Incentive Mechanisms for Cooperative Wireless ... - Springer Link

Int J Wireless Inf Networks (2016) 23:273–282 DOI 10.1007/s10776-016-0323-2

Incentive Mechanisms for Cooperative Wireless Networks with Adverse Selection and Moral Hazard Nan Zhao1 • Minghu Wu1 • Wei Xiong1 • Cong Liu1 • Li Zhu1

Received: 24 March 2016 / Accepted: 14 September 2016 / Published online: 20 September 2016 Springer Science+Business Media New York 2016

Abstract Cooperative communication is a promising technique to improve utilization of the wireless spectrum resource. However, due to the limited wireless network resource, the selfish relay nodes may be unwilling to offer their relay help without any extra incentive. In this work, we study a contract-based mechanism for incentivizing cooperative relay in the presence of the dual asymmetric information. By modelling multi-user cooperative relay as a labour market, a principal-agent model is proposed with the combination of relay power, basic wage and relay bonus in the continuous type scenario. And an optimization problem of multi-user relay incentive is formulated to achieve the twin objectives of ability-discrimination and effort-incentive. Numerical results show that the optimal contract design scheme is effective in improving the performance of cooperative communication. Keywords Relay incentive Contract theory Adverse selection Moral hazard Continuous type

& Nan Zhao [email protected] Minghu Wu [email protected] Wei Xiong [email protected] Cong Liu [email protected] Li Zhu [email protected] 1

Hubei Collaborative Innovation Center for High-efficient Utilization of Solar Energy, Hubei University of Technology, Wuhan, China

1 Introduction The limited wireless spectrum resource is becoming a bottleneck to meet the fast growing demands of wireless data services. Cooperative communication [1] has the great potential to effectively improve spectrum efficiency by exploiting spatial diversity in the wireless networks. Thus, it has been considered as a promising technique for the future wireless communication networks, such as cellular networks and wireless ad hoc networks [2–4]. In a typical cooperative communication network, one or more idle nodes can forward information as well as act as cooperative relay(s) for anothers. However, the selfish nature of wireless nodes is the main obstacle for the wide application of cooperative communication in practice. The selfish relay nodes may be unwilling to offer their relay help without any extra incentive [5]. Thus, a well-designed incentive mechanism should reward relay nodes (RNs) based on their contributions: wireless nodes that contribute more must get higher rewards than nodes with less contributions. In the ex ante relay stage, each node will attempt to harness as much reward as possible by claiming that it is a node with a good communication channel or with a good relay ability, the source may not get to know the RNs’ exact relay abilities, which would give rise to the adverse selection problem. And also during the ex post relay stage, the source may not monitor the RNs’ exact relay actions, which would influence the performance of cooperative relay and result in the moral hazard problem. The above problems are exacerbated by information asymmetry, that is, the source may not be aware of the actual RNs’ abilities and performance, which is naturally known by RNs. Therefore, our main goal is to propose an incentive mechanism by overcoming the

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dual information asymmetry simultaneously in cooperative communication networks. To resolve this obstacle, cooperative incentive mechanisms have been investigated recently with the two main categories, i.e., resource exchange-based mechanism [6, 7] and monetary exchange-based mechanism [8–13]. In the resource exchange-based mechanism, the source node can trade its resource for the other nodes’ cooperation directly. And the resource exchange-based mechanism always depends on the amount of the source’s spare spectrum resource. When the source’s own spectrum demands are high, there will be little wireless resource left for exchange. In this situation, the second mechanism with monetary exchange can be a better choice. Cong et al. [8] investigated spectrum allocation in multi-relay cooperative transmission networks using non-cooperative pricing game theory. Wang et al. [9] studied relay selection and power control mechanisms in cooperative communication networks based on pricing theory. Gao et al. [10] considered the secondary spectrum trading in a hybrid market. In [11], the pure spectrum market based on the contract theory was investigated. Zhao et al. [12, 13] developed the efficient contract models for adverse selection and moral hazard problems caused by the RNs’ asymmetric information. However, most existing works considered the single asymmetric information scenario (i.e., either the RNs’ private information or hidden action); while this work focuses on contract-theoretic cooperative relay mechanism under the dual asymmetric information scenario, which means that the source is not only unaware of the RNs’ relay static information (i.e., relay cost) but also the dynamic information (i.e., relay actions). And the above works always assumed that the RNs’ type is discrete. Practically, the RNs’ type can be continuous (i.e., the RNs’ type can be a continuous random variable depending on the realization of the channel coefficient [7]). Moreover, to the best of our knowledge, the problem of the dual asymmetric information with the RNs’ continuous type had not been studied. The main contributions of this paper are as follows: •

