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Incentive-Compatible Packet Forwarding in Mobile Social Networks via Evolutionary Game Theory Li Feng, Qinghai Yang and Kyung Sup Kwak Abstract—In the absence of end-to-end paths and without the knowledge of the whole network, packet forwarding including forwarding decision (i.e. forwarding or dropping the packet) and relaying selection is crucial to be made by the individual of the node based on the packet-forwarding protocol in autonomous mobile social networks (MSNs). In this paper, we investigate the adaptive packet forwarding in MSNs afflicted with potential selfish nodes. When considering the various selfish behaviors of network nodes in multi-hop MSNs, an incentive compatible multiple-copy packet forwarding (ICMPF) protocol is proposed to maintain a satisfied packet delivery probability whilst reducing the delivery overhead. Considering the fact that the node’s forwarding decision in the ICMPF protocol is affected by its available resources (i.e. bandwidth, location privacy) and network environment (i.e. other nodes’ actions, social ties), an evolutionary game framework is exploited for modeling the complicated interactions among nodes to guide their forwarding behaviors. Meanwhile, we portray the forwarding behavior dynamics and develop the evolutionary stable strategy (ESS) for this gametheoretic framework. Then, we prove that the strategy dynamics converge to the ESS and further develop a distributed learning algorithm for nodes to approach to the ESS. Simulation results show that our system converges to the ESS and also is robust to the learning error induced by the communication noise. Key Words: selfishness, evolutionary game theory, multiplecopy packet forwarding protocol.

I. I NTRODUCTION In autonomous mobile social networks (MSNs), nodes always move, and even there is no knowledge of future transmission opportunities, causing that a new routing paradigm, “store-carry-and-forward”, has been proposed [1]. In order to simplify the analysis in this paradigm, traditionally it always assumes that nodes forward packets in a cooperative and altruistic way [2]. However, in the real world, some or all autonomous individuals may exhibit various selfish behaviors when forwarding packets, particularly in MSNs, where nodes are constrained with limited resources. Such selfish behaviors largely degrade the packet forwarding quality [3]. Therefore, it is crucial to develop a novel packet-forwarding protocol, which enables the packets to be forwarded with a high packet delivery probability but a low information overhead in mobile network environment, while considering various selfish behaviors of nodes in MSNs. *This research was supported in part by NRF of Korea (MSIP-)(NRF2014K1A3A1A20034987) and NSF of China (61471287). L. Feng, Q. Yang are with State Key Laboratory on ISN, School of Telecommunications Engineering, Xidian University, No.2 Taibainan-lu, Xi’an, 710071, Shaanxi, China. (Email:[email protected]). K. S. Kwak is with the Graduate School of Information Technology and Telecommunications, Inha University, #253 Yonghyun-dong, Nam-gu, Incheon, 402-751, Korea. (Email: [email protected]).

There are many challenges addressed for the packet forwarding in the MSNs with selfish nodes. Firstly, due to node’s mobility and network’s non-infrastructure, network topology may change rapidly and the existing deterministic routings [4] are not applicable. Secondly, the node is selfish and may refuse to forward the received packets in order to conserve its limited resources or to protect its location privacy, which may reduce the packet delivery probability. Hence, an incentive mechanism should be raised for promoting the coordination of potential selfish nodes in order to ensure a satisfied packet transmission. Thirdly, since various social relationships exist among nodes in the MSNs where nodes with closer relationships to destination are more likely to forward the packet, we shall consider various social ties among nodes when designing the packet forwarding scheme. Besides, in traditional routing protocol [5], the packet forwarding serves as a centralized way where the source builds the end-to-end communication by collecting the network nodes’ state information. Such an approach is not fit for MSNs with selfish nodes since it brings in the significant communication overhead and the node is self-interest who takes action based on its own profit. Accordingly, it makes much sense to design a distributed scheme under continuous topology changes for autonomous node to decide its own forwarding strategy according to local information. In this paper, we investigate the node’s adaptive packet forwarding including forwarding decision and relay selection in MSNs with selfish nodes when considering network node’s internality, such as the limited resources, mobile characteristic, social tie and privacy sensitivity. In such a network, due to the complicated communication environment and nodes’ behaviors affecting each other in the packet forwarding process, the node needs to explore its optimal forwarding strategy under changing strategic environment based on its preference. Non-cooperative game theory is well fit for packet forwarding to deal with the conflicting interests among selfish nodes in wireless networks. However, for the traditional game theory where each player needs to know opponents’ strategies, when the number of players increases, it incurs a tremendous amount of information exchange and the equilibrium is difficult to achieve. Moreover, if there exists a deviation for player obtaining other’s strategy, the equilibrium is easily broken. Here, we employ evolutionary game theory (EGT) [6] to model the competition of nodes under dynamic strategic environment based on the incentive mechanism. By EGT, we can provide a strong notion of equilibrium for the system, namely evolutionary stable strategy (ESS), and apply the convergence theory of replicator dynamics stability results to our scheme. Robust

