Multi-Cap Optimization for Wireless Data Plans with Time Flexibility

An effective way for a Mobile network operator (MNO) to improve its revenue is price discrimination, i.e., providing different combinations of data caps and subscription fees. Rollover data plan (allowing the unused data in the current month to be used in the next month) is an innovative data mechanism with time flexibility. In this paper, we study the MNO's optimal multi-cap data plans with time flexibility in a realistic asymmetric information scenario. Specifically, users are associated with multi-dimensional private information, and the MNO designs a contract (with different data caps and subscription fees) to induce users to truthfully reveal their private information. This problem is quite challenging due to the multi-dimensional private information. We address the challenge in two aspects. First, we find that a feasible contract (satisfying incentive compatibility and individual rationality) should allocate the data caps according to users' willingness-to-pay (captured by the slopes of users' indifference curves). Second, for the non-convex data cap allocation problem, we propose a Dynamic Quota Allocation Algorithm, which has a low complexity and guarantees the global optimality. Numerical results show that the time-flexible data mechanisms increase both the MNO's profit (25 average) and users' payoffs (8.2

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1 Introduction

1.1 Background and Motivation

Mobile Network Operators (MNOs) profit from the wireless data services through carefully designing their wireless data plans. The pricing strategy involved in the wireless data plans has evolved from the flat-rate scheme to the usage-based scheme in the past years [3]. Now the most widely used data plan consists of a monthly data cap, a monthly one-time subscription fee, and a linear price for any unit of additional data consumption beyond the data cap. Based on this pricing strategy, MNOs usually offer multiple data caps together with different monthly subscription fees for users to choose from. For example, in the US market, AT&T charges $20 for 300MB, $45 for 1GB, $55 for 2GB, and $70 for 4GB; and the linear price for exceeding the data cap is $15/GB [4].

The purpose of MNO’s multi-cap offering is to capture more user surplus by differentiating users based on their preferences, also called price discrimination in economics [5]. To make such a price discrimination scheme work, the MNO must be able to identify the market segments by users’ preferences that are usually users’ private information, and the MNO needs to enforce the scheme through some incentive mechanism. For example, the MNO may want to offer a larger monthly data cap with a larger monthly subscription fee to businessmen, who have a stronger ability to pay and a relatively inelastic data demand comparing with other consumers (such as students). However, it is a very challenging problem to induce users to truthfully reveal their private preferences in practice, especially when users have multi-dimensional private preferences. This motivates us to ask the first key question in this paper.

Question 1.

How should the MNO optimize the multi-cap data plan offering?

Recently the growing market competition forces the MNOs to explore various innovations on their mobile data plans. For example, the rollover data plan enables users to enjoy the time flexibility over their data consumptions, by allowing the unused data from the previous month to be used in the current month. Such a rollover mechanism is attractive to users, as a user’s data demand is often stochastic and the rollover mechanism helps users balance the possible data waste within the data cap and the possible overage usage when consuming beyond the data cap.

Although based on the same rollover principle, different rollover data plans are different in terms of the consumption priority between the rollover data and the monthly data cap. For example, the rollover data plan offered by AT&T requires that the rollover data from the previous month should be consumed after the current monthly data cap [6], while China Mobile requires the other way around [7]. In our previous work [8, 9], we analyzed the MNO’s optimal data plan with time flexibility under the single-cap scheme (without price discrimination) and found that the time flexibility can increase both the MNO’s profit and users’ payoff, hence improve the social welfare. This motivates us to ask the second key question in this paper.

Question 2.

What is the impact of time flexibility under the multi-cap scheme?

In this paper, we will study the MNO’s price discrimination through the multi-cap data plans, taking into account the time-flexible data mechanisms.

1.2 Solutions and Contributions

We study how the MNO optimizes its multi-cap data plans under different data mechanisms with time flexibility. Specifically, we consider an asymmetric information scenario, where the users’ preferences for the wireless data plans are private and multi-dimensional. We formulate this problem as a multi-dimensional contract design. More specifically, the MNO needs to design a contract (with different combinations of data caps and the corresponding subscription fees) for users of different types, so that each user will truthfully reveal his type (i.e., private preferences) by selecting a contract item intended for his type.

The key results and contributions of this paper are summarized as follows:

  • Systematic Study on MNO’s Price Discrimination: To the best of our knowledge, this is the first work studying the MNO’s price discrimination through optimizing the multi-cap wireless data plans. We take into account both the time flexibility (of the rollover data mechanisms) and the realistic asymmetric information.