•

By exploiting the cooperative mechanisms and design challenges in cooperative networks, a principal-agent relay incentive model is proposed in the presence of both adverse selection and moral hazard problems with the RNs’ continuous type. By considering the different types of the RNs, the source pays the different basic wages to RNs. Moreover, in order to motivate RNs to work hard, the source offers them a bonus related to the relay performance. In light of contract theory, the optimal contract design in the presence of the dual asymmetric information is presented. The necessary and sufficient conditions for a principal-agent model with moral hazard and adverse

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•

selection to be individually rational (IR) and incentive compatible (IC) are systematically characterized to achieve the twin objectives of ability-discrimination and effort-incentive. To effectively incentivize the potential RNs to participate in cooperative communication, the optimization problem is formulated, which maximizes the source’s utility while meeting the feasible conditions of the potential RNs. The performance of the optimal contract-based cooperative communication mechanism is demonstrated through simulations.

The remainder of the article is organized as follows. The system model and problem formulation for contract-based cooperative communication are introduced in Sect. 2. Optimal contract design is investigated in Sect. 3. Numerical simulation results are given and discussed in Sect. 4, and Sect. 5 concludes the paper.

2 System Model and Problem Formulation We consider a typical cooperative network with one source and N RNs, shown in Fig. 1. With the poor channel condition between its transmitter and receiver, the source needs the RNs’ relay help. Assume that the source’ direct transmission rate is small such that the source wants to use cooperative communications. Thus, we do not consider the source’s direct transmission situation and concentrate on the source’s total increased profit due to RNs cooperative relay help. As shown in Fig. 1, the interaction between the source and the RNs involves two phases: Phase I for the source’s relay request broadcasting and Phase II for RNs’ relay communication. During the first broadcasting phase, the source transmitter broadcasts data to the RNs and the source receiver. Due to the selfish nature of wireless nodes, the source wants to get the RNs’ relay help as much as possible, which is against the RNs’ interests. And the RNs want to obtain the large reward with a little help. Therefore, in this study, we intentionally concentrate on contract theory to deal with the above conflicting demands between the source and the RNs. Contract theory [14] investigates to design the mutually agreeable contract among economic players in presence of asymmetric or incomplete information scenarios [15]. In this cooperative communication scheme, relay incentive is conceptually like the labor market. The employer, the source, recruits some RNs to cooperatively relay the traffic. As the employee, the RN, chooses one of the contract items to participate in cooperative relay. Then, during the second cooperation phase, the helping RNs forward the received data to the source receiver.


275

Fig. 1 Wireless cooperative network. a Phase I: broadcasts phase. b Phase II: relay phase

In Sects. 2.1 and 2.2, we will describe how the source and the RNs evaluate the trade-off among relay power, basic wage and performance bonus. Further, the contracttheoretic model to balance the interests of both sides is proposed in Sect. 2.3. 2.1 Source Model The source’s total achievable profit due to cooperative relay of the ith RN can be written as pi pi ðpi Þ ¼ q log 1 þ ; ð1Þ n0 where pi is the ith RN’s transmitting power at the source’s receiver, n0 is the noise power, which is normalized to be 1 for simplification in the following analysis. And q [ 0 is the equivalent profit of per unit channel capacity, which is assumed to be identical for all the RNs. Then, the payment wi ðpi ; ai ; bi Þ to the ith RN with a linear sharing scheme [16] is defined as wi ðpi ; ai ; bi Þ ¼ ai þ bi pi ðpi Þ;

ð2Þ

where ai is the basic wage of the ith RN, and bi 2 ½0; 1 is the performance-based bonus coefficient. Due to the different relay abilities and actions, the RNs may obtain the different basic wages and bonuses. Then, the source’s utility is the achievable profit gained from cooperative communication minus the reward to RNs, which can be given by US ðpi ; ai ; bi Þ ¼ ¼