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to communication noise and no requirement of complete opponents’ strategies give EGT a distinct advantage over other game theoretic mechanisms, especially in larger-scale wireless networks. The main contributions of this paper are outlined as follows. • An incentive compatible multiple-copy packet forwarding (ICMPF) protocol is proposed for ensuring packet transmission with both high packet delivery probability and a low delivery overhead in MSN with selfish nodes. • We consider both social and individual selfishness of node in the packet forwarding, and as well add the privacy sensitivity into individual selfishness of node excluding bandwidth resource. • The node’s sequential forwarding decision framework is modeled by the EGT, and the ESS of the game is obtained by a proposed distributed learning algorithm. The remainder of the paper is organized as follows. Section II presents the related works. The system model is introduced in Section III. The problem formulation is presented in Section IV. A game theoretic mechanism among different node-classes is constructed in Section V. Performance evaluation is provided in Section VI, and Section VII concludes this paper. II. R ELATED W ORKS The existing packet-forwarding schemes are mainly classified into two types: flooding-based routing and context-based routing. Flooding routing which acts as the simplest form for packet forwarding is used in many wireless networks’ routing protocols, while it results in the resource wastage and the well known “broadcast storm problem” [7]. To address these problems, efficient packet forwarding schemes have been designed. A probabilistic packet-forwarding protocol was developed in [8] where packets were mainly transmitted to the nodes who could supply the incoming packets with higher forwarding abilities. Spray and wait (SAW) forwarding protocol was proposed in [9] to balance resource consumption by controlling the number of packet copies. When considering the social ties among nodes in MSNs, there are also several algorithms proposed in existing literatures. Literature [10] proposed a distributed multistage cooperative-social-multicast protocol-aided packet forwarding scheme, where the content owners multicasted the content to their social contacts who had not received the content. The work [11] exploited small community labels to forward packets to the nodes who belonged to the same community, for achieving a high delivery probability. However, all of these solutions neglect the potential selfish nodes in the practical autonomous wireless networks. One of the most promising ways to induce the coordination of autonomous nodes in the wireless networks is using incentive mechanisms. The incentive mechanisms can be broadly divided into the following categories: the credit-based [12] and reputation-based [13] mechanisms. Compared with the existing reputation-based mechanism which requires each node to keep track of neighbors’ reputations, the credit-based mechanism, which uses virtual credits to motivate autonomous nodes to forward the packets, is more suitable for wireless networks

Mobile Social Network node ! class

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Packet forwarding process in the wireless network

