A Novel Mobile Data Contract Design with Time Flexibility

In conventional mobile data plans, the unused data will be cleared at the end of each period (e.g., month). To take advantage of consumers, heterogeneous demands across different periods and to provide more time flexibility, some mobile data service providers (SP) have offered data plans with different lengths of period. In this paper, we consider the data plan design problem for a single SP, who provides data plans with different lengths of period to consumers with different characteristics of data demands. We propose a contract-theoretic approach, where the SP offers a period-price contract consisting of a set of period-price combinations. In discrete-consumer-type model, each period-price combination is designed for a specific type of consumers. In continuous-consumer-type model, the consumers are divided into limited number of groups, and each group is assigned with a period-price combination. We systematically analyze the incentive compatibility (IC) constraint and individual rationality (IR) constraint, which ensure each consumer to purchase the data plan with the period-price combination intended for his type. We further derive the optimal contract that maximizes the SP's expected profit. Our numerical results show that our proposed optimal contract can increase the SP's profit over 35 monthly-period data plan.

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1 Introduction

1.1 Background and Motivation

The fast development and wide adoption of smart phones and tablet devices not only drive the explosive growth of mobile data consumption, but also increase the consumption fluctuation over different plan periods [2]. Conventionally, each data plan specifies data cap for a specific period, which is usually one month. The unused data will be cleared at the end of the period, and the overused data will be charged an additional fee. Hence, the consumers with large consumption fluctuation over different periods (e.g., those having frequent trips) will suffer a large utility loss, because the overused data cannot be compensated by the leftover data in the previous periods.

To deal with this problem, researchers in both academia and industry have proposed many data pricing schemes [3, 4, 5, 6, 7, 8, 9]. However, these pricing schemes do not fully take advantage of users’ heterogeneous demands across periods. Seizing this opportunity, some major mobile data service providers (SP) including AT&T [10] and T-mobile [11] have launched a novel data plan called rollover data plan, where unused data from the monthly plan allowance rolls over for one billing period.

The rollover data plan provides customers more time flexibility by decreasing the frequency of clearing unused data from once per month to once every two months. However, the two-month’s time flexibility may not be enough for the consumers with highly varying data demand. In order to provide time flexibility to more types of customers, we propose a new type of data plan, which specifies the length of period. As a simple example, an SP can offer multiple data plans to consumers, e.g., GB for every month with a price of $ per month, GB for every six months (i.e., with average data cap GB per month) with a price of $ per month, and GB for every year (i.e., with average data cap GB per month) with a price of $ per month. Such data plans can benefit different types of consumers. On one hand, the consumers with highly varying data demand may prefer the data plan with a long period (which provides more time flexibility and can potentially reduce the uncertainty of data demand). On the other hand, the consumers with rarely varying data demand may prefer the data plan with a smaller period (which can reduce the total cost due to the lower unit price). In such a scenario, a natural problem for the SP is how to design a proper set of data plans to maximize its expected profit. The problem is challenging due to (a) the information asymmetry between the SP and consumers and (b) the difficulty in discriminating consumers.

1.2 Key Results and Contributions

In the first part of this paper, we propose a contract-theoretic mechanism for a single SP for discrete-consumer-type model. The SP offers a contract consisting of a set of period-price combinations, where each period-price combination is designed for a specific type of consumers with a specific data demand distribution. Contract theory has been widely applied in solving economics, marketing and network problems [12] [13], and is a useful tool in designing incentive compatible (IC) and individual rational (IR) mechanism [14] to elicit the private information of end users. In this work, we adopt the contract theory to solve the SP’s profit maximization problem under information asymmetry. Specifically, we first provide the IC and IR constraints for the feasible contract to guarantee the truthful demand information revelation of consumers, based on which we further derive the optimal contract that maximizes the SP’s expected profit.

