1 Introduction
1.1 Motivation
SINCE Nikola Tesla invented the wireless power transfer (WPT) technology, the farfield WPT has been developing very rapidly and has emerged as a promising solution to supply energy to lowpower wireless devices. The farfield WPT allows energy users (EUs) to harvest energy remotely from the radio frequency signals radiated by energy transmitters (ETs) over the air. For example, Powercast has developed energy receivers that can harvest 10 microwatts () RF power from a distance of 10 meters, which is sufficient to power the activities of many lowpower devices, such as wireless sensors and RF identification (RFID) tags [2]. By flexibly adjusting the transmit power and time/frequency resource blocks used by an ET, the WPT can meet the dynamically changing realtime energy demands of multiple EUs simultaneously. The WPT is hence becoming an important building block of the next generation wireless systems [3, 4].
Fig. 1 shows an example of the WPT system, where an ET transmits power on three channels: GSM, 3G, and LTE. There are three EUs, each of which can harvest power on a subset of the channels. Here EUs can be heterogeneous in their channel conditions (due to different distances from the ET), energy consumption rates (due to different applications), and energy harvesting circuits (which result in different channel availabilities). Due to the heterogeneous characteristics, different EUs have different energy demands and value ET’s transmit power on different channels differently. In Fig. 1, for example, EU 3 is likely to have a higher energy demand than EU 2, since EU 3 is more energyhungry with a lower battery status. Moreover, EU 1 is only equipped with energy harvesting circuits for the 3G channel, hence he cannot harvest energy on the other two channels.
To achieve the wide deployment of the WPT technology, one needs to address the potential economic challenges in the future WPT markets. One of such challenges is how an ET should allocate the power across different channels to balance his own operation cost and EUs’ heterogeneous power demands. There has been much excellent prior work tackling this issue from a centralized optimization point of view (e.g., [5, 6, 9, 7, 8, 10, 12, 11]), assuming that EUs are unselfish and will always truthfully reveal their private information (such as the channel state information and harvested power requirements) to the ET. In practice, however, EUs may have their own interests as they may not be directly controlled by the ET, hence they may choose to misreport their private information if doing so can improve their own benefits. For example, if the ET’s goal is to ensure fairness among EUs in terms of their harvested power, an EU can report a smaller channel gain in order to receive more power than what he deserves. To our best knowledge, no existing work has addressed the network performance maximization problem in a WPT network under such a private information setting.
1.2 Challenges
To resolve the issue of private information, it is natural to consider a decentralized market solution (e.g., pricing mechanisms [13] and auctions [14]). For instance, in [14], the mobile users determine their data demands by responding to the market price and hence indirectly reveal their private information. Based on the users’ response to the market price, the mechanism would adjust the price to attain a market equilibrium. Such a mechanism works well in many network resource allocation problems, where each user only receives the benefit from the resource allocated to him. However, this may not work well in the WPT system.
More specifically, the resource in the WPT systems, the wireless power broadcast on each channel, is a nonexcludable public good that is different from many previously considered wireless resources.^{1}^{1}1An example of the nonexcludable public goods is fireworks. First, fireworks are nonexcludable because it is impossible to prevent nonpayers from enjoying them by looking up to the sky. Second, in spite of how many people are looking at them, the fireworks remain the same quality, hence are nonrivalrous. Due to the broadcast nature of wireless signals, each EU’s received power only depends on his channel condition and ET’s broadcast power.^{2}^{2}2 In practice, there are several additional considerations that may affect the public good nature of wireless power. For example, the ET may use directional antennas to enable the directional WPT, which can partially exclude an EU from effectively harvesting the power. In addition, if an EU may choose to move to a location that blocks the reception of another EU. This may affect the nonrivalrous nature. As the first work studying the public good nature of wireless power, we consider a simple scenario with one omnidirectional antenna and fixed channel conditions, and leave the aforementioned considerations for future work. Therefore, one EU harvesting power from the wireless signal does not affect the available energy to other EUs, hence wireless power is nonrivalrous and thus a public good. Furthermore, it is difficult to exclude some EUs from harvesting the energy once the wireless signal is transmitted, hence it is nonexcludable. Hence, under a conventional market mechanism approach, the paying EUs would purchase the wireless power only to a selfsatisfying level, while the remaining nonpaying EUs may silently freeride the wireless power without any payment. This eventually leads to an inefficient wireless power provision. Such a freerider problem does not occur in wireless communication networks with unicast information (not power) transmissions (e.g. [13, 14]), because the unicast information data are private goods.^{3}^{3}3To differentiate the meanings of “private” in different terms, we note that the term “private information” means information asymmetry while the term “private goods” implies excludability and rivalry. That is, they are excludable due to message encryption and rivalrous because the data dedicated to one user typically does not benefit another user.
Although the WPT technology has been extensively studied in the literature, the solution to the economic challenges (including the nature of public goods and the private information) plays an indispensable role in paving the way for widely deployment and commercialization of the WPT networks.
1.3 Solution Approach and Contributions
Among solutions to the public good provision problems in the economics literature, the Lindahl taxation [15] is one of the approaches that can achieve an efficient public good provision in a complete information setting. In the private information setting, a promising solution is the Nash mechanism implementation of the Lindahl allocation. In particular, it achieves the social welfare optimum for the public good economy [16] at every Nash equilibrium (NE). The existing mechanisms ensure economic properties such as budget balance (e.g., [17, 21, 22, 23, 18, 20]).
Nevertheless, there are several issues remaining unaddressed. First, the existing approaches cannot perfectly incentivize agents to voluntarily participate in the mechanism. Without such a property, there may exist freeriders opting out of the mechanism but still benefiting from the public goods [24]. Second, practical wireless power provision problems involve transmit power constraints (such as the maximum transmit power constraint of the ET). However, the existing distributed algorithms only guarantee to converge when there is no public good provision constraint (e.g., [21, 22, 23, 20]). The challenge for such an algorithm design lies in discontinuity of agents’ payoffs, which makes the existing algorithmic approaches inapplicable in our case. This motivates us to propose a twophase allornone scheme with proper economic mechanisms and the distributed algorithms to resolve the above two issues.
For a power transfer scenario with multiple channels (as in Fig. 1
), we need to consider power allocation across different channels. Such a scenario further brings a technical challenge: each EU’s benefit may not be strictly concave in transmit power vector over all channels. Moreover, the power allocation problem is not separable across channels, since each agent’s benefit depends on the transmit power decisions across channels. This feature complicates the design of our mechanism and distributed algorithm. Based on the above discussions, we need to consider the following questions:
Question 1.
How to design mechanisms that incentivize the participation of EUs and maximize the overall benefits of the WPT networks, for both singlechannel and multichannel settings?
Question 2.
How to design algorithms for the ET and EUs to reach an NE, considering the public goods provision constraint?
We summarize our main contributions of this work as follows:

