in Section IVB1, the type Bill APO’s payoff is
(8)  
Without loss of generality, in the rest of the paper, we normalize .
By (8), we can see that the payoff of an APO depends not only on his contract item choice, but also on other APOs’ choices. For example, the first term in depends on how many APOs choose to be Bills, the second term in depends on how other Bills choose prices, and the last term in depends on how many APOs choose to be Linus. Such a strategy coupling makes our contract design problem very different from and more challenging to analyze than traditional contract models in literatures [14, 17, 18, 19, 15, 16, 20, 21].
To make the model more practical and meanwhile facilitate the later analysis, we introduce the following assumptions:

, ;

increases with , ;

, for any types and prices (where ).
Assumption (a) implies that there is a maximum allowable price for WiFi access, which is ofter lower than the corresponding cellular access price. This is because the WiFi access is usually a complement to the cellular access, and hence a high WiFi access price may drive all APOs to the cellular network when traveling. Assumption (b) implies that the demand of APOs (on other APs) is inelastic when . When WiFi costs so little that users use it like water, increasing the price of WiFi access will not reduce revenue. Assumption (c) implies that when increasing the WiFi access price by a small value, the payment of a higher type APO (i.e., with a higher quality) will be increased by a larger value than that of a lowertype APO. This is ofter referred to as “increasing differences [28]” in economics, and is satisfied for many widelyused demand functions. One widelyused demand function that satisfies the above assumptions is:
(9) 
Here is the length of per subscription period (e.g., one month).^{7}^{7}7Function is decreasing convex in . When , , which represents the maximum length of connection time. We may also multiple a constant less than 1 to the demand function, for example, to represent that the users will not connect 24 hours a day even with zero price. Such a constant scaling will not change the analysis results. Note that our analysis is general and does not rely on the specific form of (9).
V Feasibility of Contract
In this section, we will study the sufficient and necessary conditions for a contract to be feasible. Formally, a contract is feasible, if and only if it satisfies the Incentive Compatibility (IC) and Individual Rationality (IR) constraints.
Definition 1 (Incentive Compatibility – IC).
A contract is incentive compatible, if for each type APO, he can achieve the maximum payoff when choosing the contract item intended for his type , i.e.,
(10) 
Definition 2 (Individual Rationality – IR).
A contract is individual rational, if for each type APO, he can achieve a nonnegative payoff when choosing the item intended for his type , i.e.,
(11) 
Obviously, if a contract satisfies the IC and IR constraints, each APO will choose the contract item designed for his type. Hence, we can equivalently say that each APO truthfully reveals his type (private information) to the operator, which forms a Nash equilibrium.
In what follows, we will provide the necessary and sufficient conditions for feasible contracts. Due to space limit, we will put the detailed proofs in the online technical report [29].
Va Necessary Conditions
We first show the necessary conditions for a feasible contract. For convenience, we rewrite a feasible contract as
where and are the type sets of Linus and Bills, respectively, and both are functions of .^{8}^{8}8Here we omit the parameter for notation simplicity. Then, we have:
Lemma 1.
If a contract is feasible, then
Lemma 2.
If a contract is feasible, then
Lemma 3.
If a contract is feasible, then there exists a critical APO type , such that for all and for all , i.e.,
(12) 
Lemma 1 shows that in a feasible contract, a larger WiFi access price must correspond to a larger subscription fee. Lemma 2 shows that in a feasible contract, a higher type APO will be designed with a higher WiFi access price. Lemma 3 shows that there exists a critical APO type: all APOs with types lower than the critical type will choose to be Linus, and all APOs with types higher than or equal to the critical type will choose to be Bills. When all APOs are Bills, we have ; when all APOs are Linus, we have .
VB Sufficient and Necessary Conditions
We now show that the above necessary conditions together are also sufficient for a contract to be feasible. For notational convenience, we denote as the expected number of Bills and Aliens accessing a particular AP and as the expected payment of a particular APO (for accessing other APs) when choosing to be a Bill. That is,
Then, the sufficient and necessary conditions for a feasible contract can be characterized by the following theorem.
Theorem 1 (Feasible Contract).
A contract is feasible, if and only if the following conditions hold:
(13)  
(14)  
(15)  
(16)  
(17) 
In the above theorem, the condition (15) can be derived from the IC constraints of Linus APOs, i.e., a type Linus APO will achieve a nonpositive payoff if he chooses any contract item designed for Bills, i.e., . Similarly, the conditions (16) and (17) can be derived from the IR and IC constraints of Bill APOs.
Vi Optimal Contract Design
In this section, we study the optimal contract that maximizes the network operator’s profit. We focus on finding an optimal contract from the feasible contract set characterized by Theorem 1. By the revelation principle [27], the optimal contract within the feasible set is the global optimal contract among all possible (feasible and nonfeasible) contracts.