In the second part of the paper, we extend our study to a more general mechanism, which is designed for continuous-consumer-type model. In this case, providing a period-price combination for each consumer type is equivalent to providing infinite combinations, which is not realistic and not consumer friendly. Therefore, our mechanism for continuous consumer types includes the procedure of dividing users into groups according to their types. Then, we design limited pairs of period-price combinations, where each combination is designed for a group of consumers. It is very challenging to use a limited period-price combination to model the infinite consumer types, which usually leads to an NP-hard problem [15]. Therefore, an alternative maximizing algorithm is introduced to find a sub-optimal solution. In the algorithm, we alternatively update the period assignments and group boundaries in order to maximize the SP’s total profit. The main challenge of this method lies in the step of updating group boundaries with fixed period assignment due to the non-convexity of the problem. However, by exploiting the unimodal structure of the objective function, we can obtain the sufficient condition for the optimal solution and show that sufficient condition is satisfied for different scenarios.

The main contributions of the paper are as follow.

  1. Novel Model: We study the SP’s mobile data plan design problem from the perspective of data period, which provides consumers with more time flexibility. To our best knowledge, this is the first work that systematically studies such a new data plan design perspective.

  2. Novel Method: We propose a novel period-price data plan based on the contract theory. Rather than specifying a price for each data cap in conventional data plans, our proposed data plan specifies a data price for each data period. A higher price is associated with a longer data period.

  3. Systematic Solution: We first analyze the period-price contract for discrete-consumer-type model, and then extend the analysis to a more general model with continuous-consumer-type. In continuous-consumer-type model, we assume that the consumer type follows a continuous distribution but the SP offers only a limited number of contract items, which is different from traditional continuous modeling in contract theory. In both cases, we analyze the feasibility (incentive compatibility and individual rationality) of the proposed period-price contract systematically, based on which we further derive the optimal contract that maximizes the SP’s profit.

  4. Performance Evaluation: We compare our proposed optimal contract with the conventional monthly-period scheme through numerical simulations. Numerical results show that our proposed contract can increase the SP’s profit over .

1.3 Related Literature on Data Pricing Schemes

The survey by Sen et al. in [3] reviewed the past pricing proposals and discussed several potential research problems. There are mainly three categories of methods to alleviate the problem of monthly data plan inflexibility: (1) Shared Data Plan [4] [5] allows sharing data quota among multiple devices or users, and hence to decrease the average unit usage cost. (2) Sponsored Data [6] [7] is offered by the content service providers, to sponsor the end users for the traffic of viewing their content. (3) Secondary Data Trading [8] [9] is proposed by the service providers, which allows users to trade their unused mobile data with each other. However, all the above methods have their own disadvantages. Shared data plan does not fully take advantage of the heterogeneous demands across plan periods, because there exists possibility that everyone in the shared data is in the peak month. Sponsored data is too specific to the contents, because not every content provider is willing to provide this sponsorship. Secondary data trading is not convenient for operation, since the consumer has to buy or sell every time when he is running out of data or has data left unused.

The papers [16, 17, 18, 19, 20] are the pioneer works that study the rollover data plan. Zheng et al. in [16] evaluated the benefits of rollover data for both SPs and users as well as identify the types of users who would upgrade to rollover data plans. Wang et al. in [17] and [18] analyzed the interactions between an SP and its subscribed users under both traditional and rollover data plans. In [19] and [20], they further analyzed the competitive market with multiple SPs offering rollover data plans with fixed rollover period (i.e., one month). However, none of them considers the design of data plans from the dimension of length of period. To the best of our knowledge, this work is the first paper that systematically studies a data plan design regarding the length of period.

The remainder of this paper is organized as follows. We first analyze the optimal contract for discrete-consumer-type model in Section 2 and Section 3. Specifically, we present the system model and formulate the problem in Section 2. We analyze the feasibility of the contract and propose the optimal contract in Section 3. We analyze the generalized contract for continuous-consumer-type model with group division in Section 4. Performance evaluation is illustrated in Section 5. Finally, Section 6 concludes the paper.

2 System Model

2.1 Service Provider Modeling

In the conventional data plans, the SP provides a unique period choice (e.g., one month). In those data plans, each consumer can consume data up to a quantity of during one period. In our proposed data plans, the SP offers multiple data plans with different plan periods, and we denote the length of the period as (). 111 For presentation convenience, in the rest of the paper, we use “period ” to refer to “period with length ”, and use ”unit period” to refer to ”period with length 1”. We assume that can be any positive number, so that it is possible to provide any time flexibility.