N X i¼1 N X

½pi ðpi Þ wi ðpi ; ai ; bi Þ ð3Þ ½ð1 bi Þpi ðpi Þ ai :

i¼1

Note that the source only chooses to offer a cooperativerelay contract when its cooperative utility is larger than zero. In that case, the number of the RNs is at least one in cooperative communication. Without loss of generality, we

omit the parameters (pi ; ai ; bi ) in US ðpi ; ai ; bi Þ for simplicity of writing. 2.2 Relay Node Model In this subsection, the RN’s model related with its relay effort and the reward obtained is considered. Considering that the ith RN have the relay channel gain (hST i ;SR ) between its transmitter ST i and the source’s receiver (SR), if the ith RN wants to reach a received power pi at SR, the RN needs to transmit with a power pi =hST i ;SR . Then, the relay communication cost of the ith RN can be represented as pi Ci ðpi Þ ¼ c; ð4Þ hST i ;SR i where the ci is the relay cost per unit transmission power of the ith RN. To facilitate following discussions, the ith RN’s type is defined as ci hi ¼ ; ð5Þ hST i ;SR which describes all the hidden information of this RN. A lower hi means that the RN has a better relay channel condition (a larger channel gain hST i ;SR ), or it has a lower relay cost (a smaller ci ). Obviously, hi is a continuous random variable depending on the realisation of the channel coefficient and the relay cost. Then, we define the ith RN’s utility as URNi ðpi ; ai ; bi Þ, which is the difference between its received rewards and its cost of cooperative communication. That is, URNi ðpi ; ai ; bi Þ ¼ wi ðpi ; ai ; bi Þ Ci ðpi Þ ¼ ai þ bi pi ðpi Þ hi pi :

ð6Þ

Notice that the RN’s utility URN i ðpi ; ai ; bi Þ is increasing in the basic wage (i.e., ai ) and performance-based bonus (i.e., bi ). And the RN’s relay ability (i.e., relay cost hi ) and relay action (i.e., relay effort pi ) also affect its utility. In

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this work, we focus on the contract-theoretic model considering both the RN’s relay ability and actions asymmetric information scenarios. Similarly, we omit the parameters (pi ; ai ; bi ) in URNi ðpi ; ai ; bi Þ for simplicity of writing.

3 Optimal Contract Design

information scenario, the source does not know each RN’s relay ability or action, but has the knowledge of the distribution of each hi . In this case, the source needs to design a contract to achieve the twin objectives of ability-discrimination and effort-incentive. Once the optimal contract is designed, the complete representation of relay incentive mechanism can be described as follows.

In cooperative communication markets, the RNs often have some prior information about the average cost of cooperative relay before offering the source any relay effort. The RNs’ hidden relay information gives rise to the adverse selection problem. Moreover, due to the selfishness of RNs and the finiteness of spectrum resources, the RNs may deviate from the contract to maximize their own benefit, and the relay actions of the RNs are unobservable to the principal (the source). Thus, the RNs’ hidden relay actions give rise to the moral hazard problem. Moreover, since the above two problems are often intertwined, the source often needs to deal with both adverse selection and moral hazard problems simultaneously. Therefore, in this section, the optimal contract design under the dual asymmetric information scenario is presented to achieve the twin objectives of ability-discrimination and effort-incentive. The contract model with adverse selection being followed by moral hazard is illustrated in Fig. 2. First, considering that the type-hi RN chooses the contract item (ai ðhi 0 Þ; bi ðhi 0 Þ), then its utility URNi in (6) can be written as

•

URNi ðhi ; hi 0 jpi Þ ¼ ai ðhi 0 Þ þ bi ðhi 0 Þpi ðpi Þ hi pi :

2.3 Contract Formulation In this subsection, we will give an analysis about the continuous type case with type hi . Suppose each RN knows hi , which is a random variable and distributed on a strictly positive interval H 2 ½hL ; hH , with a probability density function fi ðhÞ and the corresponding distribution function Fi ðhÞ, where hL \hH , and fi ðhÞ [ 0 for all hi 2 H. Let h ¼ ½h1 ; h2 ; . . .; hN T . And to facilitate following discusðhÞ sions, let us define Hi ðhÞ ¼ FfiiðhÞ . In the dual asymmetric