Via Optimal Contract Formulation
By Theorem 1 and the revelation principle, we can formulate the following contract optimization problem:
Problem 1 (Optimal Contract).
In Problem 1, is the critical AP type; is the WiFi access price on Linus APs, where the network operator will charge Bills or Aliens with this price ; and , are the WiFi access prices and subscription fees for Bills.
Next we define the feasible price assignment, which we will use in later discussions.
Definition 3 (Feasible Price Assignment).
A price assignment is feasible if it satisfies constraint (14).
We will solve Problem 1 in the sequential manner. First, we derive the best subscription fees , given the feasible price assignment and critical type . Then, we design an algorithm to compute the optimal price assignment , given the critical type . Finally, substituting the best subscription fees and price assignment, we search the critical type for the optimal contract.
ViB Optimal Subscription Fee
We first derive the best subscription fees, given the feasible price assignment and critical type. The subscription fee optimization problem is given below.
Problem 2 (Optimal Subscription Fees).
It is easy to see that Problem 2 is a linear programming. The optimal solution is given in the following lemma.
Lemma 4.
Given a critical type and a feasible price assignment , the optimal subscription fees are given by:
(18)  
(19) 
for all .
Note that and depends on the critical type and the price assignment , and can be calculated by (6) and (IVB2), respectively. For convenience, we denote
Then, we can rewrite the network operator’s profit from Bill APOs in the following way:
where
It is easy to see that is related to only, while independent of for all .
ViC Optimal Price Assignment
Next we derive the optimal price assignment, given a fixed critical type . Substitute the best subscription fees derived in Lemma 4, we can formulate the following optimal price assignment problem.
Problem 3 (Optimal Price Assignment).
We denote the solution of Problem 3 as , which depends on the critical type . Hence, we will also write as when we want to emphasize such dependances.
It is easy to see that the optimal price for Linus is always the price upperbound , i.e., . This is because monotonically increases with . The optimal price assignment for Bills can be solved by the following optimization problem:
(20)  
Since only depends on , we can compute the maximizer of each individually, and check whether the feasible price condition in (20) is satisfied. Let denote the price that maximizes , i.e.,
(21) 
Note that if satisfies the constraint in (20), i.e., it is a feasible price assignment, then it is exactly the optimal solution of Problem 3, i.e., . In some cases, however, may not satisfy the constraint in (20), and hence is not feasible. In such cases, we can adopt a dynamic algorithm (similar as that in [14]) to adjust the price assignment gradually, such that in each step, at least one infeasible price subsequence will be removed.^{9}^{9}9An infeasible price subsequence refers to a subset of prices that do not satisfy the constraint in (20). The key idea is to optimize all prices in an infeasible price subsequence together, rather than optimizing them individually. Due to space limit, we put the detailed algorithm in our online technical report [29].
ViD Optimal Critical Type
Vii Simulation Results
In this section, we first simulate a network where all APOs have the homogeneous mobility pattern in Section VIIA, and then simulate a network where APOs have heterogeneous mobility patterns in Section VIIB.^{10}^{10}10For more details about the heterogeneous mobility scenario, please refer to our online technical report [29]
Viia Optimal Contract and Impact of Alien Number
We simulate a network with APOs and Aliens. The APOs are classified into types with WiFi qualities . Each APO stays at home with probability , and travels to other APs with the same probability (i.e., ). The maximum allowable WiFi access price is . We study the optimal contracts under three different APO type distributions illustrated in Figure 4: In case I, the population is low type APO dominant; In case II, the population is medium type APO dominant; and In case III, the population is high type APO dominant. Figures 4 and 4 illustrate the optimal contract items offered to Bills (i.e., the subscription fees and WiFi access prices) in three cases. Figure 7 shows the payoff of each APO in three cases.
By Figures 4–7, we have the following observations for the optimal contracts in the above three cases.
Observation 1.
The subscription fee can be negative, in which the operator gives bonus to APOs for sharing WiFi. Moreover, the subscription fee in Case I (low type APO dominant) is the highest one among all cases (Figure 4).
As a lower type APO charges a smaller price at his own AP but pays higher prices to higher type APOs, his payoff can be negative without proper compensation from the operator. Hence, the network operator will give bonus to the lower type APO for joining the community and sharing WiFi. In the low type APO dominant case (Case I), each type APO’s payoff as a Bill is higher comparing with other two cases, since lower type APOs charge smaller prices and on average it is less expensive to use other APs. Hence, the network operator will set higher subscription fees to extract more profit.
Observation 2.
The WiFi access price in Case I (low type APO dominant) is the highest one among all cases (Figure 4).
The highest price in Figure 4 corresponds to the highest subscription fee in Figure 4. An APO will choose a higher price since he needs to pay a higher subscription fee.