with mobile nodes. The credit-based incentive mechanism was proposed in [14], [15] for encouraging autonomous node to participate in the forwarding process for two-hop Delay Tolerant Networks (DTNs). A two-player packet forwarding game was formed and the optimal packet forwarding strategies were derived in [16]. An adaptive repeated game theoretic scheme was designed in [17] for wireless networks to ensure the cooperations among nodes, and then a self-learning algorithm was employed to find the optimal cooperation probability. The schemes above indeed stimulate the selfish nodes to cooperate with each other, but they may be impractical for large-scale wireless networks. Evolutionary game has been adopted in [18-20] to model node’s strategy update in dynamic strategic environment. A non-cooperative forwarding control was proposed in [18] to regulate the participation of the relays to the delivery of messages for DTNs based on EGT. An evolutionary game theoretic scheme was developed in [19] to model the dynamic information diffusion process in social networks, while an indirect reciprocity framework was presented in [20] to enhance the cooperations of packet forwarding strategies via EGT in mobile Ad hoc networks. However, the literatures above assume that the interactions occur among homogeneous nodes, while nodes always have different resources and social ties in practical MSNs. III. S YSTEM M ODEL Autonomous MSN is a class of resource-constrained multihop mobile networks, where nodes, who have limited bandwidth resources and computing capacity, will make autonomous packet-forwarding decisions in terms of their own interests. All nodes adopt a store-carry-forward paradigm to provide communications among mobile devices with the absence of the infrastructure, relying on short range communication technologies such as Wi-Fi and Bluetooth. Moreover, individuals usually have regular mobility patterns, making the mobile devices come into MSNs in a regular fashion. There are m mobile attraction points (APs), which delegate the sources for nodes in the MSN. The APs, {S1 , S2 , · · · , Sm } ⊆ S, can generate the data packets (e.g. commercial promoting information, political information) and require to deliver them

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to the destination community. Due to the limited communication range, the AP needs to recruit the intermediate nodes to forward the packet to the destination, as shown in Fig. 1. Furthermore, since the intermediate nodes rely on the wireless channel to forward the packets originated from the APs in the MSN, the exposures of those intermediate nodes can be easily detected by periodically broadcasting beacon messages and monitoring data flows, thus exposing their location privacies [21]. Therefore, the autonomic intermediate node typically does not want to forward the packet for saving its limited bandwidth resource and/or for protecting its location privacy, which indeed affects the packet delivery from source to destination. Accordingly, we make the following Assumptions in this paper. Assumption 1. The MSN has K kinds of node-classes (e.g. private cars, taxis, and smart phones carried by people) in the MSN and each one, denoted by ψi , i ∈ K = {1, 2, · · · , K}, contains ri mobile nodes. We denote the node within nodeclass ψi as the node-class ψi ’ node. Assumption 2. Nodes within the same node-class have the same bandwidth capacity as well as the sensibility to location privacy, often do similar things and appear in the particular area during a specific period of time. Different node-classes have various bandwidth resources and privacy sensibilities, e,g. private car has higher privacy sensitivity than taxi. Assumption 3. Node is selfish and may be reluctant in the cooperation with other neighboring nodes, if this is not directly beneficial to it, owning to the fact that the node should consume a certain bandwidth resource and location privacy when performing forwarding packet. Assumption 4. Contacts between two nodes are symmetrical and sufficient for transferring the packet. Moreover, the contact process between node-class ψi ’ node and node-class ψj ’ node follows a Poisson process with a constant rate which depends on their social ties. Assumption 5. Time is slotted into the discrete intervals indexed by t ∈ T. For the sake of analysis, at each timeslot, there is one AP, who is randomly selected as the source, delivers one of its packets, and the other mobile nodes act as the potential relays. An incentive mechanism needs to be integrated with the packet forwarding scheme for encouraging selfish intermediate nodes to cooperate. If the AP has some packets to be delivered, it needs to stimulate intermediate nodes to help to forward the received packets by using a number of virtual currencies. In this paper, each packet needs to be transmitted to the destination, and is associated with a fixed virtual currencies by the AP, denoted by R, which is positively related to the packet properties (e.g. time to live T T L). Moreover, APs also provide some entertainment information for the relay node who can download its interesting information if it moves into the APs’ communication ranges. The relays must pay some currencies for downloading its interesting data while it can

