Contract design for relay incentive mechanism under dual asymmetric information in cooperative networks

Abstract

Cooperative relay network can effectively improve the wireless spectrum efficiency and extend the wireless network coverage. However, due to the selfish characteristics of wireless nodes, spontaneous cooperation among nodes is challenged. Moreover, wireless nodes may acquire the different network information with the various nodes’ location and mobility, channels’ conditions and other factors, which results in information asymmetry between the source and relay nodes. In this paper, the incentive issue between the relay nodes’ cooperative service and the source’s relay selection is investigated under the asymmetric information scenarios. By modeling cooperative communication as a labour market, a contract-theoretic model for relay incentive is proposed to achieve the twin objectives of ability-discrimination and effort-incentive. Considering the feature of asymmetric information, the static and dynamic information of the relay nodes are systematically discussed. To effectively incentivize the potential relay nodes to participate in cooperative communication, the optimization problems are formulated to maximize the source’s utility under the multiple information scenarios. A sequential optimization algorithm is proposed to obtain the optimal wage-bonus strategy with the low computational complexity under the dual asymmetric information scenario. Simulation results show that the optimal contract design scheme is effective in improving the performance of cooperative communication.

Appendix

1.1 Proof of Theorem 1

1.1.1 Proof of necessary conditions

Part (b) of (17) is the necessary IR constraint for the highest type ${\theta _N}$ in a feasible contract, and part (c) and (d) are obtained by combining Propositions 1 and 2. Then, the left inequality of part (a) can be derived from the necessary IC constraint for the type-${\theta _i}$ in a feasible contract, that is ${\alpha _i} + G({\beta _i},{\theta _i}) \ge {\alpha _{i + 1}} + G({\beta _{i+1}},{\theta _i})$. Thus, ${\alpha _i} \ge {\alpha _{i + 1}} + G({\beta _{i+1}},{\theta _i})-G({\beta _i},{\theta _i})$.

And the right inequality of part (a) can similarly be derived from the necessary IC constraint for the type-${\theta _{i+1}}$.

1.1.2 Proof of sufficient conditions

Let $\varPhi (n) = \{ ({\alpha _i},{\beta _i}),|i = N-n+1,\ldots ,N\}$ denote as a subset with the last n wage-bonus combinations.

First, we show that $\varPhi (N)$ is feasible. The contract is feasible if it satisfies the IR constraint in (5). This is true due to part (b) in Theorem 1.

Secondly, we prove the IC constraint for the type-${\theta _{i-1}}$. Since the contract $\varPhi (i)$ is feasible, the IC constraint for a type-${\theta _i}$ RN must hold

$$\begin{aligned} {\alpha _i} + G({\beta _i},{\theta _i}) \ge {\alpha _k} + G({\beta _k},{\theta _i}),\quad k=i,\ldots ,n. \end{aligned}$$

(20)

Also, the right inequality of (17) in part (a) can be transformed to

$$\begin{aligned} {\alpha _i} + G({\beta _i},{\theta _{i-1}})-G({\beta _{i-1}},{\theta _{i-1}}) \le {\alpha _{i - 1}}. \end{aligned}$$

(21)

By combining the above two inequalities, we have

$$\begin{aligned} {\alpha _{i - 1}} + G({\beta _{i-1}},{\theta _{i-1}})\ge {\alpha _k} + G({\beta _k},{\theta _i})+G({\beta _i},{\theta _{i-1}})-G({\beta _i},{\theta _i}). \end{aligned}$$

(22)

Notice that ${\theta _{i-1}}<{\theta _i}$ and $k \ge i$, thus $\beta _i \ge \beta _k$. Then, we have

$$\begin{aligned} G({\beta _k},{\theta _{i}})-G({\beta _i},{\theta _{i}}) \ge G({\beta _k},{\theta _{i-1}})-G({\beta _i},{\theta _{i-1}}). \end{aligned}$$

(23)

By combining the above two inequalities, we have

$$\begin{aligned} {\alpha _{i - 1}} + G({\beta _{i-1}},{\theta _{i-1}})\ge {\alpha _k}+G({\beta _{k}},{\theta _{i-1}}), \end{aligned}$$

(24)

which is actually the IC constraint for the type-$\theta _{i-1}$.