To sharpen the insights of plan periods, we assume that all the data plans are with the same data cap for a unit time period. Figure 1 is an example of the contract of data plans provided by the SP. In the contract, the SP offers a set of combinations, where each combination (also called a contract item) corresponds to one data plan. In each combination, there is a period and a corresponding unit period price . In other words, a consumer who chooses the contract item , needs to pay a price , and can consume data up to a quantity of in a period of . Intuitively, the unit period price is an increasing function of , because a larger period provides more time flexibility for consumers.

Fig. 1: An example of the contract of data plans.

We define the cost for the SP as the average expense of providing a data plan of period with data cap . Here, we denote the total expense of a data plan of period as , where the cost (i.e., average expense for one period) is formulated as follows

where is the fixed cost (e.g., the fixed monthly spectrum license fee for providing service and the infrastructure maintenance cost) and is the time-specific cost. We can show that is monotone increasing on . This is because with a smaller , the SP can better predict and hence schedule the demand of consumers. We further assume that grows more rapidly with larger period , which means .222 denotes the first order derivative of with respect to (). and denotes and respectively. Then, we can see and .

The SP’s profit comes from selling data plans with different periods. We use to denote the unit period profit of SP from the data plan , which is the gap between the unit period price and the unit period cost , i.e.,

2.2 Consumer Modeling

We assume consumer

’s data demand per unit period follows normal distribution with density function

, where is the mean and

is the standard deviation

333 Normal distribution is widely adopted in modeling consumers’ demand in different areas. For example, [21] applied normal distribution to model consumers’ connectivity of mobile network, while [22] applied normal distribution to model consumers’ demand of electricity.. In our paper, we focus on the demand fluctuation over unit periods, so our design of contract is based on , i.e., standard deviation of consumers’ data demand per unit period. Therefore, in our modeling, we assume that consumers are divided into different groups with different average monthly demand , and we only design contract for a particular group with a certain . Hence, we assume that every consumer has the same and different . For writing convenience, we call a consumer as a type- consumer if the standard deviation of his data demand is . We first assume that the consumer types follow a discrete distribution, and the SP aims to design a specific contract item for each consumer type.444 Mathematically, when we choose a large enough number of types, the discrete-consumer-type model can well approximate a continuous-consumer-type model. In reality, since consumers are heterogeneous, it is more reasonable to assume that consumers’ types follow a continuous distribution [23]. Hence, we will introduce the case of continuous consumer type distribution in Section 4. We denote the set containing all consumer types as . Due to the properties of the normal distribution, when a period of the data plan is changed from unit period to a period of , a consumer’s total data demand within a period still follows normal distribution. The parameters of this normal distribution are as follows: the mean value of the consumer’s total data demand within months is , and the standard deviation is . 555 Intuitively, a consumer with a larger is with a higher data fluctuation, and naturally needs a data plan with higher time flexibility.

We use to denote the valuation of a type- consumer for the contract with period . Similar to [24], for a given period with data cap , we define the unit period valuation as a linear function on the average data consumption per unit period:

(1)

where is a predefined parameter, which represents the valuation of unit data, and is identical for all consumers. For writing convenience, we define . We assume the cost of usage exceeding data cap is very large, so that the consumption will not exceed the data cap . Hence, the average data consumption per unit period equals to the consumer’s average data demand , which is the consumer’s maximum average data consumption per unit period, minus average unsatisfied data demand. In a period data plan, the total unsatisfied demand of a type- consumer in a period of is , then the average unsatisfied data demand per unit period is . For example, if a consumer with average data demand GB and standard deviation consume monthly data plan with a quota of GB, and his demands of consecutive two months are GB and GB, then his unsatisfied data demand of these two months are GB and GB, respectively. Since we assume the consumer’s demand per unit period follows normal distribution with and , the average unsatisfied data demand equals to GB, which means his average total data consumption is . As shown in Figure 2, if all the consumers choose the plan of period and , only the consumer with (the black dashed line) can reach an average consumption of . On the contrary, the consumer with (the red dashed line) can satisfy his demand in the , , and periods, but only consumes in the other periods due to the data cap.

Fig. 2: A data demands example of consumers with different types-(, ).