Problem Formulation: To our best knowledge, this is the first work that addresses a wireless resource allocation problem from the perspective of nonexcludable public goods. In particular, we solve the effective WPT provision problem by considering the EUs’ private information and selfish behaviors.

Mechanism Design: We propose a twophase allornone allocation scheme, and design a Power And Taxation (PAT) Mechanism that is significantly simpler than the existing Nash implementation schemes for public goods provision. Our scheme can incentivize the EUs to voluntarily participate in the mechanism, and can achieve several desirable economic properties such as efficiency and budget balance.

Distributed Algorithm Design: For the case of a singlechannel wireless power transfer, we propose a distributed PAT (DPAT) Algorithm under which the decisions of ET and EUs are guaranteed to converge to an NE. We prove its convergence by mapping the NE to the saddle point of the Lagrangian of a related distributively solvable optimization problem. The proof methodology suggests a general distributed algorithm design methodology for computing the NE of our game.

MultiChannel Extension: For the more general case of multichannel transfer, we present a MultiChannel Power and Taxation (MPAT) Mechanism which maintains the properties of the PAT Mechanism. We also design a distributed MPAT (DMPAT) Algorithm and show its convergence to the NE even if agents have nonstrictly concave payoffs.

Performance Evaluation: Compared with a benchmark mechanism, the EUs’ average payoff under our proposed schemes significantly increases in the number of EUs increases. In addition, the proposed schemes are most beneficial when EUs have more homogeneous channels.
We organize the rest of the paper as follows. In Section 2, we review the related work. In Section 3, we introduce the system model and the problem formulation. In Section 4, we propose the constrained Lindahl allocation scheme. We propose the PAT Mechanism and the DPAT Algorithm for a singlechannel system in Section 5. We further propose the MPAT Mechanism and the DMPAT Algorithm for a multichannel system in Section 6. In Section 7, we provide numerical results to validate our analysis. Finally, we conclude our work in Section 8.
2 Related Work
Most of the early studies on WPT networks focused on system optimization with unselfish users (e.g., [5, 6, 7, 9, 8, 10, 12, 11]), where ETs and EUs are willing to truthfully report their private information and follow the optimization result. Specifically, references [5, 6] proposed efficient schemes to optimize the longterm ET placement, and references [7, 9, 8, 10, 12, 11] considered the wireless resource allocation in WPT networks to optimize the communication performance.
To our best knowledge, there is only one recent work considering the gametheoretical analysis of the power provision problem in WPT networks with selfish EUs[25]. Specifically, Niyato et al. in [25] formulated a bidding game for a simple WPT system and analyzed the NE of the game. The equilibrium of the game does not maximize the system performance. Our work differs from [25] in that we aim to achieve a socially optimal system performance through mechanism design.
There are several related works on Nash implementation for public goods (e.g., [17, 18]). Specifically, Hurwicz in [17] presented a Nash mechanism that yields the social optimum for a public good economy, and the mechanism also ensures individually rationality and budget balance. Sharma et al. in [18] extended the results in [17] to a more general local public goods scenario with the CDMA networks as an example. However, references [18, 17] did not consider how the agents should iteratively update the messages to converge to the NE under private information. Another related study [19] considered the network security investment as a nonexcludable public good and studied a mechanism. However, the studied scheme cannot always achieve the efficient public good provision as we do in our paper.
Only few papers proposed public goods mechanisms together with the corresponding updating processes that converge to the NE [21, 22, 23, 20], where the best response dynamics [20, 21, 22] and the gradientbased dynamics [23] can provably converge to the NE under some technical conditions. However, these prior algorithms are not directly applicable in our model. This is because algorithms in [21, 22, 23, 20] relied on the continuity of the best response dynamics or the gradientbased dynamics. Our constrained public good setting introduces the discontinuity, and hence requires new distributed algorithms for the constrained public good provision problem (due to the ET’s transmit power constraint).
Moreover, all the above studies did not consider the voluntary participation issue for nonexcludable public goods. In our work, we will show that if the ET knows the total number of EUs, then a twophase scheme can ensure voluntary participation.
3 System Model and Problem Formulation
In this section, we introduce the system model that captures several unique characteristics of the WPT problem. Accordingly, we formulate the public good provision problem, with the goal to maximize the social welfare.
3.1 System Model