Thirdly, we show the IR constraint for the type-$\theta _{i-1}$. Since $\theta _{i - 1} \le \theta _i$, and ${{\alpha _i} + G({\beta _i},{\theta _i}) \ge \bar{U}}$. Then, we get

$$\begin{aligned} \bar{U} \le {\alpha _i} + G({\beta _i},{\theta _i}) \le {\alpha _i} + G({\beta _i},{\theta _{i-1}}). \end{aligned}$$

(25)

Using (24), we also have

$$\begin{aligned} {\alpha _i} + G({\beta _i},{\theta _{i-1}}) \le {\alpha _{i - 1}} + G({\beta _{i-1}},{\theta _{i-1}}). \end{aligned}$$

(26)

By combining the last two inequalities, we have

$$\begin{aligned} {\alpha _{i-1}} + G({\beta _{i-1}},{\theta _{i-1}}) \ge \bar{U}, \end{aligned}$$

(27)

which is the IR constraint for the type-$\theta _{i-1}$.

Finally, we show that if the contract $\varPhi (i)$ is feasible, then the new contract $\varPhi (i-1)$ can be constructed by subtracting the item $({\alpha _i},{\beta _i})$ and the new contract is also feasible.

Since contract $\varPhi (i)$ is feasible, the IC constraint for the type-${\theta _k}$ holds,

$$\begin{aligned} {\alpha _k} + G({\beta _k},{\theta _i}) \le {\alpha _i} + G({\beta _i},{\theta _i}),\quad \forall k = i,\ldots ,N. \end{aligned}$$

(28)

Also, we can transform the left inequality of (17) in part (a) to

$$\begin{aligned} {\alpha _{k + 1}} + G({\beta _{k+1}},{\theta _k})-G({\beta _k},{\theta _k}) \le {\alpha _k}. \end{aligned}$$

(29)

By combining the above two inequalities, we conclude

$$\begin{aligned} {\alpha _{k + 1}} + G({\beta _{k+1}},{\theta _k})-G({\beta _k},{\theta _k}) \le {\alpha _i} + G({\beta _{i}},{\theta _i})-G({\beta _k},{\theta _i}). \end{aligned}$$

(30)

Note that $k\ge i$, thus ${\theta _k} \ge {\theta _i}$ and ${\beta _{k}} \ge {\beta _{k+1}}$. Then, we have

$$\begin{aligned} G({\beta _k},{\theta _k})-G({\beta _{k+1}},{\theta _k}) \le G({\beta _k},{\theta _i})-G({\beta _{k+1}},{\theta _i}). \end{aligned}$$

(31)

By combining the above two inequalities, we conclude

$$\begin{aligned} {\alpha _{k + 1}} + G({\beta _{k+1}},{\theta _i}) \le {\alpha _i} + G({\beta _{i}},{\theta _i}),\quad \forall k = i,\ldots ,N, \end{aligned}$$

(32)

which is actually the IC constraint for the type-${\theta _i}$. This completes the proof.

1.2 Proof of Theorem 2

First, the basic wages in (18) can be easily proofed to be satisfied the sufficient conditions of contract feasibility in Theorem 1. Then, we will prove the optimality and uniqueness of the solutions in (18).