  • Exploring Time Flexibility in Price Discrimination: We investigate three different data mechanisms (i.e., one traditional data mechanism and two rollover data mechanisms) and analyze the MNO’s multi-cap data plan optimization under the three data mechanisms in a common design framework.

  • Solving the Optimal Contract: The MNO’s contract problem involves user’s multi-dimensional private information, hence is challenging to solve. We exploit the separable structure (between users’ types and quota allocation) of our problem and develop a tractable approach to solve the MNO’s contract problem. First, we find that the slope of a user’s indifference curve on the contract plane corresponds to his willingness-to-pay, and a feasible contract (satisfying the incentive compatibility and individual rationality conditions) should allocate the data caps according to users’ willingness-to-pay. This enables us to obtain the optimal prices for a particular data cap allocation in closed-form. Second, for the non-convex data cap allocation problem, we propose a Dynamic Quota Allocation Algorithm, which guarantees the global optimality with a low computational complexity.

  • Performance Evaluation based on Empirical Data: We evaluate the optimal contract under different data mechanisms based on the empirical data. The numerical results show that the time-flexible data mechanisms increase both the MNO’s profit (25% on average) and users’ payoffs (8.2% on average) under the multi-cap price discrimination, hence improves the social welfare.

The remainder of this paper is organized as follows. In Section 2, we review the related works. Section 3 introduces the system model. Section 4 analyzes the contract feasibility and Section 5 studies the contract optimality. In Section 6, we present the numerical results. Finally, we conclude this paper in Section 7.

2 Literature Review

There have been many excellent studies on the wireless data plan optimizations (e.g., [10, 11, 12, 13]). However, they did not take into account the recently introduced rollover mechanism or the ubiquitous multi-cap scheme.

The rollover mechanisms have been studied in [14, 15, 8, 9, 16]. Zheng et al. in [14] found that moderately price-sensitive users can benefit from subscribing to the rollover data plan compared with the traditional data plan. Wei et al. in [15] studied the rollover period length from a profit-maximizing MNO’s perspective. In our previous works, we studied the optimization of the time-flexible data plans in [8] and investigated the impact of the market competition in [9] and the trading market in [16]. However, all of these studies were based on the single-cap scheme without considering the ubiquitous multi-cap adoption.

The MNO’s multi-cap offering was seldom studied in previous literature. Dai et al. in [17] considered a case where the MNO offers two different data caps, i.e., a cap of basic rate and a cap of premium rate. However, the analysis was difficult to be generalized to more than two data caps. Therefore, there is no existing systematic study on the MNO’s optimal multi-cap design, let alone under the time-flexible data mechanisms. A key challenge for this problem is that different users make their data cap choices based on their individual preferences, which are often private information and can be multi-dimensional. Hence the MNO needs to properly design multiple data caps to differentiate users without knowing their exact private information and maximize the MNO’s profit. Such a problem naturally leads to a contract design problem [18].

Literature Rollover Considered? Multi-Cap Considered?
[10]-[13] No No
[8][9][14]-[16] Yes No
[17] No Yes (but limited)
This Paper Yes Yes
TABLE I: Comparing Related Literature.

Users’ multi-dimensional private information leads to a multi-dimensional contract design problem. Such a problem is often very challenging, since the multi-dimensional private information makes it difficult to achieve the global incentive compatibility [19]. To address this problem, McAfee and McMillan in [20] proposed the generalized single-crossing condition to ensure the globally incentive compatibility for a contract problem with multi-dimensional private information, but such a strong condition is not satisfied in many models (including ours). Rochet and Chone in [21] developed a sweeping procedure which adjusts the solution to ensure the global incentive compatibility. Such an approach requires that the dimension of the type space and allocation space coincide (which is not applicable to the MNO’s multi-cap data plans optimization), and cannot be solved analytically except in very special cases. In this paper, we introduce users’ willingness-to-pay by investigating their indifference curves, based on which we can develop a tractable approach for the MNO to provide the global incentive to all user types and solve its optimal contract under multi-dimensional private information.

3 System Model

We formulate the MNO’s multi-cap data plan design as a three-step process as shown in Fig. 1

. In Step I, the MNO collects data from the user market to estimate the

statistical information of users’ individual preferences (i.e., a user’s type), which are often private and multi-dimensional information (hence difficult to predict on a per user basis). In Step II, the MNO chooses a data mechanism to provide the subscribers with time flexibility. Then in Step III, the MNO proceeds with the multi-cap contract design to induce users truthfully revealing their types and hence maximize the MNO’s profit. Generally speaking, the MNO should periodically (e.g., every year) repeat the three steps to capture users’ varying requirements (due to, for example, technology changes).