From (1), we can find that

and

which means that 1) without considering price, every consumer prefers a larger period and 2) the consumer with larger type has a smaller valuation. Furthermore, is negative through direct calculation, meaning that grows more slowly in a larger period.

The utility of the consumer with type- who accepts the data plan with period is defined as the gap between his valuation and payment of the data plan:

2.3 Contract Formulation

In this paper, we aim to design an optimal contract for the SP to maximize its expected profit. The contract contains a set of combinations, each of which includes a period and a corresponding unit period price . Each consumer can only select one combination. Therefore, for each consumer type , the SP will assign a period with unit period price . The set of period-price combinations shown above is a period-price contract. We denote the contract as .

A feasible contract should satisfy the following two constraints: 1) For any type- consumer, he prefers the contract item with period at the price than any other contract items; 2) The SP should guarantee that the contract designed for any type- consumer leads to non-negative utility so that the consumer is willing to accept the contract designed for him. These two constraints are named as incentive compatibility (IC) constraint and individual rationality (IR) constraint correspondingly. Specifically, we define,

Definition 1.

IC constraint:

Definition 2.

IR constraint:

Any feasible contract satisfies IC and IR constraints, and any contract satisfying IC and IR constraints is feasible. The overall profit of the SP from a feasible contract can be written as:

(2)

where is the number of consumers with type-.

Symbol Meanings
the average data cap per unit period
the time length of the data plan’s period
the unit period price of the data plan
the data item with price
and data quota in a period of
the SP’s unit period expense of offering data plan
with period , i.e., the total expense of a data plan
of period is
the unit period profit of the SP
from the data plan item
consumers’ average data demand in a unit period
In discrete-consumer-type model:
represents the standard deviation
of the unit period data demand of consumer ;
In continuous-consumer-type model:
represents the largest standard deviation
among the unit period data demand
of the consumers in the group
the valuation of a type- consumer
for the contract with period
the total number of consumers
the number of consumers in group

the probability density function

of the distribution of consumer type

the cumulative distribution function

of the distribution of consumer type
TABLE I: Notation

3 Contract Feasibility and Optimality

In this section, we first show the necessary and sufficient conditions for the contract to be feasible. Then we derive the best period assignments and price assignments for the optimal contract that maximizes the SP’s overall profit, which is defined in (2).

According to our assumption in Sec. 2.2, there is a finite number of consumer types . Without loss of generality, we let . Then, we rewrite the period assigned to the type- consumers as , and rewrite the price corresponding to the period as for simplicity. Accordingly, we can rewrite the SP’s profit function and cost function as as and cost function as , respectively. For convenience, we summarize the key notations in Table I.

Therefore, the contract optimization problem can be written as:

Problem 1.

where .

3.1 Feasibility

According to (1), we have the following property: for a given period length increment, the consumers with larger type will have a larger valuation increment than the consumers with smaller type. We call this property as increasing preference (IP) property.666Due to space limit, we put all of the detailed proofs in the online technical report [32].

Proposition 1 (IP property).

For any consumer types and any data plan periods , the following condition holds:

(3)

Now, we try to find the necessary and sufficient conditions for the contract to be feasible, i.e., the necessary and sufficient conditions of IC and IR constraints. We show the first necessary condition in the following lemma.

Lemma 1.

For any contract , if it is feasible, then the following condition holds:

Lemma 1 shows that the consumer with larger type should be assigned a longer period.

We show the second necessary condition in the following lemma.

Lemma 2.

For any contract , if it is feasible, then the following condition holds:

Lemma 2 shows that a longer period must be assigned with a higher price. If there is a service with longer period and a lower price, then everyone will select this data plan, and the data plans with shorter periods are meaningless. Together with the observations from Lemma 1, we can find that the data plan with higher price will be assigned to the consumer with larger consumer type (i.e., standard deviation).

From the above two lemmas and IP property, we have the following theorem, which shows the necessary and sufficient conditions for a feasible contract.

Theorem 1 (Necessary and Sufficient conditions for a feasible contract).

For any contract , its IC and IR constraints are equivalent to the following conditions:

(4)
(5)
(6)
(7)

The feasible regions of price assignments are then

From IP property, we have . Therefore, the feasible regions of price assignments are not empty.

Then, the IC and IR constraints in Problem 1 can be substituted by the conditions shown in Theorem 1.