Observation 3.
A higher type Bill APO gains more payoff than a lower type Bill APO under all three cases. Moreover, for each particular APO, it can achieve the largest payoff in Case I (low type APO dominant) among all cases (Figure 7).
The reason is that the higher type APO receives more revenue on his own AP and pays less to other lower type APOs, hence achieves a payoff higher than that of a lower type APO. In the low type APO dominant case (Case I), on average each Bill APO pays less to access other APs than in other two cases. This turns out to be the dominant factor in determining the payoff, although every Bill APO pays more to the operator (Observation 1) and charges other users more (Observation 2).
Next, we study how the ratio of Aliens and APOs (i.e., ) affects the optimal contract. Figure 7 shows the critical AP type under different values of in three cases, where we fix and change from to (hence changes from to ). Figure 7 shows the operator’s profit per user (including Aliens) under different values of in three cases, where we fix and change from to (hence changes from to ).
By Figures 7 and 7, we can obtain the following observations for the optimal contracts in the above three cases.
Observation 4.
More APOs will choose to be Linus in Case III (high type APO dominant) than in other cases (Figure 7).
The reason is that higher type APOs charge higher prices (and many of them charge ). In the high type APO dominant case (Case III), on average it is more expensive for a Bill to access other APs. As a result, a lower type APO’s payoff as a Bill is smaller due to the higher payment on other APs. Hence, more APOs will choose to be Linus.
Observation 5.
The number of APOs choosing to be Linus increases with the number of Aliens (Figure 7).
Recall the operator’s total profit defined in (3), i.e.,
The operator sets the same price on all Linus APs as . As increases, the term becomes increasingly important for the operator’s total revenue. Hence, in the optimal contract, the operator increases to gain more revenue from Aliens’ WiFi access on Linus APs.
Observation 6.
The network operator’s profit per user increases with the number of Aliens. Moreover, given the number of Aliens, the operator’s profit per user in Case III (high type APO dominant) is the largest one among all cases (Figure 7).
The operator’s profit per user increases with , as more Aliens bring more revenue. The operator’s profit is the largest in the high type APO dominant case (Case III). This is because when the WiFi quality increases, users generate more demand in the network, and hence the operator gains more revenue.
ViiB Heterogeneous Mobility Scenario
We now provide simulation results for the heterogeneous mobility scenario. We simulate a network with Aliens and APOs with mobility patterns ( and ) and WiFi qualities . The maximum allowable WiFi access price is .
In our online technical report [29], we have theoretically proved that if , then . Here, we numerically show how the critical AP types and change with the mobility parameters . Figure 8 shows the critical AP types under different values of when .
Observation 7.
The critical AP type decreases with , i.e., more APOs who stay at home with probability choose to be Bills as increases (Figure 8).
The reason is that as increases, APOs with stay at home with a larger probability, while travel to other APs with a smaller probability. So APOs with prefer to be Bills to gain profit at his own APs rather than be Linus to enjoy free WiFi at other APs. Hence, more APOs with choose to be Bills as increases.
Observation 8.
(a) The critical AP type first decreases with , i.e., more APOs who stay at home with probability choose to be Bills as increases. (b) When is large enough, i.e., , increases with , i.e., less APOs who stay at home with probability choose to be Bills as increases (Figure 8).
The reason for Observation 8(a) is that as increases from small to medium values, more APOs with choose to be Bills (Observation 7). Since the number of paying users (Bills) increases, APOs with can gain larger profit if they choose to be Bills. Hence, more APOs with choose to be Bills. The reason for Observation 8(b) is that as increases from medium to large values, Bill APOs with stay at home with a larger probability, so the profit achieved by Bills from these paying users (Bill APOs with ) will decrease. Hence, less APOs with choose to be Bills.
Viii Conclusion
In this paper, we proposed a novel contract mechanism for crowdsourced wireless community networks under incomplete information. Different from existing contract mechanisms in the literature, the proposed contract considers the coupling among users’ contract item choices, hence is more complicated to design. We analyzed the feasibility and optimality of the proposed contract systematically based on the user equilibrium analysis. We also provided simulation results to illustrate the optimal contract and the profit gain of the operator. Our analysis helps us to understand how different users choose their WiFi sharing schemes, which facilitates the network operator to better optimize her profit in different network and information scenarios. As for the future work, it is important to study a more general model with heterogeneous users, where different users may have different traffic demands and mobility patterns. It is also interesting to study the competition among multiple network operators.
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Appendix A Algorithms for Optimal Price Assignment
As discussed in Section VIC, the optimal price assignment for (20) can be obtained by solving each price individually according to (21), as long as the solutions of (21) satisfy the constraint in (20). In the general case, however, may not satisfy the constraint in (20), and hence may not be feasible.