•

•

•

•

•

Step 1: The source broadcasts a set of contract items (ai ; bi ) to all the geographically potential RNs. Step 2: After receiving and evaluating the contract items, each RN will send feedback signals to indicate whether they are willing to participate in according the estimated utility or not. If the RN is willing to accept certain contract item, it informs the source its choice. Step 3: After getting the feedback from RNs, the source will sign the contract with the RNs who accept it. If all the RNs reject the contract, the source will get zero utility gained from cooperative communication. Step 4: After signing the contract, the source informs the involved RNs cooperative instructions and messages (i.e., the space-time codes). And the source’s transmitter broadcasts its data to the involved RNs’ transmitters. Step 5: After receiving the data from the source’s transmitter, the involved RNs’ transmitters decode the data and forward to the source’s receiver simultaneously using the space-time codes assigned by the source from its random codebook. Through proper choice of space-time codes, RNs’ simultaneous relay signals do not interference with each other at the source’s receiver [17]. Step 6: If the cooperative transmission is successful, the source will reward the involved RNs based on their contract. Otherwise, if the cooperative transmission failed, the employed RN will not receive the reward.

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ð7Þ

In order to incentive RNs to participate in relay, the source first needs to make sure that each RN can obtain the utility at least as much as it would achieve by declining the contract. And this minimum utility is called the retained Thus, the contract needs to satisfy the following utility U. IR constraint ðIRÞ

URNi ðhi ; hi 0 jpi Þ U;

1 i N:

ð8Þ

Moreover, as the source knows little information about the RNs’ relay effort or action, it needs to ensure that each RN accepts the contract to maximize its own utility, that is, the contract needs to satisfy the following IC constraint ðICÞ

max URNi ðhi ; hi 0 jpi Þ ¼ max½ai ðhi 0 Þ þ bi ðhi 0 Þpi ðpi Þ hi pi : pi 0

pi 0

ð9Þ From the IC constraint, the optimal relay power 0

pi ðhi Þ ¼ maxðqbihðhi i Þ 1; 0Þ can be obtained. Especially, pi ðhi Þ ¼ 0 means that the RN is not willing to offer its

Fig. 2 The contract model with adverse selection being followed by moral hazard


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relay help, which results in ai ðhi Þ ¼ 0, bi ðhi Þ ¼ 0, URNi ðhi Þ ¼ 0, and the source obtains nothing from the RN. Then, (7) can be simplified as qbi ðhi 0 Þ URNi ðhi ; hi 0 Þ ¼ ai ðhi 0 Þ þ bi ðhi 0 Þq log hi ð10Þ 0 qbi ðhi Þ þ hi : Next, considering that the type-hi RN chooses the contract item (ai ðhi Þ; bi ðhi Þ), then its utility can be written as URNi ðhi Þ ¼ URNi ðhi ; hi Þ ¼ ai ðhi Þ qbi ðhi Þ þ bi ðhi Þq log qbi ðhi Þ þ hi : hi

ð11Þ

And in order to attract RNs to truthfully reveal their private types, the source needs to ensure that each type-hi RN gets the maximum utility by choosing the contract item designed for its type hi . That is, the contract needs to satisfy the following IC constraint ðICÞ

URNi ðhi Þ URNi ðhi ; hi 0 Þ;

hi ; hi 0 2 ½hL ; hH :

ð12Þ

Then, the source’s expected utility of (3) with the RNs’ continuous type can be rewritten as E ½US ¼

N Z X i¼1

hH

ð1 bi ðhi ÞÞq log

hL

qbi ðhi Þ ai ðhi Þ dFi ðhi Þ: hi

ð13Þ Therefore, the contract design optimization problem under the dual asymmetric information scenario is to maximize the source’s expected utility under the RNs’ IC and IR constraints, which is defined as max

ai ðhi Þ;bi ðhi Þ

s:t: ðIC1Þ ðIC2Þ ðIRÞ

N R hH P hL

i¼1

ð1 bi ðhÞÞqlog

qbi ðhÞ ai ðhÞ dFi ðhÞ; h

URNi ðhi Þ URNi ðhi ; hi 0 Þ; qbi ðhi 0 Þ 1; 0 ; pi ðhi Þ ¼ max hi URNi ðhi Þ U; hi ; hi 0 2 ½hL ;hH :

ð14Þ The first two constraints of (14) correspond to the IC constraints under the hidden relay type and hidden relay action scenarios, respectively. Specifically, the IC constraint (IC1) under the hidden relay type scenario ensures that each type-hi RN gets the maximum utility by choosing the contract item fai ðhi Þ; bi ðhi Þg designed for its type. And the IC constraint (IC2) under the hidden relay action scenario ensures that each RN will get the maximum utility by choosing the optimal relay power pi ðhi Þ. The third constraint (IR constraint) ensures that each RN get a retained utility by choosing the optimal relay power.