Furthermore, the MNO should extract as many dimensions of the user type as possible to characterize users’ private information precisely, which leads to a contract problem with multi-dimensional private information. As mentioned as Section I, a multi-dimensional contract is challenging to solve. In this paper, we exploit the separable structure (between the user’s types and the data cap allocation), and propose to characterize each type of users’ willingness-to-pay by investigating their indifference curves. To provide a clear demonstration, we use a two-dimensional user type to illustrate our approach.111In reality, the MNO can further introduce more dimensions and solve the multi-dimensional contract using our method if the users’ types and the data cap allocation exhibit a similar structure.

Next we describe three data mechanisms in Section 3.1. Then we introduce users’ two-dimensional characteristics and derive users’ payoffs under different data mechanism in Section 3.2. Finally, we formulate the MNO’s optimal contract problem in Section 3.3.

Fig. 1: System model for the MNO’s multi-cap design.

3.1 Data Mechanisms

A mobile data plan can be characterized by the tuple , where a subscriber pays a lump-sum subscription fee for a data usage up to the monthly data cap , beyond which the MNO will charge an additional fee for each unit of data consumption.222We assume that all data plans have the same additional unit usage fee . This is often true in practice. For example, for AT&T, /GB. Here represents different data mechanisms that offer subscribers different time flexibilities on their data consumption over time.

The key differences among the three data mechanisms are the rollover data and consumption priority, both of which will affect the subscriber’s expected overage data consumption [8]. First, the rollover data from the previous month can enlarge a user’s effective data cap of the current month, within which no additional fee involved. Second, the consumption priorities of the rollover data and the monthly data cap further affect how much the effective cap is enlarged. In Table II, we use to denote a user’s rollover data from the previous month. More specifically,

  • The case of denotes the traditional data plan. The subscriber has no rollover data, and the effective cap of each month is ;

  • The case of denotes the rollover data plan offered by AT&T. The rollover data from the previous month is consumed after the current monthly data cap . Thus the effective cap of the current month is ;

  • The case of denotes the rollover data plan offered by China Mobile. The rollover data from the previous month is consumed prior to the current monthly data cap . Thus the effective cap of the current month is ;

Plan Rollover Data Consumption Priority
0 Cap
CapRollover
RolloverCap
TABLE II: .

As we mentioned above, the time flexibility can enlarge the subscriber’s effective data cap. According to Table II, the effective data cap of the traditional data mechanism is always . However, for , the effective data cap is , which is no smaller than in the traditional data mechanism. Although and lead to the same expression , the stationary distribution of is different for .333We refer interested readers to Section 4 of [8] for more details. Moreover, when we consider the -month rollover period, the rollover data has an even larger range, i.e., . Intuitively, the larger the effective data cap is, the less additional payment is incurred, which will further change users’ subscription choices.

3.2 User Model

3.2.1 User Characteristics

Next we introduce users’ stochastic data demand and the two-dimensional preferences: for the valuation of unit data and for the network substitutability.

To capture the stochastic nature of a user’s data demand over time, we model a user’s data demand as a discrete random variable with a probability mass function

, a mean value of , and a finite integer support .444In practice, the MNO can estimate users’ demand distributions based on their historical data usage, and incorporate such a difference among users into the user type modeling. In this paper, we focus on the user differences in data evaluation and network substitutability, and assume homogeneous demand distribution [22, 23]. Notice that users’ demand realizations can still be different. Here the data demand is measured in the minimum data unit (e.g, 1KB or 1MB according to the MNO’s billing practice). Accordingly, we denote as a user’s utility from one unit of data consumption, i.e., his valuation for unit data [10, 24].

Furthermore, a user’s data consumption behavior might change after exceeding the effective cap, since it incurs additional payment. Intuitively, the user will still continue to consume data in this case, but may reduce his data consumption by utilizing alternative networks (e.g., Wi-Fi) instead. Therefore, we follow [25] by incorporating users’ network substitutability as one of the user’s characteristics. Mathematically speaking, denotes the fraction of overage usage shrink. A larger value represents more overage usage cut (thus, a better substitutability). A user’s mobility pattern can significantly influence the availability of alternative networks, which will further change a user’s data plan choice. For example, a businessman who is always on the road may have a poor network substitutability (hence a small value of ), hence prefers to a large data cap; while a student can take advantage of the school Wi-Fi network (hence a large value of ), hence will be fine with a small data cap.