ET and EUs: We consider a WPT system consisting of one ET transmitting power over channels (bands) to a group of EUs.^{4}^{4}4We consider lowmobility EUs such as sensors and IoT tags that are typically deployed as fixed locations. Furthermore, we consider the wireless power transfer by dedicated base stations: each having a fixed and stable coverage range and operating independently with others. Such an arrangement can help ensure good QoS for the EUs, i.e., their received power levels can be ordersofmagnitude higher than that achieved by the traditional environmental RF energy harvesting, where each device harvests from multiple sources randomly and has to leverage mobility to improve average power harvested. Let be the set of channels and be the set of EUs. We further use agent to denote the ET, hence the total set of decision makers in the system is . For the purpose of presentation, we will refer to the ET as “she” and an EU as “he”. The ET has an omnidirectional antenna and transmits wireless energy over the channels.^{5}^{5}5For the directional multiantenna WPT and/or multiET networks, it is possible to extend our idea by further formulating a multiple public goods provisioning problem. We will consider it in the future work. We discuss the potential multiET extensions in Appendix C.2. Each EU has one energy receiver, but can potential receive energy from multiple channels. Different EUs can experience different timevarying channel conditions due to shadowing and fading. We consider a time period long enough such that the smallscale channel fading effects are averaged out and the channel conditions are stationary.
Power and Cost: The ET transmits a power of (Watts) over channel . We denote as the ET’s transmit power vector of all channels in . The ET incurs a cost of , which is a positive, increasing, continuously differentiable, and (not necessarily strictly) convex in . The cost function can capture, for example, both the energy consumption cost and the maintenance cost for the ET’s operation. A strictly convex cost function may capture an increasing marginal cost of power transmission, which is due to several reasons such as
Transmit Power Constraints of the ET: Let represent the ET’s total power constraint over all channels, and let represent the ET’s peak power constraint for channel . These two types of constraints capture the limitation of the physical circuits and regulations.^{6}^{6}6For example, Federal Communications Commission (FCC) in the US imposes a Maximum Effective Isotropic Radiated Power (EIRP) of 4 W (36 dBm) in the band of 902 to 928 MHz [30]. The transmit power thus lies in the set of
(1) 
Both and are ET’s private information and are not known by the EUs.
Utility of EUs: For EU , let denote his channel gain over channel , let denote his overall channel gain vector, and let denote the received ambient wireless power apart from the the ET’s transmitted wireless power (e.g. the wireless signal generated by the cellular networks). Here we assume that represents EU ’s longterm average channel gain. Given the ET’s transmit power , EU ’s total received power is
(2) 
Each EU has a utility function , which is strictly concave, increasing, and continuous, which can reflect

an EU’s overall valuation of the received power considering his battery status and energy consumption level;

the nonlinear energy conversion efficiency of the physical circuits, such as the (concave) logistic functions [31];
Throughout this paper, we assume for all EUs without loss of generality.^{7}^{7}7This is because for any arbitrary positive and , we can always find another increasing, strictly concave, and continuous utility function such that for all . Both and are EU ’s private information and are not known by the ET or other EUs.
3.2 Problem Formulation
We assume that a network regulator operates the ET and aims to optimize the system performance. Our analysis also applies to the case where the ET is a selfinterested decision maker. This is because the ET also participates in the PAT Game defined in Section 5 (or the MPAT Game defined in Section 6) as a rational player to maximize her payoff.^{8}^{8}8However, in the case where the ET is a selfinterested decision marker, we need to introduce a thirdparty network regulator, who serves as a coordinator that collects the price and power proposals from the agents, for implementing the DPAT Algorithm and the DMPAT Algorithm. It is possible to further design profitmaximizing mechanisms by integrating the existing profitmaximizing mechanisms for private goods (e.g. [33]) and our proposed mechanisms. We will study this interesting direction in the future work. Specifically, the ET is interested in choosing the transmit power to solve the following Social Welfare Maximization (SWM) Problem
(3a)  
(3b) 
The objective function of the SWM Problem is concave and the constraint set is compact and convex.
To solve the SWM Problem in a centralized fashion, the ET needs to know the complete information of the EUs (i.e., utility functions and for all ). This is difficult to achieve in practice, since EUs may not want to report their utility functions or their channel gains, as doing so may not maximize their own benefits. Hence, we need to design economic mechanisms to solve the SWM Problem, considering EUs’ private information and the public good nature of wireless power.
3.3 Desirable Mechanism Properties
In this paper, we will design an economic mechanism that satisfies the following four desirable economic properties:

(E1) Efficiency: Maximizes the social welfare, i.e., achieves the optimal solution of the SWM Problem.

(E2) Incentive compatibility: An EU should (directly or indirectly) truthfully reveal his private marginal utility.

(E3) Voluntary participation: An agent should get a nonnegative payoff by participating in the market mechanism.