1.2.1 Proof of optimality

First, we show that given fixed bonus coefficients, the basic wage $\{ \alpha _i^*\}$ in (18) maximize the source’s utility

$$\begin{aligned} {U_{S}}({\alpha _i},{\beta _i},{p_i}) = \rho \log \left( 1 + \sum \limits _{i \in \varOmega } {{N_i}{p_i}} \right) - \sum \limits _{i \in \varOmega } {{N_i}({\alpha _i} + \beta {p_i})}. \end{aligned}$$

(33)

Here, we prove by contradiction. Suppose these exists another feasible basic wage $\{ {{\bar{\alpha }}_i},\forall i\}$ which achieves a better solution than $\{ \alpha _i^*,\forall i\}$ in (18). Since ${U_{S}(\alpha _i^*)}<{U_{S}({\bar{\alpha }} _i)}$, and $U_{S}$ is decreasing in the total basic wage, we must have $\sum \nolimits _{i \in \varOmega } {{N_i}{{\bar{\alpha }}_i}} < \sum \nolimits _{i \in \varOmega } {{N_i}\alpha _i^*}$. Thus, there is at least one reward allocation ${{\bar{\alpha }}_i} < \alpha _i^*$ for one type-${\theta _i}$.

If $i = N$, then ${{\bar{\alpha }}_N} < \alpha _N^*$. Since $\alpha _N^* = {\bar{U} - G({\beta _N},{\theta _N})}$, then ${{\bar{\alpha }}_N} < {\bar{U} - G({\beta _N},{\theta _N})}$. But this violates the IR constraint for the type-${\theta _i}$. Then we must have $1 \le i < N$.

Since $\{ {{\bar{\alpha }}_i},\forall i\}$ is feasible, then $\{ {{\bar{\alpha }}_i},\forall i\}$ must satisfy the left inequality of part (a) in Theorem 1. Thus we have

$$\begin{aligned} {{\bar{\alpha }}_{i + 1}} + G({\beta _{i+1}},{\theta _i})-G({\beta _i},{\theta _i}) \le {{\bar{\alpha }}_i}. \end{aligned}$$

(34)

Also, from (18), we can have

$$\begin{aligned} \alpha _i^* - \alpha _{i + 1}^* = G({\beta _{i+1}},{\theta _i})-G({\beta _i},{\theta _i}). \end{aligned}$$

(35)

By substituting the above equality into (34), we get

$$\begin{aligned} {{\bar{\alpha }}_{i + 1}} + \alpha _i^* - \alpha _{i + 1}^* \le {{\bar{\alpha }}_i}. \end{aligned}$$

(36)

Since ${{\bar{\alpha }}_i} < \alpha _i^*$, then we have ${{\bar{\alpha }}_{i + 1}} < \alpha _{i + 1}^*$. Using the above argument repeatedly, we finally obtain that ${{\bar{\alpha }}_N} < \alpha _N^*$, which violates the IR constraint for the type-${\theta _N}$ again.

1.2.2 Proof of uniqueness

We next prove that the relay powers in (18) is the unique solution that maximizes (33). We also prove by contradiction. Suppose there exists another $\{ {{\tilde{\alpha }}_i},\forall i\} \ne \{ \alpha _i^*,\forall i\}$ such that $\sum \nolimits _{i \in \varOmega } {{N_i}{{\tilde{\alpha }}_i}} = \sum \nolimits _{i \in \varOmega } {{N_i}} \alpha _i^*$ in (33). Then there is at least one reward allocation ${{\tilde{\alpha }}_k} > \alpha _k^*$ and one reward allocation ${{\tilde{\alpha }}_l} < \alpha _l^*$. We can focus on the type-${\theta _l}$ and ${{\tilde{\alpha }}_l} < \alpha _l^*$. By using the same argument before, we have ${{\tilde{\alpha }}_N}< \alpha _N^* < {\bar{U} - G({\beta _N},{\theta _N})}$. But this violates the IR constraint for the type-${\theta _N}$.

1.3 Optimal contract design with the RNs’ continuous types

In this subsection, we will give an analysis about the continuous type case with type ${\theta _i}$. Suppose each RN knows ${\theta _i}$, which is a random variable and distributed on a strictly positive interval $\varTheta \in [{\theta _L},{\theta _H}]$, with a probability density function $f_i$ and the corresponding distribution function $F_i(\theta )$, where ${\theta _L} < {\theta _H}$, and $f_i(\theta ) > 0$ for all ${\theta _i} \in \varTheta$.