Different from our previous works in [8, 9], in this paper, we consider a more realistic asymmetric information scenario, i.e., the parameters and are each user’s private information that the MNO does not know precisely. As a result, we propose to use a contract-theoretic approach to cope with users’ multi-dimensional private information and optimize the MNO’s multi-cap data plans.

3.2.2 User Payoff

A user’s payoff is defined as the difference between his utility and payment. Specifically, for a type- user with units data demand and an effective cap , his realized data consumption is where . Hence a type- user’s utility is . In addition, the user’s total payment consists of the monthly subscription fee and the overage charge . Therefore, the (monthly) payoff of the type- user with a data demand and an effective cap is

(1)

Here both and are random variables, and we take the expectation over them to obtain a user’s expected payoff as

(2)

where is the type- subscriber’s expected overage data consumption, as follows:

(3)

Note that the differences among the three data mechanisms are entirely captured by in (3). Specifically, in (3) represents the distribution of the subscriber’s rollover data under data mechanism , which is the key difference among the three data mechanisms. In our previous work, we have introduced how to compute and in details (see Section 4 of [8]). In this paper, we directly summarize the key conclusion from [8] in Proposition 1.

Proposition 1.

For an arbitrary data demand distribution , for any .

Proposition 1 indicates that a user incurs less overage data consumption under the rollover mechanism than the traditional one . Moreover, among the two rollover mechanisms , is more time-flexible than , since . This is why we say that the rollover mechanism offers the best time flexibility, while offers the worst.

The above discussion indicates that a user’s expected payoffs under different mechanisms have a similar expression. The difference is only in terms of the expected overage usage . Thus, for notation simplicity, we will focus on a generic data mechanism and express the expected payoff of a type- user as

(4)

where is the subscriber’s utility, and is the overage payment. In economics, the subscription fee is a user’s sunk cost (incurred in advance and often independent of the user’s actual consumption), while the overage payment is the prospective cost (depending on the user’s actual consumption). Therefore, we call the user’s payoff without the sunk cost as the “virtual payoff”, defined as

(5)

which will be used in Section 4 and Section 5.

So far we have generalized the users’ expected payoffs under different data mechanisms into a unified expression. Our later analysis for the MNO’s optimal contract problem is based on this general framework.

3.3 MNO’s Contract Formulation

Next we formulate the MNO’s optimal contract problem.

3.3.1 Feasible Contract

The MNO offers a contract (with different combinations of data caps and corresponding subscription fees) to a group of users who are distinguished by two-dimensional private information: the data valuation and the network substitutability . Recall that in Step I (of Fig. 1), the MNO collects the statistical information from the user market. For example, we consider a set of data valuation types and a set of network substitutability types. Hence there are a total of types of users in the market, characterized by a joint probability mass function for each type- user.555The MNO can flexibly divide users’ into several categories through some data mining techniques such as -means [26, 27]. The choices of parameters and determine the trade-off between contract complexity and profit. Without loss of generality, we assume that users’ types are indexed in the ascending sort order in both dimensions, i.e., and .

According to the revelation principle [28], it is enough for the MNO to consider a class of contracts that enables users to truthfully reveal their types. In other words, it is enough for the MNO to design a contract, denoted by that consists of contract items , one for each user type. Formally, a contract is feasible if and only if it ensures that each user selects the contract item intended for this type. It is obvious that a contract is feasible if and only if it satisfies the Individual Rationality (IR) and Incentive Compatibility (IC) conditions, defined as follows:

Definition 1 (Individual Rationality).

A contract is individually rational if for all and , the type- user achieves a non-negative payoff by choosing the contract item intended for this user type, denoted by , i.e.,

(6)
Definition 2 (Incentive Compatibility).

A contract is incentive compatible if for all and , the type- user maximizes its payoff by choosing the contract item intended for this user type, i.e.,

(7)

Our later analysis for the contract feasibility in Section 4 involves the concept of Pairwise Incentive Compatibility (PIC) in Definition 3. Basically, PIC consists of the all IC conditions in the two-user scenario. That is, the IC conditions are equivalent to the PIC conditions for all the two-user pairs.

Definition 3 (Pairwise Incentive Compatibility).

The contract items and are pairwise incentive compatible, denoted by , if and only if

(8)

3.3.2 MNO’s Profit

Next we derive the MNO’s revenue, cost, and profit under a feasible contract .

The MNO’s revenue from a subscriber consists of the subscription fee and the overage fee. Based on the above discussion of the feasible contract, the MNO’s expected revenue under a feasible contract is

(9)

Furthermore, we consider two kinds of costs experienced by the MNO, i.e., the capacity cost and operational cost.