(E4) (Strong) Budget balance: The total payment from the EUs equals the revenue obtained by the ET. In other words, if the mechanism is administrated by a thirdparty, then there is no need to inject money into the system.
We would like to clarify some aspects with respect to the existing mechanism design literature. First, for a direct mechanism, (E2) usually indicates that the mechanism can fully reveal the participants’ utility functions. However, [34] proved that there does not exist any public goods mechanism that satisfies (E1) and induces direct and truthful revelation at the same time. Hence, we focus on the indirect revelation, in the sense that the mechanism reveals EUs’ marginal utility at NEs.
Existing results on Nash mechanisms have focused on achieving (E1), (E2), and (E4), without considering (E3) [17, 20, 21, 22, 23, 18]. The common assumption in these prior work is that there is a “government” that has enough power to force all agents to participate in the mechanism [17, 20, 21, 22, 23, 18]. However, such mandatory participation may not be easily enforceable in reality.^{9}^{9}9In Japan, for example, it is mandatory to pay a TV public broadcasting fee if one owns a device that can receive a TV broadcast signal. The punishment for nonpaying individuals, however, is practically nonexistent due to the difficulty of enforcement.
Saijo and Yamato in [24] showed that there does not exist a mechanism that guarantees (E1), (E3), and (E4) simultaneously for nonexcludable public goods. Nevertheless, the key assumption in [24] is that the consumers (which correspond to EUs in our model) can directly access the production technology and produce the nonexcludable public goods themselves (even without participating in the mechanism). This assumption does not hold in our model, as we assume that EUs cannot access other energy sources other than the ET. Hence it is possible to design a mechanism that achieves (E1)(E4), as shown in Section 5.
4 Constrained Lindahl Allocation Scheme
Before explaining our proposed mechanism, we first propose a constrained Lindahl allocation scheme in this section. We then show that, in a complete information setting, such an allocation scheme can lead to the selfenforcing and socially optimal transmit power in the WPT systems. Under incomplete information, we will further design mechanisms to implement the constrained Lindahl allocations (in Sections 56).
Consider a linear taxation scheme, under which there is a tax rate for each agent on channel . Let denote the tax rate vector for agent . Based on the corresponding tax rate, each EU pays to the ET; the ET receives a reimbursement . For presentation simplicity, we let denote the payment for the ET.
Suppose that the agent can determine the transmit power considering his/her payment . Agent aims to find the transmit power to solve the payoff maximization problem:
(4) 
For each EU , we have that the optimal solution satisfies^{10}^{10}10It is possible that there does not exist an optimal solution if is poorly designed (e.g. some for some ). However, any optimal solution, if exists, should satisfy (4).
(5) 
In the following, we introduce a specific taxation scheme, namely Lindahl tax [15]. Such a taxation scheme incentivizes agents to agree on provisioning the public goods to the socially optimal level, which constitutes a Lindahl allocation. However, the Lindahl tax scheme in the existing literature does not consider the public good provision constraint. Therefore, we propose the constrained Lindahl allocation scheme as follows.
Scheme 1 (Constrained Lindahl Allocation Scheme).
The constrained Lindahl allocation is a tuple :

The transmit power vector is an optimal solution to the SWM Problem.

Lindahl tax rate for each agent satisfies

, for each EU ;

, for the ET.

Intuitively, being charged the corresponding constrained Lindahl tax , each EU ’s optimal solution to (4) is the socially optimal transmit power , as shown from (5). Similarly, we can also show that the socially optimal transmit power is also the optimal solution to the ET’s problem in (4). This indicates that under the constrained Lindahl tax, every agent prefers the socially optimal transmit power to all other power choices.
In Fig. 2, we present illustrative examples of the constrained Lindahl allocation scheme for a singlechannel WPT system, with one ET and two EUs. The horizontal axis corresponds to the transmit power. The vertical axis represents each agent’s marginal utility (for the EUs) or cost (for the ET). Fig. 2(a) describes the case where the maximal transmit power is sufficiently large. In this case, from (3), the socially optimal transmit power corresponds to the intersection of the ET’s marginal cost function and the two EUs’ aggregate marginal utility functions . Fig. 2(b) corresponds to the more complicated case where is not large enough. More specifically, is smaller than the intersection point of the two curves and , and the socially optimal transmit power equals in this case.
As shown in both Fig. 2(a) and Fig. 2(b), the constrained Lindahl tax rate for EU is exactly his marginal utility at the optimal transmit power. Note that if we substitute into in (4), we have from (5). That is, each EU’s best power purchase under the constrained Lindahl tax rate is to choose the optimal transmit power . In Fig. 2(a), when the constraint is slack, ET’s reimbursement per unit of power is , which is equal to her marginal cost . From (4), we see that . In Fig. 2(b), however, the constraint is tight. This means that ET’s reimbursement is larger than the ET’s marginal cost , which means that the ET aims to transmit as much as possible, i.e., .
The interpretation of the constrained Lindahl allocation scheme is hence as follows. Under his constrained Lindahl tax rate, each agent experiences a “personalized market”. That is, each agent sees himself/herself capable of deciding the total amount of transmit power he/she will purchase/transmit (through solving (4)). In each agent’s personalized market, the socially optimal transmit power is the optimal decision according to (4). Therefore, the constrained Lindahl allocation scheme enables all agents to agree on the social optimum.
The above discussions assume that the ET knows the complete information regarding EUs’ utility functions. However, in reality, the ET cannot readily obtain such information. As a result, each EU has an incentive to underreport his utility to the ET to reduce his payment. This leads to the freeriding EUs. The underreported utilities may distort the taxation scheme and lead to an inefficient transmit power (comparing with the socially optimal value). The consideration of such a realistic incomplete information scenario motivates us to design new mechanisms in Section 5 for the singlechannel scenario and in Section 6 for the multichannel case, to achieve the constrained Lindahl allocation at an equilibrium without relying on complete network information.
5 SingleChannel Nash Mechanism
In this section, we start with a singlechannel scenario with incomplete information. For a better readability, we will drop the index in and in this section. We can express the feasible transmit power region as , where is the ET’s maximum transmit power. We further focus on the case where there are EUs. Appendix C discusses the singleEU case.
We propose a twophase allornone scheme, including a PAT Nash Mechanism in Phase II. We show that under the proposed scheme, the agents’ equilibrium strategies coincide with constrained Lindahl allocation (given in Definition 1) and achieve (E1)(E4). We further propose a DPAT Algorithm that converges to an NE of an induced game, of which the challenge mainly lies in the transmit power constraint.
5.1 TwoPhase AllorNone Scheme
We propose a twophase allornone scheme, as shown in Fig. 3. In Phase I, each EU sends a 1bit message to the ET indicating whether or not he will participate in the Power and Tax (PAT) mechanism (to be described in Section 5.2). In Phase II, if all agents are willing to participate, the ET and the EUs will execute the PAT Mechanism in Phase IIY; otherwise, the ET will transmit zero power and no trading occurs in Phase IIN. Here we adopt the following assumption throughout the paper:
Assumption 1.
The ET knows the total number of EUs, .
Under Assumption 1, the ET will know whether some EU keeps silent without sending any indication at the end of Phase I. Assumption 1 is satisfied, for example, when all EUs also actively transmit information (to the ET or other EUs) (e.g., in wireless sensor networks) and hence can be detected by the ET[3]. Even for passive (silent) EUs, it is possible to detect their existence from the local oscillator power inadvertently leaked from their (receiving) communication circuits [35]. We provide the detailed discussions of a potential direction to extend our mechanism to the case of the unknown in Appendix C.3.^{11}^{11}11 If Assumption 1 is not satisfied, the results in the literature assert that no mechanism can achieve the properties (E1)(E4) simultaneously [24].
5.2 Nash Mechanism
Next, we describe the PAT Mechanism to be executed in Phase IIY.
Mechanism 1.
Power And Taxation (PAT) Mechanism