From (11), we have dURNi ðhi Þ qb ðhi Þ ¼1 i ¼ pi ðhi Þ 0; dhi hi

ð15Þ

which means that the RN’s utility URN i ðhi Þ is decreasing in hi . Then, by considering the IR constraint (14), we have Thus, URNi ðhi Þ with the URNi ðhH Þ ¼ min URNi ðhi Þ ¼ U. continuous type can be rewritten as Z hH qbi ðsÞ URNi ðhi Þ ¼ URNi ðhH Þ 1 ds s hi ð16Þ Z hH qb ðsÞ i ¼U 1 ds: s hi Then, by combining (11) and (16), we have qbi ðhi Þ þ qbi ðhi Þ hi ai ðhi Þ ¼ URNi ðhi Þ bi ðhi Þq log hi qbi ðhi Þ ¼ URNi ðhH Þ bi ðhi Þq log hi Z hH qbi ðsÞ þ qbi ðhi Þ hi 1 ds: ð17Þ s hi Then, the source’s expected utility can be rewritten as N Z hH X qbi ðhi Þ qbi ðhi Þ E½US ¼ q log U hi i¼1 hL ð18Þ Z hH qbi ðsÞ þ hi þ 1 ds dFi ðhi Þ: s hi By changing the integration order of (18), we have hH Z hH qb ðsÞ 1 i dsdFi ðhi Þ s hL hi Z hH Z hi qbi ðhi Þ ð19Þ ¼ dFi ðsÞ 1 dhi hi hL hL Z hH qb ðhi Þ ¼ Hi ðhi Þ 1 i dFi ðhi Þ: hi hL

Z

Then, by combining the above two equalities, we have N Z hH X i ðhi Þ; ð20Þ E½US ¼ ½Rðbi ðhi ÞÞ UdF i¼1

hL

where qbi ðhi Þ Rðbi ðhi ÞÞ ¼ q log qbi ðhi Þ þ hi hi qb ðhi Þ þ Hi ðhi Þ 1 i : hi

ð21Þ

At this point, the source’s expected utility optimization problem is simplified to obtain the maximum Rðbi ðhi ÞÞ in

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Parameter

Setting

800

bi ðhi Þ

hi hi þHi ðhi Þ

700

max hi þHqi ðhi Þ 1; 0

pi ðhi Þ

qbi ðhi Þ U hi þ qbi ðhi Þ qbi ðhi Þ log hi Z hH q 1 ds þ s þ Hi ðsÞ hi i Rh h qb ðsÞ U hiH 1 is ds q log hi þHqi ðhi Þ q þ hi þ Hi ðhi Þ

ai ðhi Þ

URN ðhi Þ i

Rðbi ðhi ÞÞ

E US

N R P hH i¼1

hL

Source’s optimal utility US*

Table 1 Optimal contract design setting

Information symmetry Information asymmetry (Optimal) α=0 β=0

600 500 400 300 200 100 0 0

i ðhi Þ ½Rðbi ðhi ÞÞ UdF

5

10

15

20

25

30

ρ

Fig. 3 Comparison between the source’s optimal utility for the different equivalent profit q under the different incentive mechanisms

(21). Any local optimal solution (denoted as b^i ðhi Þ) to maximize Rðbi ðhi ÞÞ satisfies dRðbi ðhi ÞÞ q qHi ðhi Þ q ¼ ¼ 0: ð22Þ dbi ðhi Þ b ðh Þ¼b^ ðh Þ b^ ðhi Þ hi i

i

i

i

d Rðbi ðhi ÞÞ obi ðhi Þ2

¼

q bi ðhi Þ2

\0;

ð23Þ

which means that the local optimal solution to maximize Rðbi ðhi ÞÞ is unique and globally optimal. Thus, bi ðhi Þ ¼ hi þHhii ðhi Þ. Thus, the optimal contract design setting and the optimal expected utility of the source and RNs are summarized in Table 1. Since the contract design optimization problem in (14) can be transformed into a single-variable optimization problem, the computational complexity is O(N). The computational complexity increases with the number of RNs. This is because that the source may want to involve more than one RNs in the contract to mitigate the information asymmetry and avoid risk of having no relay in cooperative communications.