In this section, we propose a twostage algorithm to solve the optimal price assignment for (20). In Stage I, we use a Dual Algorithm to solve (20), where computed in (21) will serve as the initial price choices. If the solution returned by the Dual Algorithm is feasible, then it is optimal to (20); otherwise, we use it as an initial point for a Dynamic Algorithm in Stage II, which will return a feasible (but suboptimal) solution of (20).
Aa Dual Algorithm
(22) 
Let be the Lagrange multiplier associated with the linear constraint . We denote , and . Then the Lagrangian of problem (20) is:
(23) 
and it can be written as
where only involves , and is defined as:
(24)  
(25)  
(26) 
The dual algorithm aims to solve the convex but possibly nondifferentiable dual problem of (20):
(27) 
where
(28) 
Computing involves separate subproblems, , where each subproblem is a single variable minimization problem and is easy to solve.
Note that the righthand side of (28) might have multiple solutions, hence the dual function may not be differentiable. We apply the subgradient method to solve the dual problem (27), i.e., we update the dual variable according to (22) in Algorithm 1, where is the iteration index. The update rule (22) is intuitive: the dual variable increases if the constraint is violated. In Algorithm 1, is a tolerance parameter.
The sequence generated by Algorithm 1 converges to an optimal dual solution. Let be the limiting point of the sequences of , and
(29) 
By the weak duality theory, we have
(30) 
Recall that is the optimal solution of Problem 3. Hence, if is feasible to problem (20), it is the optimal solution to problem (20), i.e., If is not feasible to problem (20), we will further compute a feasible solution using the dynamic algorithm introduced later.
To summarize, the dual algorithm fully takes advantage of the separable structure of the objective function in problem (20); it is easy to implement, since the optimal solution of the Lagrangian can be obtained easily; it provides an upper bound of problem (20); it is possible to return the global solution of problem (20) with a certificate (if the returned solution is feasible); if not, it returns a good approximate solution, which can be used as an initial point for local searching.
AB Dynamic Algorithm
Now we deal with the case where is not feasible.
Definition 4 (Infeasible Subsequence).
We denote a subsequence of , say , as an infeasible subsequence, if and .
We propose to use a Dynamic Algorithm in Algorithm 2 (based on a similar algorithm proposed in [14]) to make all the infeasible subsequences in feasible, by enforcing all prices in the infeasible subsequence to be equal (Line 4 of Algorithm 2). Specifically, for an infeasible subsequence , we combine and select an identical price for by:
(31) 
It is easy to solve (31), as it has only a single variable . We denote the solution returned by Algorithm 2 as .
The Dynamic Algorithm will terminate within times, as we combine at least two different at each time, hence we need to do at most times.
AC Complexity Analysis
We now analyze the complexity of the above algorithms. Specifically, we first need to derive the best subscription fees , given the feasible price assignment and the critical type . Note that the closedform expressions of the best subscription fees are given in (18) and (19), and hence can be computed without involving complicated optimization.
Then, we adopt a twostage algorithm to solve the price assignment , given the critical type . The complexity of the Dual Algorithm in Stage I is . Specifically, the complexity to calculate the dual objective function value and the corresponding subgradient value once is . The stopping criterion of the Dual Algorithm is that the difference between the objectives of two consecutive iterations is no larger than the tolerance parameter . The complexity of such iterations is [30]. The complexity of the Dynamic Algorithm in Stage II is , which is negligible compared with the complexity of the Dual Algorithm.
Finally, substituting the obtained subscription fees and price assignment, we can find the critical type through exhaustive search. Obviously, we only need to search times for . In summary, the overall complexity of our proposed algorithm is
Appendix B Heterogeneous Mobility Scenario
In the previous discussion, we consider the homogenous mobility scenario, where different APOs have the same mobility pattern. In this section, we will study a more general scenario with the heterogeneous mobility, where different APOs may have different mobility patterns, i.e., different probabilities of staying at home (or traveling outside).
Similar as the APO quality model, We consider a finite and discrete set of possible probabilities (of staying at home) for APOs, and denote as the th probability. Without loss of generality, we assume:
Recall that in our previous analysis, all APOs are divided into types based on their provided WiFi access qualities, and a type APO’s WiFi access quality is denoted by . In this section, each APO is characterized not only by his provided WiFi access quality, but also by his mobility pattern. Hence, all APOs can be divided into types based on their provided WiFi access qualities and their mobility patterns, where a type APO provides the WiFi access with quality and stays at home with probability . Note that and are private information of each APO, and are not known by the network operator.
Ba Contract Formulation
To induce APOs to reveal their private information truthfully, the operator needs to design a contract item for each type of APO (characterized by and ). Hence, the contract that the operator offers is
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