The MNO’s capital expenditure is mainly due to its investment on the network capacity [3]. Imposing the data cap would help manage the network congestion and arrange the scarce network capacity [17]. Motivated by this phenomenon, we model the MNO’s capacity cost caused by a type- subscriber as an increasing function in his data cap [24]. Intuitively, a larger data cap corresponds to a severer network congestion on average that requires the MNO’s more investment on the network in advance.

The MNO’s operational cost is mainly due to the system management [29]. After the MNO decides which data plan to implement, the subscribers’ total data consumption will influence the MNO’s operational expense. Therefore, the MNO’s operational cost caused by a type- subscriber with data cap can be formulated as , where is the MNO’s marginal cost for the system management [17], and is the type- subscriber’s expected data consumption.666Such a linear-form cost has been widely used to model an operator’s operational cost, e.g., [30, 31].

Therefore, the MNO’s expected cost under a feasible contract can be calculated as

(10)

The MNO’s expected profit under a feasible contract is the difference between its revenue and cost, given by

(11)

3.3.3 MNO’s Multi-dimensional Contract Problem

Based on the above discussion, we formulate the MNO’s contract problem as follows:

Problem 1 (Optimal Contract Design).
(12)

The key idea of the contact design problem is to ensure the individual rationality and the incentive compatibility of all user types, so that each user is willing to participate and truthfully reveals his type by selecting the contract item intended for this type of users. Problem 1 makes it clear, where the MNO needs to address a total of IR constraints (condition (6)) and a total of IC constraints (condition (7)).

The main difficulty of Problem 1 is twofold:

  1. The non-monotonicity of the allocation rule. A monotonic allocation rule usually requires the satisfaction of the single-crossing property, under which two indifference curves of any two different user types cross only once [18]. That is, the user’s marginal utility should be monotone increasing (or monotone decreasing) in the user type. When this condition holds, an allocation rule is incentive compatible only if the rule is monotonic in the user type [32]. In Problem 1, we have

    (13)

    which indicates that the marginal utility increases in the data valuation for any . Therefore, the higher valuation user deserves a larger allocation for any . However, for the network substitutability , we have

    (14)

    which can be positive or negative, depending on the relationship between the data valuation and the per-unit fee . Therefore, the allocation rule in terms of the network substitutability is not monotonic and hence is challenging to analyze.

  2. Two-dimensional user types. A contract design involving multi-dimensional user types is also very challenging in general. For contract problems involving only one-dimensional user types, the satisfaction of single-crossing condition guarantees a monotone allocation rule. Therefore, the approach used in [33, 34, 35, 36] can significantly reduce the unbinding IC and IR constraints so that the contract problem is more tractable. However, the approach in [33, 34, 35, 36] cannot be easily generalized to the two-dimensional user type case, even if the allocation rule is consistent (and we have shown that it is not in our problem).

Next we will exploit the special structure in Problem 1 and propose a new approach of solving the problem. This is a key contribution of this paper. Specifically, we will investigate the contract feasibility and optimality in Section 4 and Section 5, respectively. Table III summarizes the key notation in this paper.

Symbol Physical Meaning
The monthly data cap.
The fixed monthly subscription fee.
The overage usage fee when exceeding the data cap.
The data mechanism .
The user’s data valuation.
The user’s network substitutability.
A total of different , i.e., .
A total of different , i.e., .
The -th () user type after sorting as (17).
The smallest-payoff user type defined in (21).
The user’s willingness-to-pay, defined in (16).
The user’s monthly expected payoff, defined in (4).
The user’s virtual payoff, defined in (5).
The user’s virtual payoff increment, defined in (26).
The user’s virtual payoff differences, defined in (32).
MNO’s expected revenue, defined in (9).
MNO’s expected cost, defined in (10).
MNO’s expected profit, defined in (11).
MNO’s contract .
Contract item for type- user.
Contract item for type- user after sorting.
TABLE III: Key Notation

4 Contract Feasibility

To study the feasibility of the two-dimensional contract, we will first introduce a user’s marginal rate of substitution (which also represents the user’s willingness-to-pay) and the new user ordering in Section 4.1 and Section 4.2, respectively. Then we investigate the necessary and sufficient conditions for a feasible contract in Section 4.3 and Section 4.4, respectively.