3.2 Optimality

To solve the contract optimization problem, we first solve the optimal price assignments given the fixed period assignments, and then solve the optimal period assignments by substituting the derived price assignments. From the conditions in Theorem 1, we can get the following lemma, which leads to the optimal price assignments.

Lemma 3.

For any feasible contract with fixed periods , the set of optimal price assignments that maximizes under the conditions in Theorem 1 is given by:

(8a)
(8b)
Proof.

We can observe that the price assignments in Lemma 3 satisfy the conditions in Theorem 1.

Since the period assignments are fixed, the total cost of the SP is fixed. Therefore, if there is another set of price assignments that leads to a larger profit (i.e., ), then there is at least one price . According to Theorem 1, to guarantee the feasibility of the contract, the following constraint on must be satisfied:

(9)

From (8) we have

(10)

By substituting (10) into (9), we have , which implies . Since , the IR condition is violated. Therefore, there does not exist any set of feasible price assignments with a larger profit than . ∎

From Lemma 3, we can find that for fixed period assignments, the optimal price assignments are:

(11)

The maximum overall profit is obtained by solving the following optimization problem

(12)

where is the overall profit of the optimal contract with fixed period assignments . By subsituting the derived optimal price assignments (11) into (2), we have:

(13)

where and .

We define as and find that is only based on the period , which is designed for type consumers. Therefore, the contract optimization problem can be divided into the following optimization problems.

(14)

We use to indicate the period that maximizes , i.e., . Since is concave for all 777It is because both and are negative. , the optimal is either of the boundary points or the critical point (the point satisfying and ).

We can show that if the period assignments are in increasing order, then they are the optimal solution of problem (12). However, it is possible that are not in increasing order, which means that they may not be feasible. Each set of infeasible period assignments must have at least one infeasible sub-sequence, which is defined in the following definition:

Definition 3.

A sub-sequence is an infeasible sub-sequence if it satisfies the following two conditions:

Next, we design a mechanism to replace each infeasible sub-sequence by a feasible sub-sequence. We apply the following proposition to design the mechanism.

Proposition 2.

There are K concave functions and . If , then the optimal solution

(15)

satisfies .

The proposition is proved in [25].

Based on Proposition 2, we can see that Algorithm 1, which is an iterative algorithm, can be used to adjust infeasible sub-sequences in into feasible sub-sequences. The details of Algorithm 1 are shown as follows.

1:  Initialization for all .
2:  repeat
3:     Find an infeasible sub-sequence .
4:     Let , for all .
5:  until  are feasible.
Algorithm 1 Iterative Algorithm to deal with infeasible sub-sequences

4 Continuous-Consumer-Type with Group Division

In general, since consumers are mutually independent, the probability that each two consumers have the same standard deviation (i.e., ) approaches to zero. Therefore, it is more realistic to assume that the consumer types follow a continuous distribution. Under such an assumption, to design a contract item for each consumer type is equivalent to providing infinite contract items, which is not realistic and not consumer friendly. Thus, we propose a novel contract design mechanism for continuous-consumer-type model. The mechanism divides the consumers into limited number of groups according to their types and give a contract item for each group. Specifically, in this mechanism, we optimize the group boundaries as well as the period and price assignments in order to maximize the SP’s overall profit.

4.1 Contract Formulation

We assume that the SP divides the consumers into groups. Instead of designing a distinct contract item for each consumer type, the SP offers a single contract item for each group of consumer types. We denote the set of group indices as , where the minimum consumer type in the () group is denoted as and the maximum consumer type in the group is denoted as . We assume that the consumer type follows a continuous distribution and the probability density function is . We use (where ) and (where ) to denote the minimum and maximum value of the feasible interval, i.e., only when .888 To better illustrate the insights, we assume that for all in this paper. For the case that for some in the feasible interval, we can also show that our following analysis is valid. Without loss of generality, we assume . Since the consumer types follow a continuous distribution, we have for all . The set of variables is named as group boundaries. In this paper, we let for simplicity. We use to denote the total number of consumers, and denote the number of consumers in the group as , where

(16)

In (16), is the cumulative distribution function of consumer type , and

In other words, denotes the number of consumers with type less than or equal to .