The message space: Each agent sends a message to the ET:
(6) where and are agent ’s power proposal and price proposal, respectively. Note that the ET (agent 0) also needs to send a message (to herself). We denote the message profile as .

The outcome function: The ET computes the transmit power based on the agents’ power proposals:
(7) The ET further computes the tax rate for agent based on the agents’ price proposals:
(8) where , and is the modulo operator.^{12}^{12}12For example, when , we have = . The ET will announce to agent , and the agent needs to pay the following tax to the ET,
(9)
Next we mention several key features of the PAT Mechanism. First, the determination of the transmit power in (7) depends on every agent’s power proposal. Second, agent ’s tax rate in (8) does not depend on his own price proposal . Finally, the agents’ taxes in (9) cancel each other, i.e.,
(10) 
which means that the mechanism achieves the budget balance property (E4).
The PAT Mechanism is motivated by the Hurwicz mechanism [17]. In [17], each agent can only select the price proposal from , and the tax function is computed in a complicated function form: . In our proposed scheme, the power is computed in the same way in [17] as in (7). A key contribution of this paper is that our proposed PAT Mechanism is considerably simpler than the Hurwicz mechanism in [17] but achieves the same desirable economic properties, as explained next.
5.3 Properties of the PAT Mechanism
In this subsection, we will prove that the PAT Mechanism achieves the economic properties of (E1) and (E3). Specifically, we will first analyze agents’ decisions in Phase II, assuming that every agent chooses to participate in Phase I. Then we return to Phase I to analyze agents’ participation decisions.
5.3.1 Analysis of Phase II
The PAT Mechanism induces a game among agents in Phase II, which we simply refer to as the PAT Game.
Game 1.
PAT Game (Induced by the PAT Mechanism in Phase II)

Players: all agents in .

Strategy: described in (6) for each agent .