4.1 Marginal Rate of Substitution (Willingness-to-Pay)

In economics, a consumer’s indifference curve connects those good bundles that achieve the same consumer satisfaction (payoff). In our problem, we can plot a user’s indifference curve over the contract plane (i.e., the data cap and the subscription fee ) as in Fig. 2. On the plane, a type- user’s indifference curve with a fixed payoff satisfies

(15)

Fig. 2 shows that the indifference curve is increasing and concave777Showing the increasing and concave property for the indifference curve is equivalent to showing that is decreasing and convex in , which has been proved in our previous work (see Section 5.2 of [8]). in the data cap , which indicates that the subscription fee would increase (with a diminishing marginal increment) as the data cap increases to maintain the same payoff. Moreover, as a user’s indifference curve shifts downward, his payoff increases because of the decreasing subscription fee.

Fig. 2: Two indifference curves of the same user type with two different expected payoffs, i.e., and .
(a)
(b)
(c)
Fig. 3: Three market modes.

The slope of an indifference curve is called the marginal rate of substitution (MRS), which is the rate at which a consumer is ready to give up one good in exchange for another good, while maintaining the same level of satisfaction. In our problem, we denote the MRS of a type- user on a data cap as

(16)

which depends on the user’s private information and the data cap . The MRS indicates a type- user’s willingness-to-pay for an additional unit of data on a data cap . In the rest of the paper, we will use the three phrases “marginal rate of substitution”, “slope of the indifference curve”, and “willingness-to-pay” interchangeably.

4.2 User Ordering Based on Willingness-to-Pay

Without loss of generality, now we sort and index the user types based on the corresponding willingness-to-pay in an ascending order as follows:

(17)

where for some and . In this case, under the data cap , we have

(18)
Lemma 1.

The new user ordering in (17) does not depends on the data cap. That is, for any , we have

(19)

Lemma 1 indicates that the user ordering in (17) does not change, even though the value of would change with the data cap . Intuitively, this is because that a user’s willingness-to-pay in (16) has a separable structure between the user types (i.e., and ) and the data cap . The proof of Lemma 1 is given in Appendix A.

For notation simplicity, in the following, we will directly use to denote a user type under the ordering specified in (18), and denote the contract item intended for the type- users.

To have a better understanding on the new user ordering, we use Fig. 3 to illustrate how maps to . There are three different market modes depending on the relationship between the extreme valuations ( and ) and the overage fee , i.e., as in Fig. 3(a), as in Fig. 3(b), and as in Fig. 3(c). Specifically, the arrows in Fig. 3 point to the direction where the user’s MRS increases, the blue square denotes the minimum willingness-to-pay user type-, and the red star denotes the largest willingness-to-pay user type-. The following proposition summarizes the mapping from to and . The proof is given in Appendix A.

Proposition 2.

Under the three market modes, the type- and type- users have their private information as follows:

(20)

Furthermore, the green triangles in Fig. 3 denote the smallest-payoff user type given the contract item , defined as follows

(21)

Lemma 2 indicates that the smallest-payoff user type does not change with data cap or subscription fee. Similar to Lemma 1, this is because the separable structure between the user types (i.e., and ) and the contract item (i.e., and ). For notation simplicity, we will use in the following. The proof of Lemma 2 is in Appendix A.

Lemma 2.

The smallest-payoff user defined in (21) does not depends on the data cap or the subscription fee, i.e.,

(22)

Proposition 3 presents the mapping from to . The proof is in Appendix A.

Proposition 3.

Under the three market modes, the type- user has the private information as follows:

(23)

Next we study the necessary conditions for a contract to be feasible based on users’ willingness-to-pay.

4.3 Necessary Conditions

Lemmas 3 and 4 present two necessary conditions for a contract to be feasible (satisfying IC and IR conditions). The proofs are given in Appendix B.

Lemma 3.

For any feasible contract , if and only if .

Lemma 4.

For any feasible contract , if for all , then .

Lemma 3 reveals that a larger data cap corresponds to a higher subscription fee in the feasible contract, which is intuitive. Lemma 4 shows that a user with a stronger willingness-to-pay for the data cap deserves a larger data cap in the feasible contract. Next we provide a proof sketch for Lemma 4 to show the key insights.

Proof Sketch of Lemma 4.

We illustrate the key insights of Lemma 4 based on the contract plane in Fig. 4.

  • For a type- user, we assume that the red dot in Fig. 4 is the contract item intended for this user type, and the red circle curve represents his indifference curve with a payoff equal to that of selecting .

  • For a type- user, the blue square curve is his indifference curve with a payoff equal to this user choosing the red dot contract item (not intended for his type).