earn currencies by relaying packets for APs. Importantly, if a relay does not have any currency, it cannot receive any entertainment information from APs, then its journey is boring. To improve the packet’s successful delivery probability from AP to destination, multiple copies of the packet are injected into the network across different parallel paths. In this way, the AP along with the nodes of different node-classes forms a multi-hop packet forwarding network during a timeslot. The destination is merely interested in one copy of a packet and pays for the first received copy only. That is to say, when a node acts as a forwarder, can it gain reward only if its forwarding packet has been successfully delivered to destination as the first copy. To enable nodes to pay and/or earn virtual currency, a Credit Clearance Center (CCC) [22] is employed to manage the currency for each node, which allows the currency to be distributed by the current relay without the involvement of the source. Specifically, each node holds a digital signature, and it can register itself to the CCC to obtain its currency. Moreover, when a source sends the packet, it generates a bundle which indicates the nodes who have forwarded such packet, except for setting the packet head with necessary information. At each subsequent hop during the packet forwarding, the relay who takes part in forwarding will add its signature to the packet. By this way, it is easy to track the packet forwarding path and determine forwarding nodes by checking the signatures of the packet. When one packet is successfully forwarded to the destination as the first copy, the destination will submit the signatures imbedded to CCC and the nodes involved will be paid after the CCC verifies the receipt. If a node does not forward the packet or participates in packet forwarding but fails, it will not obtain the virtual currency. This means that it may not be able to afford for its own data in the future. Additionally, we assume that selfish nodes do not collude with each other. IV. P ROBLEM F ORMULATION Note that, if there exist many nodes executing forwarding actions when having received the packets, the packet delivery probability from source to destination will increase while the number of released copies may also increase which results in higher bandwidth consumption. On the other hand, if there are few nodes executing forwarding actions, the copies’ number will decrease but the delivery probability may decrease. Therefore, the objective of our packet forwarding scheme is to perform a good tradeoff between the number of released copies and the packet delivery probability to ensure the efficient packet forwarding. An intuitive way of accomplishing this objective is to select the appropriate relays and make the relay autonomously schedule its forwarding strategy based on the proposed incentive mechanism to keep a satisfied number of the cooperative forwarding nodes in the MSN with selfish nodes. In this paper, we consider a generalized spray-and-focus packet forwarding architecture as the benchmark, and then propose a novel incentive compatible multi-copy packetforwarding scheme (Algorithm 1) by considering various

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selfish behaviors for different node-classes. In Algorithm 1, one AP is randomly selected as the source and forwards the packet to the first κ distinct nodes it encounters at the beginning of each timeslot, which can be regarded as the spray phase. Then, by terming the nodes with packets as carriers, the carrier makes forwarding decision, selects the next-hop relay and executes the packet transmission, hop by hop until the packet is delivered to the destination, which can be considered as the focus phase. Specifically, at each hop, the carrier makes packet-forwarding strategy (forwarding or drop) based on its selfish nature (first step). If it decides to forward the packet, the carrier keeps the packet until it encounters with and picks up the node who has never received the packet before (this timeslot) and as well has the higher forwarding capability as the next-hop relay (second step). Furthermore, the carrier executes the packet transmission to the selected relay at the cost of its limited resources (third step). For our protocol above, carrier’s autonomous forwarding strategy and the nexthop relay selection are crucial. Algorithm 1 Incentive Compatible Multi-copy Packet Forwarding (ICMPF) Protocol 1: for timeslot t = 1 to ∞ do 2: A randomly selected source Si sends a packet to the destination; 3: The carrier source Si forwards the packet copies to κ distinct nodes it encounters NSi . Then, the carrier set for one hop is C1 =Si , and the carrier set for second hop is C2=NSi . 4: for hop count n = 2 to T T L do 5: if any one of the carriers Cn encounters destination D then 6: Carrier forwards the packet to destination D; 7: break; 8: else 9: for carrier v = 1 to Cn do 10: Carrier v makes forwarding decision based on its profit. 11: if carrier v chooses dropping action d then 12: break; 13: else 14: if carrier v meets with one node who has the estimated higher forwarding capability then 15: carrier v moves the packet’s copy to that node and deletes the copy of the packet it holds. 16: else 17: carrier v keeps the packet’s copy. 18: end if 19: end if 20: end for 21: end if 22: Cn+1 ← Cn . 23: end for 24: end for