Payoff function for each EU
(11) for the ET (agent 0)
(12)
The value can be interpreted as an infinite penalty for the ET if she violates the maximum power constraint. Note that the ET’s payoff function in (12) is discontinuous due to the constraint . This leads to a key challenge for the distributed algorithm design discussed later in Section 5.4.
Definition 1 (Nash Equilibrium (NE)).
An NE of the PAT Game is a message profile that satisfies the following condition:
(13) 
where is the NE message profile of all other agents except agent .
Traditionally, an NE describes the agents’ stable strategic behaviors in a static game with complete information [37]. However, the agents in our WPT system do not know the private information (i.e., utilities, cost, or the transmit power constraint) of others. Here, we adopt the interpretation of [38], i.e., NE corresponds to the “stationary” messages profile of some message exchange process (to be described later in Section 5.4) that possesses the equilibrium property in (13).
To analyze an NE of the PAT Game, we summarize the sufficient and necessary conditions for an NE in Lemma 1, with its proof presented in Appendix A.2.
Lemma 1.
A message profile is an NE if and only if the following conditions are satisfied,
(14) 
for every agent , where is the NE tax rate for agent .
To understand Lemma 1, let denote the NE transmit power. From (7), we have . This allows us to rewrite (14) as,
(15) 
Equation (15) implies that under the NE tax rates , the common NE transmit power maximizes every agent’s payoff. Otherwise, one agent would have the incentive to adjust to change and improve his payoff. Therefore, an NE only occurs when all agents agree on the same transmit power.
We can show that there are multiple NEs for the PAT Game. To see this, given any , we can add every by the same constant, while the new message profile still satisfies the conditions described in (15) and thus is also an NE. However, we can show that the NE allocation is the same for all NEs, where corresponds to the unique optimal solution of the SWM Problem. We can show that all NEs yield the unique constrained Lindahl Allocation in the following theorem (with its proof presented in Appendix A.3):
Theorem 1 (Implementing Constrained Lindahl Allocation).
There exist multiple NEs in the PAT Game, and all NEs correspond to the unique constrained Lindahl allocation.
Proving the existence of NEs involves constructing an NE based on the optimal solution to the SWM Problem. Intuitively, the tax rates defined in (8) ensure that the sum of all tax rates is zero, i.e., . Together with (15), we can derive the KarushKuhnTucker (KKT) conditions for the SWM Problem based on all agents’ optimality conditions of (15). The remaining part of the proof for Theorem 1 involves showing that the NE condition in (15) can lead to a unique tax rate for each agent.
The significance of Theorem 1 is threefold. First, Theorem 1 shows that every NE of the PAT Game induced by the PAT Mechanism yields the socially optimal transmit power level as suggested in Definition 1. Second, Theorem 1 implies that the PAT Mechanism is incentivecompatible (E2). This is because the NE tax rates are the Lindahl taxes, which reveal every EU’s marginal utility at the NEs by Definition 1. Third, each agent receives the same payoff at every NE due to the uniqueness of the Lindahl allocation. This means that each agent is indifferent to the choice among multiple NEs.
However, there should be an effective approach of selecting one NE from the multiple ones. Without such an agreement, the agents’ distributed choices may not lead to an nonNE message profile. We will resolve the above issue through a distributed algorithm design in Section 5.4.
5.3.2 Analysis of Phase I
We now proceed to analyze agents’ decisions in Phase I, where each agent compares the unique NE allocation (where everyone participates the PAT Mechanism) to the allocation (where at least one agent chooses not to participate). We can show that each EU voluntarily participates, with its proof in Appendix A.4.
Theorem 2 (Voluntary Participation).
All agents will not be worse off by participating in the PAT Mechanism.
Theorem 2 implies that all EUs choose to participate in the PAT Mechanism in Phase I. The intuition is that, given arbitrary messages from other agents in the PAT Game, an EU or the ET can always choose a power proposal so that the transmit power is zero (hence his tax is zero due to (9)). Such a choice is equivalent to the outcome where someone chooses not to participate in Phase I. Hence choosing to participate in Phase I is a weakly dominant strategy for each EU. Here we assume that each EU will voluntarily participate if he is not worse off by doing so. This is without loss of generality, since in practice we can let the ET offer an additional arbitrarily small amount of benefit to every EU who chooses to participate to break the tie. In other words, this ensures that every EU can receive a strictly positive payoff improvement by choosing to participate. In addition, successfully inducing voluntary participation also relies on one of the fundamental assumptions of neoclassical economics: agents make decisions rationally; irrational EUs may make the allornone scheme fragile. We may tackle this issue using theories from behavioral economic (e.g., cognitive hierarchy [36]).
We note that the EUs’ participation involves the energy consumption due to the communication overhead of the proposed algorithms. In this paper, we assume that such participation energy overhead is negligible for each EU. Specifically, the participation process is of a finite duration and the energy cost is onetime only. Once the ET decides the policy, the resultant wireless power transfer is over a much longer duration. Therefore, the energy harvested is much larger than the energy overhead for each EU (if he decides to participate) and thus we ignore the latter in our work for simplicity.
To summarize, we have shown that the twophase allornone scheme and the PAT Mechanism together can achieve the desirable economic properties of (E1)(E4). We will next propose a distributed algorithm, under which the agents can achieve the NE of the PAT Game.
5.4 Distributed Algorithm to Achieve the NE
As we mentioned previously, the private information setting and the NE selection issues make it difficult for the ET and EUs to directly compute their messages at an NE. Hence, we will propose an iterative distributed algorithm for the ET and EUs to exchange information and compute the NE. To prove the convergence of the algorithm, we will establish the connection between the NE of the PAT Game and the optimal primaldual solution of a reformulation of the SWM Problem.
(16)  
(17)  
Algorithm 1 illustrates the proposed iterative DPAT Algorithm, with the following key steps. Each agent initializes his arbitrarily chosen message (line 1). Then, the algorithm iteratively computes the messages until convergence. In each iteration, first each EU sends his message to the ET (line 1). Then the ET computes each agent ’s tax rate , and sends together with agents and ’s price proposals (lines 11) to EU . Accordingly, each agent updates his power proposal and his price proposal (line 1), where . Finally, the ET checks the termination criterion (line 1). The termination happens if both the relative changes of agents’ power proposals and price proposals are small, determined by the positive constants and . The ET finally computes the transmit power and taxes (line 1).