Then, 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

$$\begin{aligned}&\mathop {\max }\limits _{\{ \{ {\alpha _i},{\beta _i}\} \ge 0\} } \;\;\;\int _{{\theta _L}}^{{\theta _H}} {\left[\rho \log \left(1 + \sum \limits _{i \in \varOmega } {p_i} \right) - \sum \limits _{i \in \varOmega } ({\alpha _i} + \beta _i {p_i})\right] {f_i}d{\theta _i}} , \\&s.t. \quad (IC1) \quad \mathop {\max }\limits _{{p_i} \ge 0} \quad {\alpha _i} + {\beta _i}{p_i} - {\theta _i}{T}({p_i}), \\&\qquad (IC2) \quad {\alpha _i} + {\beta _i}{p_i} - {\theta _i}{T}({p_i}) \ge {\alpha _j} + {\beta _j}{p_j} - {\theta _i}{T}({p_j}), \\&\qquad (IR) \quad {\alpha _i} + {\beta _i}{p_i} - {\theta _i}{T}({p_i}) \ge \bar{U}, \end{aligned}$$

(37)

From the first IC constraint, we have ${\beta _i} = {\theta _i}{T}^\prime ({p_i^{*}})$, then, the optimal relay power $p_i^{*}$ can be written as $p_i^{*}({\beta _i},{\theta _i})$. Then, the optimization problem in (37) can be rewritten as

$$\begin{aligned}&\mathop {\max }\limits _{\{ \{ {\alpha _i},{\beta _i}\} \ge 0\} } \;\;\;\int _{{\theta _L}}^{{\theta _H}} {\left[\rho \log \left(1 + \sum \limits _{i \in \varOmega } {p_i^{*}({\beta _i},{\theta _i})} \right) - \sum \limits _{i \in \varOmega } ({\alpha _i} + \beta _i {p_i^{*}({\beta _i},{\theta _i})})\right] {f_i}d{\theta _i}} , \\&s.t. \quad (IC1) \quad \mathop {\max }\limits _{{p_i^{*}({\beta _i},{\theta _i})} \ge 0} \quad {\alpha _i} + {\beta _i}{p_i^{*}({\beta _i},{\theta _i})} - {\theta _i}{T}({p_i^{*}({\beta _i},{\theta _i})}), \\&\qquad (IC2) \quad {\alpha _i} + {\beta _i}p_i^{*}({\beta _i},{\theta _i}) - {\theta _i}{T}(p_i^{*}({\beta _i},{\theta _i})) \ge {\alpha _j} + {\beta _j}{p_j^{*}({\beta _j},{\theta _i})} - {\theta _i}{T}({p_j^{*}({\beta _j},{\theta _i})}), \\&\qquad (IR) \quad {\alpha _i} + {\beta _i}p_i^{*}({\beta _i},{\theta _i}) - {\theta _i}{T}(p_i^{*}({\beta _i},{\theta _i})) \ge \bar{U}, \end{aligned}$$

(38)

From (4), we have

$$\begin{aligned} \frac{{d{{U_{R{N_i}}}({\theta _i})}}}{{d{\theta _i}}} = - {T}({p_i^{*}({\beta _i},{\theta _i})}) \le 0, \end{aligned}$$

(39)

which means that the RN’s utility ${U_{{RN}_i}}({\theta _i})$ is decreasing in ${\theta _i}$.