4 Results and Discussions In this section, we present the simulation results on the feasibility and performance of the proposed contract-based cooperative communication method developed in ¼ 3, and the RN MATLAB. We assume that N ¼ 21, U type hi is uniform distribution in [1, 5]. First of all, we will evaluate the feasibility of the proposed contract. For comparison purposes, we introduce another three incentive mechanisms. The first one is the optimal contract under the information symmetry (i.e., the

123

7 RN

And the second order derivative of Rðbi ðhi ÞÞ is 2

8

RNs’ optimal utility U*

i

9

Type−6 Type−12 Type−18

6 5 4 3 2 1 0 2

4

6

8

10

12

14

16

18

20

Type of RNs Fig. 4 RNs’ optimal utility URN with the different type for abilitydiscrimination design (q ¼ 15)

source is aware of the RNs’ relay information), which is the optimal outcome that we can achieve and serve as the upper bound. The second contract scheme is the linear pricing with a ¼ 0 for all the RNs. And the three contract mechanism is the linear pricing with b ¼ 0 for all the RNs. And the optimal contract solved in the previous section is denote as information asymmetry (optimal). Figure 3 shows the source’s optimal utility under the different incentive mechanisms and the different equivalent profit q. Among the four incentive mechanisms, the source’s optimal utility under information symmetry achieves the maximum value, followed by the optimal contract under information asymmetry. And the source with the other two contract design strategies obtains the less utilities.


279

9 8

*

RNs’ optimal utility URN

7 6 Type−6 Type−12 Type−18

5 4 3 2 1 0 0

1

2

3

4

5

6

7

8

RNs’relay power Fig. 5 RNs’ optimal utility URN with the different type for effortincentive design (q ¼ 15)

Moreover, Fig. 3 shows that the source’s optimal utility is increasing in the equivalent profit q. As q increases, the RNs has more incentive to provide the source the relay help, and the source obtains more utility from the RNs’ cooperative relay. And when q is smaller than hi þ Hi ðhi Þ, the relay power pi ðhi Þ of RNs is reduced to zero; thus, the RNs chooses not to participate in cooperative communication, which also makes the sources utility close to zero. And the sources zero utility means that all the RNs are not

Fig. 6 RNs’ optimal relay power p and the optimal utility for each RN (q ¼ 15) URN

willing to offer their relay help, that is the non-cooperative incentive scenario. Next, we evaluate the ability-discrimination of our proposed optimal contract scheme. We show the utilities of type-6, type-12, and type-18 RNs when selecting all the contracts offered by the source. The utility of each user is a concave function. Each RN can achieve their maximum utility if and only if it selects the type of contract that is designed for its own type, as shown clearly in Fig. 4. Thus, by designing a contract in this form, the type of an RN will be automatically revealed to the source after its selection. In other words, the optimal contract under hidden relay type enables the source break the information asymmetry and retrieve the information related to RN type. Moreover, Fig. 4 shows that when the three types of users select their optimal contracts, their utilities follows the inequality URN 6 [ URN 1 2 [ URN 1 8 . As hi increases, the RNs of that type have poorer relay channel condition, or the RNs of that type have a higher relay cost for relay communication; thus, they obtains less utility URN . Then, we evaluate the effort-incentive of our proposed optimal contract scheme. We show the utilities of type-6, type-12, and type-18 RNs when selecting all the contracts offered by the source. Each RN can get their maximum utility if and only if it selects the optimal relay power pi ðhi Þ, as shown in Fig. 5. Thus, by designing a contract in this form, the RN will be incentivized to work hard after its selection. In other words, the optimal contract under hidden

6

θ

4 2 0

0

2

4

6

8

10

12

14

16

18

20

22

12

14

16

18

20

22

12

14

16

18

20

22

Optimal power p*

RN 15 10 5 0

0

2

4

6

8

10

Optimal utility U*

RN

RN 20 15 10 5 0

0

2

4

6

8

10 RN

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relay action enables the source break the information asymmetry and motivate RNs to offer their maximum relay effort. And the corresponding RNs’ relay type hi , the optimal relay power p and the optimal utility URN are shown in Fig. 6. As a lower hi means that the RN has a better relay channel condition, or it has a lower relay cost, the source’s receiver can obtain a higher power from the RN with a lower hi , thus the RN achieves a more utility in return. And then, we study the system performance when the RNs’ number increases, while the other parameters are fixed. Figure 7 shows the source’s optimal utility US with the various numbers of RNs. As the RNs’ number increa-