Next, we discuss in details regarding the updates of messages in line 1. First, for the power proposal update in (16), each agent selects the power proposal equal to the transmit power that maximizes his payoff. For each EU, we impose a power upper bound (which is a large enough constant^{13}^{13}13For example, an EU can set the upper bound to be the maximal transmit power of the local TV broadcast (e.g. kW for the TV Tokyo).), such that the power proposal does not go to infinity (which can happen when the tax rate is negative and there is no power upper bound). Second, the price proposal update in (17) is motivated by Lemma 1, which suggests that the NE transmit power should maximize every agent ’s payoff. As we can see, a larger increases EU ’s tax rate and decreases ’s tax rate, respectively, hence may reduce the gaps in their power proposals according to (16).
Finally, we discuss the synchronization and overhead issues of the DPAT Algorithm. First, the DPAT Algorithm should be executed in a synchronous fashion, which requires a common clock of all agents and a negligible delay for passing messages. This can be achieved in a practical WPT network, since the ET and the EUs are often physically closeby. Second, the distributed algorithm has small communication and computation overheads. Specifically, each agent needs to send and to the ET; the ET needs to send each EU her tax rate and two other agents’ proposals, for all . Hence the communication overhead is per iteration. The computational complexity per iteration is for each EU and for the ET, since she computes the tax rates for all EUs with a complexity of .
5.5 The Convergence of the DPAT Algorithm
There are two classes of existing dynamics that have been shown to converge to the NE of various public good provision mechanisms: the bestresponse dynamics (e.g. [21, 22]) and the gradientbased dynamics (e.g. [23]). These existing approaches, however, all assume that there are no constraints on public good provision. This assumption does not hold in our model, since we need to consider the maximum total transmit power constraint. Hence we need to find a new way to prove the convergence of our proposed DPAT Algorithm.
The approach we take is to first reformulate the SWM Problem with a decomposition structure, then connect the saddle point of the Lagrangian of the reformulated problem and the NE of the PAT game. We will show that the DPAT Algorithm converges to a saddle point and thus an NE of the PAT game.
5.5.1 Problem Reformulation
Inspired by [39], we reformulate the SWM Problem by introducing auxiliary variables , which decouple agents’ utility and cost functions:
(18a)  
(18b)  
(18c) 
We can verify that the RSWM Problem is equivalent to the SWM Problem and has a unique optimal solution.
5.5.2 Lagrangian
5.5.3 Dual Decomposition
The Lagrangian in (19) has a nice dual decomposition structure, i.e., , where is the decomposed Lagrangian for each agent as follows,
(20) 
Define , where . Thus, the dual problem of the RSWM Problem is
(21) 
We define the saddle point of as a tuple that satisfies:
(22) 
For such a saddle point, we can show that is the unique optimal solution to the RSWM Problem and is the optimal solution to the dual problem in (21) [40, Chap. 5.4].^{15}^{15}15There are multiple optimal dual solutions . To see this, given any saddle point of , we can add every by the same constant, and the new tuple still satisfies the conditions described in (22) and thus is also a saddle point.
5.5.4 Relation between the Saddle Point and the NE
If we set and , for all agents , then in the PAT Game the in (11)(12) becomes exactly the decomposed Lagrangian , i.e.,
(23) 
Proposition 1 characterizes the relation between a saddle point for the Lagrangian in (19) and an NE of the PAT Game.
Proposition 1.
For any saddle point defined in (22), the message profile is an NE of the PAT Game.
We present the proof of Proposition 1 in Appendix A.5. Intuitively, Lemma 1 asserts that an NE only occurs if all agents have the same payoffmaximizing transmit power, given the equilibrium tax rate . On the other hand, we attain the optimal dual solution only when the maximizer of the Lagrangian satisfies the equality constraint in the constraint in (18b). Together with the relation of and in (23), we can see that Proposition 1 holds.
The significance of Proposition 1 is twofold. First, Proposition 1 provides a new interpretation of the messages of the PAT Mechanism: the power proposal for each agent plays a role of the auxiliary variable, while the price proposal plays a role of the consistency price that pulls the auxiliary variables together.
Second, Proposition 1 also implies that for any distributed algorithm with a provable convergence guarantee to a saddle point of the Lagrangian in (19), we can design a corresponding distributed algorithm that converges to an NE of the PAT Game. This property allows us to exploit the convergence properties of welldesigned optimization algorithms that the traditional approaches in [21, 22, 23, 20] may not possess. Such a property also facilitates overcoming the additional technical challenge introduced in the multichannel model in Section 6.
We are ready to show the convergence of the DPAT Algorithm in the following theorem with the proof in Appendix A.6.
Theorem 3.
When is strictly convex and the step size is diminishing^{16}^{16}16An example of the diminishing step size is for a constant ., the DPAT Algorithm converges to a saddle point of the Lagrangian in (19), hence an NE of the PAT Game.
The proof of Theorem 3 involves showing that the DPAT Algorithm is the gradient method for solving the dual problem in (21). We can guarantee its convergence [41] if we employ the bounded gradients, which is satisfied due to the bounds in (16).
Note that the gradient method requires the strict concavity of each decomposed Lagrangian . Thus, a linear cost function cannot meet this requirement. However, we can adopt the algorithm to be introduced in Section 6.2, which guarantees its convergence even if is linear.
6 MultiChannel Nash Mechanism
We now turn to the problem for a general multichannel WPT network, where the ET can transmit over orthogonal channels. The multichannel WPT network brings a new consideration of allocating power across available channels. We will consider the multichannel extension to achieve (E1)(E4).
Such a new algorithm design is nontrivial, because each agent’s payoff function couples the transmit power decision across all channels. In addition, agents’ payoff functions may not be strictly concave. Note that even if is a strictly concave in , it may not be strictly concave in when for some channel . For example, consider a system with channels and (i.e., EU only operates on channel ). The Hessian matrix of EU ’s utility function with respect to is given by For every , the Hessian is negative semidefinite but not negative definite. Hence, EU ’s utility is concave but not strictly concave in . Thus, we cannot directly adopt a gradientbased algorithm similar to the DPAT Algorithm. Instead, we consider an algorithm based on the augmented Lagrangian method to distributively compute the NE.
6.1 Nash Mechanism
In this subsection, we design the Nash mechanism for the multichannel network. We then show the proposed twophase allornone scheme together with a new mechanism achieves the economic properties (E1)(E4) for the multichannel network.
We propose Mechanism 2, which is a generalization of the PAT Mechanism. Specifically, each agent submits a message for every channel even if he can only operate on a subset of all channels.
Mechanism 2.
MultiChannel Power and Taxation (MPAT) Mechanism