Then, by considering the IR constraint in (38), we have ${U_{R{N_i}}}({\theta _H}) = \min {U_{R{N_i}}}({\theta _i}) = \bar{U}$. Thus, ${U_{R{N_i}}}({\theta _i})$ with the continuous type can be rewritten as

$$\begin{aligned} {U_{R{N_i}}}({\theta _i}) = {U_{R{N_i}}}({\theta _H}) + \int _{{\theta _i}}^{{\theta _H}}{T}({p_i^{*}({\beta _i},\tau )})d\tau . \end{aligned}$$

(40)

Then, by combining (4) and (40), we have

$$\begin{aligned} {\alpha _i}({\theta _i})&= {U_{R{N_i}}}({\theta _i}) - {\beta _i}p_i^{*}({\beta _i},{\theta _i}) + {\theta _i}{T}(p_i^{*}({\beta _i},{\theta _i})) \\&= \bar{U} - {\beta _i}p_i^{*}({\beta _i},{\theta _i}) + {\theta _i}{T}(p_i^{*}({\beta _i},{\theta _i})) + \int _{{\theta _i}}^{{\theta _H}}{T}({p_i^{*}({\beta _i},\tau )})d\tau . \end{aligned}$$

(41)

Then, the source’s expected utility can be rewritten as

$$\begin{aligned} E\left[ U_S \right] = \int _{{\theta _L}}^{{\theta _H}}\mathrm{{ }}[\rho \log (1 + \sum \limits _{i \in \varOmega } {p_i^*({\beta _i},{\theta _i})} ) - \sum \limits _{i \in \varOmega } {(\bar{U} + {\theta _i}T(p_i^*({\beta _i},{\theta _i})) + \mathrm{{ }}\int _{{\theta _i}}^{{\theta _H}} {T(p_i^*({\beta _i},\tau ))d\tau } } ]{f_i}d{\theta _i}. \end{aligned}$$

(42)

By changing the integration order of (42), we have

$$\begin{aligned}&\int _{{\theta _L}}^{{\theta _H}}\int _{{\theta _i}}^{{\theta _H}}T(p_i^*({\beta _i},\tau )){f_i}d\tau d{\theta _i} \\&\quad = \int _{{\theta _L}}^{{\theta _H}}\int _{{\theta _L}}^{{\theta _i}}T(p_i^*({\beta _i},\tau )){f_i}d{\theta _i} \\&\quad = \int _{{\theta _L}}^{{\theta _H}}{H_i}({\theta _i})T(p_i^*({\beta _i},\tau ))d{F_i}({\theta _i}), \end{aligned}$$

(43)

where $H_i(\theta ) = \frac{{F_i(\theta )}}{{f_i(\theta )}}$.

Then, by combining the above two equalities, we have the source’s expected utility

$$\begin{aligned} E\left[ U_S \right] = \int _{{\theta _L}}^{{\theta _H}}\mathrm{{ }}[\rho \log (1 + \sum \limits _{i \in \varOmega } {p_i^*({\beta _i},{\theta _i})} ) - \sum \limits _{i \in \varOmega } {(\bar{U} + {\theta _i}T(p_i^*({\beta _i},{\theta _i})) + \mathrm{{ }} {H_i}({\theta _i})T(p_i^*({\beta _i},\tau ))} ]{f_i}d{\theta _i}. \end{aligned}$$

(44)

At this point, the source’s expected utility optimization problem is simplified to obtain the maximum $E\left[ U_S \right]$ in (44). And we can use an efficient one dimensional exhaustive search algorithm to find the global optimal solution.

Next, simulation result is given to analyze the performance of the optimal contract design method with two RNs ($N_1 = N_2= 1$). The relay cost ${T({p_i})}$ of the ith RN is defined as ${T_i}({p_i}) = {p_i^2}$. And the RN type ${\theta _i}$ is uniform distribution in [0.1, 0.5]. Figure 12 demonstrates that the source’s optimal utility is increasing in the equivalent profit $\rho$, which is similar to that of the discrete types. As $\rho$ increases, the RNs has the more incentive to provide relay communication with the source, and the source obtains the more utility from the RNs’ cooperative relay.

Fig. 12

The source’s optimal utility $U_{S}^*$ versus the equivalent profit $\rho$ for fixed $\bar{U} = 0.1$