Similar to our discussion on the contract for discrete-consumer-type model in Sec. 2 and 3, for each consumer group , the SP will assign a combination including a period and a unit period price . We denote the contract for continuous-consumer-type model as . The expected profit of the SP can be written as:

(17)

To guarantee the feasibility of the contract, it should satisfy the IC and IR constraints: 1) For any consumer in group , he prefers the contract item that with period at the price than any other contract items; 2) The SP should guarantee that the contract item designed for any consumer group leads to non-negative utility for each consumer in this group so that the consumers are willing to accept the contract designed for them. Specifically, we define,

Definition 4.

IC constraint:

Definition 5.

IR constraint:

Then, the SP’s profit maximization problem becomes finding the optimal group boundaries, period assignments and price assignments, i.e.,

(18)

subject to the IC and IR constraints in Definition 4, 5 and the following boundary condition:

(19)

First, we can find that IP property in Proposition 1 is still satisfied in continuous-consumer-type model. Then, we try to find the necessary and sufficient conditions for the IC and IR constraints. We show the first necessary condition in the following corollary.

Corollary 1.

For any contract , if it is feasible, then the following condition holds:

Corollary 1 is directly obtained from Lemma 1. Hence, the proof of the corollary is structurally the same as Lemma 1 and is omitted.

The second necessary condition is shown in Lemma 2, which shows that a longer period must be assigned with a higher price.

From Lemma 2, Corollary 1 and IP property, we have the following theorem, which shows the necessary and sufficient conditions of the IC and IR constraints for the contract for continuous-consumer-type model with group division.

Theorem 2.

For any contract , its IC and IR constraints are equivalent to the following conditions:

(20)
(21)
(22)

The contract optimization problem is then to maximize the SP’s overall profit under Theorem 2 and boundary condition (19). In this problem, the variables needed to be optimized are: 1) the period and price of each contract item, i.e., and ; 2) the boundary of each group, i.e., .

4.2 Contract Optimization

According to Theorem 2, we can obtain the optimal price assignments as follows:

which implies

(23)

By substituting (23) into (17), the overall profit of the SP can be rewritten as follows

(24a)
(24b)
(24c)
(24d)

where and .

(32)
(36)

4.2.1 Introduction of The Alternative Maximizing Algorithm

Finding the optimal period assignments and group boundaries that can maximize the SP’s overall profit with price assignments in (23) is very challenging because problem (18) is NP-hard [15]. Therefore, we introduce an alternative maximizing algorithm to find a sub-optimal solution. In this algorithm, we divide the variables into two groups, where the first group contains all the period assignments , and the other group contains all the group boundaries . At the beginning of the algorithm, we divide the consumers into groups by randomly generating group boundaries such that . Then, we iterate the following two steps. In the first step, we keep the group boundaries unchanged and maximize the overall profit by tuning period assignments . In the second step, we keep period assignments (which are obtained by solving the problem in the previous step) unchanged and update group boundaries to maximize . For the rest of the paper, we will simply use to denote which is the threshold between group and group .

The details of these two steps are as follows.

  • Step I: In this step, we find the optimal period assignments that can maximize the overall profit with fixed group boundaries . Specifically, we have the following problem

    (25)

    We can show that the overall profit in (24c) is structurally similar to the overall profit (13) in Sec. 3. Therefore, this optimization problem can be solved through the same method that solves problem (12).

    For writing convenience, we define

    (26)

    where and . Then, we divide the problem (25) into optimization problems

    (27)

    Since is concave for all , the period assignment that can maximize is at the boundary points or at the critical point, i.e.,

    (28)

    where is the solution of .

    By denoting the optimal solution of problem (25) as , we can see that if the period assignments from (27) are in increasing order, then for all . However, if are not in increasing order, which means that they may not be feasible, we need to use Algorithm 1 to adjust infeasible period assignments to make them feasible. In the input of the algorithm, we define as in (26) and let .

  • Step II: In this step, we find the optimal group boundaries that can maximize the overall profit with fixed period assignments . Specifically, we have the following optimization problem:

    (29)

    By defining as

    (30a)
    (30b)

    we can find that the overall revenue in (24d) can be represented as the summation of . (30) implies that