The message space: Each agent sends a message to the ET of the following form:
(24a) (24b) where and are agent ’s power proposal and price proposal for channel , respectively. We denote the message profile as .

The outcome function: The ET announces the transmit power on each channel to every agent:
(25) The ET further computes the tax rate for agent based on the agents’ price proposals: for every and every ,
(26) The ET announces EU ’s tax : for every and every ,
(27)
Equation (27) implies that . Thus, the MPAT Mechanism achieves the budget balance (E4). Similarly, the MPAT Mechanism induces the following MPAT Game among agents in Phase II.
Game 2.
MPAT Game (Induced in Phase II)

Players: all agents in .

Strategy: described in (24) for each agent .

Payoff function for each EU ,
(28) for the ET (agent 0),
(29)
Different from the singlechannel scenario, the multichannel scenario may not admit a unique constrained Lindahl allocation. This is mainly due to the nonstrict concavity of the objective of the SWM Problem (3a). However, we present the following theorem with the proof in Appendix A.7:
Theorem 4.
Each agent receives the same payoff across different constrained Lindahl allocations.
We prove Theorem 4 by establishing the uniqueness of the received power for each EU, which leads to the uniqueness of each agent’s utility/cost and each agent’s total tax. This theorem indicates that each agent is insensitive to different constrained Lindahl allocations.
In the light of Theorem 4, we can show that the MPAT Mechanism achieves properties (E1)(E3), with proofs in Appendices A.8 and A.9, respectively.
Proposition 2 (Implementing Constrained Lindahl Allocations).
There exist multiple NEs in the MPAT Game, and each NE corresponds to a constrained Lindahl allocation.
Proposition 3 (Voluntary Participation).
Each agent will participate in the MPAT Mechanism in Phase I.
To summarize, we have shown that every agent is indifferent to the choices of the NEs. Moreover, the twophase allornone scheme and the MPAT Mechanism can achieve the desirable economic properties (E1)(E4) for the multichannel system. We next introduce the distributed algorithm.
6.2 Distributed Algorithm to Achieve the NE
In this subsection, we design the distributed algorithm for agents to compute an NE of the MPAT Game. We have shown that every NE leads to a constrained Lindahl allocation (Proposition 2) and all constrained Lindahl allocations are equivalent for every agent (Theorem 4). Therefore, agents just need to agree on reaching any of the NEs.
However, we cannot directly adopt a dual gradientbased algorithm similar to the DPAT Algorithm, which requires the strict concavity of every agent’s payoff function. Instead, we propose an alternative approach in Algorithm 2, which ensures the convergence even if an agent’s payoff is not strictly concave. We will show Algorithm 2 is based on the Accelerated Distributed Augmented Lagrangians (ADAL) method [42] and prove its convergence in Section 6.3.
(30) 
(31) 
(32)  
Algorithm 2 illustrates the proposed iterative DMPAT Algorithm. The key difference compared with Algorithm 1 mainly lies in the updated of messages, as described in the following. First, for the power proposal update (30) (line 2), each agent maximizes his/her payoff minus a quadratic penalty (due to inconsistency with the agent ’s power proposals); each agent updates the power proposals by (31).^{17}^{17}17Here the parameter is the step size, which should be chosen in the interval , where is the number of agents coupled in the “most populated” constraint of the problem [42]. As we can observe, in our case, so we set . One main benefit of including the penalty term is that it ensures a unique solution of (32), therefore admits gradientlike updates of proposals as in (30) and (31). This resolves the drawback of the DPAT algorithm of requiring the strict concavity to make the gradientbased algorithms feasible. Second, the price proposal update in (31) is similar to the DPAT algorithm, which is designed to reduce the gaps of their power proposals according to (30).
Similar to the DPAT Algorithm, the DMPAT Algorithm should be executed in a synchronous fashion. In addition, the distributed algorithm has a small communication complexity of per iteration.
6.3 Convergence of the DMPAT Algorithm
Similar to the approach in Section 5.5, we prove the convergence of the DMPAT Algorithm by reformulating the SWM Problem in (3a)(3b). Then we demonstrate the connection between the saddle point of the augmented Lagrangian of the reformulation and the NE of the MPAT game. We next show that the DMPAT Algorithm is an ADALbased algorithm that converges to a saddle point and thus an NE of the MPAT game.
6.3.1 Problem Reformulation
We reformulate the SWM Problem by introducing auxiliary variables , where . They will help decouple agents’ utility/cost functions in the following reformulated problem:
(33a)  
(33b)  
(33c) 
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