120

Source's optimal utility U*S

100

80

60

40

5 Conclusion

20

0 0

5

10

15

20

25

Number of RNs

Fig. 7 Source’s optimal utility US as a function of the RNs’ number N (q ¼ 15)

1400

1200

Case A Case B Case C

1000

Source’s optimal utility U*S

ses, the number of potential RNs for participating in cooperative communications increases and the more source’s optimal utility are expected. Finally, we take a close look at the system performance with the different distributions of RNs type hi . We investigate the optimal contracts in the three scenarios, which differ from one another in the distribution of RNs type. In case A, all types are uniformly distributed on the [1, 5] with fi ðhi Þ ¼ 1=4. And in case B, the small type has larger probability than the large type with fi ðhi Þ ¼ ð6 hi Þ=12. And in case C, the small type has smaller probability than the large type with fi ðhi Þ ¼ hi =12. Figure 8 shows the source’s optimal utilities US with the different distributions of RNs’ type of the above three cases. As shown in Fig. 8, the source’s optimal utility is increasing in the equivalent profit q in the three cases. And the source’s optimal utility in case B is always more than that in case A and case C. If the probability of lower type sources become larger, e.g., case B, the source’s receiver can obtain a higher power from the RN with a lower hi , and the source will achieve the more optimal utility US . Otherwise, if the probability of lower type sources become smaller, e.g., case C, the source will achieve the less optimal utility.

800

In this paper, the relay incentive mechanism between the sources and multiple RNs is studied in the wireless cooperative communication networks under the dual asymmetric information scenario. To deal with the conflicting objectives between the source and the RNs, cooperative communication is modelled as a labour market. A principal-agent model is proposed to examine characteristics of optimal relay selection and incentive schemes in the presence of both adverse selection and moral hazard problems with the RNs’ continuous type. The optimal contract design is formulated to obtain the sources’ maximum utility subject to the RNs’ individually rational and incentive compatibility restraints. The performance of the optimal contract relay incentive mechanism is demonstrated through simulations.

600

Acknowledgments This work was supported by National Natural Science Foundation of China (Nos. 61501178, 61471162 and 61601177). The author would like to acknowledge the anonymous reviewers whose constructive criticism, comments, and suggestions led to a greatly improved manuscript.

400

200

0

0

5

10

15

ρ

20

25

30

Fig. 8 Source’s optimal utilities US with the different distributions of RNs’ type

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Compliance with Ethical Standards Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this paper.


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Nan Zhao received the B.S., M.S. and Ph.D. degrees from Wuhan University, Wuhan, China in 2005, 2007 and 2013, respectively. She is currently an associate professor in Hubei University of Technology, Wuhan, China. Her research interests include wireless signal processing and cognitive radio.

Minghu Wu received the B.S. degree in electronic information engineering from Communication University of China, Beijing, China, the M.S. degree in communication information system from Huazhong University of Science and Technology, Wuhan, China, and the Ph.D. degree in Nanjing University of Posts and Telecommunications, Nanjing, China, in 1998, 2002 and 2013, respectively. He is currently a professor in Hubei University of Technology. His major research interests include communication signal processing, video coding and compressive sensing. Wei Xiong received the B.S., M.S. and Ph.D. degrees from Wuhan University, Wuhan, China in 2002, 2005 and 2010, respectively. He is currently an associate professor in Hubei University of Technology, Wuhan, China. His research interests include wireless signal processing.

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282 Cong Liu received the B.S. degree from Wuhan University of Science and Engineering, Wuhan, China, the M.S. degree from Wuhan University of Technology, Wuhan, and the Ph.D. degree from Huazhong University of Science and Technology, Wuhan in 2004, 2008 and 2014, respectively. He is currently a lecturer in Hubei University of Technology, Wuhan, China. His research interests include cognitive radio, multimedia transmission, wireless network, and many other related areas.

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Int J Wireless Inf Networks (2016) 23:273–282 Li Zhu received the B.S. degree from Wuhan University of Science and Technology in 2004 and M.S. degree in Physical Electronics from Huaqiao University in 2007. She received the Ph.D. degree in Communication and Information System from Wuhan University in 2011. She is currently an assistant professor in Hubei University of Technology. Her research interests include cloud computing, QoS and computer vision.