It is obvious that is steeper than ; mathematically speaking, for all (which is the condition in Lemma 4). That is, comparing with the type- users, the type- users have a stronger willingness-to-pay under any data cap. Moreover, as a user’s indifference curve shifts downward, his payoff increases because of the decreasing subscription fee.

Fig. 4: An illustration for Lemma 4.

Next we will show that to ensure the PIC condition , the contract item (intended for the type- users) must locate below (or on) the blue square curve and above (or on) the red circle curve , i.e., in the blue region of Fig. 4. We prove this by contradiction. Assuming that this is not true, then we need to consider the following two scenarios:

  • Scenario 1: The contract item is above the blue square curve , such as the green squares labeled 1, 2, 3 in Fig. 4. In this case, the indifference curve for the type- should shift upward (with a decreasing payoff) to touch one of the three green squares. However, the type- user can achieve a higher payoff (comparing with selecting ) by selecting the red dot contract item , which violates the PIC condition for the type- user.

  • Scenario 2: The contract item is below the red circle curve , such as the green squares labeled 4 and 5 in Fig. 4. In this case, the indifference curve for the type- user should shift downward (with an increasing payoff) to touch one of the three green squares. Therefore, the type- user can achieve a higher payoff by selecting the green square contract item , which violates the PIC condition for the type- users.

The above discussion indicates that the contract item must locate in the blue area, which is on the right of the dash line. Thus , as Lemma 4 implies.   

According to Lemma 3 and Lemma 4, we summarize the necessary conditions for a feasible contract as follows:

Theorem 1 (Necessary Conditions for Feasibility).

The feasible contract has the following structure

(24)

4.4 Sufficient Conditions

Next we derive the sufficient conditions for the feasible contract though the following two transitivity properties for Pairwise Incentive Compatibility (PIC) and Individual Rationality (IR). The proofs are given in Appendix C.

Lemma 5 (PIC-Transitivity).

Suppose the necessary conditions in Theorem 1 hold, then for any , the following is true

(25)

The above PIC transitivity property makes the contract problem (i.e., Problem 1) more tractable. It shows that we can reduce a total of PIC conditions to a total of PIC conditions for the neighbor user type pairs, i.e., .

We presents the IR transitivity in Lemma 6.

Lemma 6 (IR-Transitivity).

Suppose the necessary conditions in Theorem 1 and all PIC conditions hold, then the following is true,

Recall that the user type-, defined in (21), achieves the smallest payoff among all the user types for any given contract item. Lemma 6 implies that once we can guarantee all the PIC conditions, then we only need to further ensure that the IR constraint for the smallest-payoff type- users. This allows us to reduce a total of IR conditions to one IR condition .

Before we present the sufficient conditions for the feasible contract, we first introduce a user’s virtual payoff increment. Recall that defined in (5) denotes the type- user’s virtual payoff. We define and as the type- user’s virtual payoff increments between selecting the contract item and the contract items intended for his neighbor user types (i.e., and ), as follows

(26a)
(26b)

Based on Lemmas 36, we derive the following sufficient conditions for a contract to be feasible.

Theorem 2 (Sufficient Conditions for Feasibility).

The contract is feasible if all the following conditions hold,

  1. ,

  2. for ,

    (27)
  3. for all ,

    (28a)
    (28b)
  4. for all ,

    (29a)
    (29b)

Now we discuss the intuitions of Theorem 2. Condition 1) satisfies the necessary conditions in Theorem 1. Condition 2) guarantees the IR condition for the type- users, i.e., , which is sufficient for the IR conditions of all other user types according to Lemma 6. Condition 3) and Condition 4) guarantee the PIC condition for the neighbor user types, i.e., , which is sufficient for the global IC condition according to Lemma 5. Specifically, the inequality (28a) ensures that the type- user will not select the contract item , i.e., ; the inequality (28b) ensures the type- user will not select the contract item , i.e., . Similar intuitions apply to (29).

So far we have derived the necessary and sufficient conditions for a feasible contract. Next we will analyze the optimality of the contract.

5 Contract Optimality

We will study the MNO’s optimal contract problem (i.e., Problem 1) based on the necessary and sufficient conditions for a feasible contract. To reveal the key insights, we will investigate the contract optimality in the following two steps.

  • First, in Problem 2, we derive the MNO’s optimal prices given a feasible choice of data caps where .

  • Second, in Problem 3, we substitute the optimal prices to the MNO’s profit function and derive the optimal data cap .

5.1 Optimal Pricing

In Problem 2, we compute the MNO’s optimal prices, denoted by , given a feasible data cap allocation , i.e., . Note that the constraints (27), (29), and (28) are the sufficient conditions in Theorem 2. Hence the solution together with the given data cap must be a feasible contract.

Problem 2 (Optimal Prices).
(30)

Next we characterize the optimal prices in Theorem 3. The proof is given in Appendix D.

Theorem 3 (Optimal Pricing Policy).

Given a set of feasible data caps satisfying . The optimal pricing policy for the MNO, denoted by , is

(31a)
(31b)
(31c)

Comparing Theorem 2 and Theorem 3, we notice that, given a set of feasible data caps , the MNO should charge the highest prices satisfying the IC and IR conditions.

Next we further study the MNO’s optimal data caps based on the optimal prices in (31).

5.2 Optimal Data Caps

For notation simplicity, we first introduce the concept of virtual payoff difference. For a given data cap , the virtual payoff differences between the type- user and his neighbor user types (i.e., and ) are defined as

(32a)
(32b)

We substitute the optimal prices (31) derived in Theorem 3 into the objective function of Problem 2, and write the MNO’s objective function (i.e., the total profit) as follows:

(33)

where is given by (34), and and are two constants related to the distribution of the user types. Thus we get the following optimization problem over the data caps.

(34)
Problem 3 (Optimal Data Caps).
(35a)
(35b)
(35c)
(35d)

Problem 3 is a nonlinear integer programming with two special structures. First, the objective function has a separable structure over each decision variable . Second, the decision variables are monotonic. Moreover, the convexity of Problem 3 depends on all user types for all and the corresponding distribution for all .

In previous literature (e.g., [33, 34, 35, 36]), the commonly used approach to solving Problem 3 is monotonicity relaxation. The main idea is to first relax the monotonicity constraints (35b) and maximize each over the corresponding decision variable . If the solution obtained under the relaxation violates the monotonicity constraints (35b), then one needs to adjust the solution according to the algorithm proposed in [33] to become feasible. We refer interested readers to Appendix E for more details. In general, the monotonicity relaxation approach is very efficient, since it only needs to deal with several single-variable optimization problems. However, the adjusted solution is only a locally optimal solution when the problem is not convex [1]. Moreover, it is difficult to analytically characterize the sub-optimality gap of the solution. To obtain the globally optimal solution of Problem 3 efficiently, in Section 5.3, we will propose the Dynamic Quota Allocation Algorithm, which is one of the major contributions in this paper.

5.3 Dynamic Quota Allocation (DQA) Algorithm

5.3.1 Basic Idea

The basic idea of the DQA Algorithm comes from dynamic programming, i.e., breaking the original problem down into simpler sub-problems in a recursive manner [37]. Specifically, we will decompose Problem 3 by utilizing the separability of objective (35a) and the monotonicity constraints (35b). Next we introduce how to define the proper sub-problems.

5.3.2 Level-() Subproblem

In the DQA Algorithm, we refer to Problem 4 as the level-() sub-problem of Problem 3. Basically, the level-() sub-problem focuses on the optimal data caps for the smallest user types (i.e., type- to type-, where ) under the data cap upper bound (). Recall that there are a total of types of users and is users’ maximal possible monthly data demand. The special case of the level-() sub-problem is equivalent to Problem 3, since the MNO does not need to offer any data cap larger than .

Problem 4 (Level- Sub-problem).

Given and , the level- sub-problem is

(36a)
s.t. (36b)
(36c)
var: (36d)

Here we denote and as the optimal value and the optimal solution of the level-() sub-problem (36), respectively. Since the level-() sub-problem is equivalent to Problem 3, we have

  • The optimal value of Problem 3 is .

  • The optimal data caps in Problem 3 is , i.e., for all .

In the following, we will show that if we know for all and , then we can directly find . To present this connection clearly, we first introduce some properties of in Propositions 4 and 5. The proofs are given in Appendix F.

Proposition 4.

For any and , has the following recursive relation

(37a)
s.t. (37b)

The proof of Proposition 4 follows the definition of the level-() sub-problem in (36).

Proposition 5.

Given any , we have

  • Function is non-decreasing in the data cap .

  • There exists a critical point such that does not change for any .

The intuitions behind Proposition 5 are two-fold.

  • First, the non-decreasing property of results from the constraints (36b). Mathematically, in (36b) defines the domain upper bound of the level-() sub-problem. That is, a larger in (36b) corresponds to a larger feasible domain, hence a no smaller optimal value .

  • Second, will not increase in anymore if the optimal solution of the level-() sub-problem is smaller than the domain upper bound . Basically, equals to the -th element of the optimal solution for the level-(