A. Carrier’s Autonomous Forwarding Strategy Generally, the node’s autonomous forwarding behavior (the first step of the focus phase in the ICMPF protocol) is affected by its selfishness. Addressing the selfishness of a node in WSNs will benefit not only the design of packetforwarding protocol, but also handling of privacy problem. In the context of WSN, two different types of selfishness need to be considered: individual and social ones. The individual selfishness is the peculiarity of a node that looks out for its own interests. From a social point of view, a selfish node chooses to cooperate with other nodes to whom it is familiar. 1) Individual Selfishness: The individual selfishness is affected by the following factors. • Privacy Sensitivity: Nodes worry about their location privacies when forwarding the packet, and they would rather stay “quiet” and not cooperate with other nodes. The higher privacy sensitivity means that the node-class will pay more attention to its local privacy. • Resource Limitation: Note that, a node makes use of its limited bandwidth to receive and then forward those packets originated from the APs. Generally, the node is unwilling to participate in the packet forwarding process for saving its limited bandwidth resource which can be used to receive its interesting entertainment information from the nearby APs. In this paper, we assume that different node-classes are allocated by various bandwidth resources corresponding to different QoS requirements, e.g. a private car may have more bandwidth resource than the taxi and smart phone due to its higher QoS requirement. • Reward Value: Since the node will gain the reward if its forwarded packet reaches the destination as the first copy, the award value also influences node’s selfish behavior. Note that, the nodes within one node-class have the same resources (Assumption 2 in system model). Individual selfishness Sindi for node-class ψi ’s node can be denoted as i i , R, B ) where Pri , Bi and R are ψi ’s privacy sen(P Sindi i r i sitivity, allocated bandwidth and reward value, respectively. 2) Social Selfishness: The social selfishness is affected by the social ties among APs, destination and intermediate nodes. The social selfishness for node-class is affected by the following factors. • Friendship with APs: In this paper, a node-class having better friendship with APs means that it is of higher possibility to move into APs’ communication ranges and has much more chances to download its interesting data using its currency, which is also defined as higher currencies efficiency. Then, they are more willing to help APs’ packet transmission. For instance, the taxi has more chance to move into APs’ ranges, and is more likely to forward the packet for APs. • Familiar with Destination: The familiar degree ρi,D between node-class ψi ’ node and destination means the meeting chance with each other. Naturally, a larger familiar degree indicates that the packet has higher possibility

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to be delivered to the destination as the first copy and that the node has the more chances to receive the reward. Then, social selfishness Ssoci can be denoted as i Ssoci (ρi,D , ρi,S ) where ρi,S is node-class ψi ’ friendship i with APs. In this paper, we use the social pressure metric (SPM) [23], which is obtained by the social characteristics of the nodes’ movements, to denote the social ties of the nodes. Since social ties are relatively stable, and they vary less frequently than that of the transmission link between mobile nodes [20], the nodes’ social ties are assumed to keep constant. After having analyzed nodes’ selfish characters, we employ a game-theoretic mechanism to model the node’s packet forwarding decision (participate to forward or not) based on its social and individual selfishness. The work is pivoted around a new approach: by modeling the competition of the nodes as a coordination game, we show that it is possible to enforce the cooperations within the node-classes through competition. Here, we rely on the evolutionary game theory (EGT) to model the competitions among nodes. The EGT rules the performance of competing nodes and the number of packet copies, and thus configures as an appropriate tool to drive the network to a desired operating point. In the proposed EGT, the node within one node-class updates its forwarding strategy for each timeslot and finally obtains the optimal strategy which is related to its selfishness (Sindi and Ssoci ). i i Based on Assumption 2 in system model, the nodes within the same node-class have the same property and may present the same forwarding strategy. Additionally, the number of packet’s copies is controlled by nodes’ autonomous strategies. The detail is illustrated in Section V. B. Relay Selection In the proposed protocol, the carriers select the next-hop relay based on the node-class’s forwarding capability which is the likelihood that the nodes within the node-class cooperatively forward the packet. Motivated by [3], node v measures t at timeslot t based node-class ψj ’s forwarding capability Fv,j on its observation, which is ∑ f t−1 1 (t − 1) ψ k j k∈C t−1 t v Fv,j = e−λτ Fv,j + (1 − e−λτ ) , (1) t−1 ψ |Cv j | t−1 where Fv,j is ψj ’ forwarding capability at the last timeslot, τ is the duration of each timeslot, λ is the update parameter, ψ t Cv j is the set of the encountering node-class ψj ’ nodes for node v at timeslot t and { 1 if node k forwards packet at timeslot t, 1fk (t) = 0 if node k drops packet at timeslot t. (2) The numerator on the right hand side of Eq. (1) denotes the number of nodes within node-class ψj who take cooperative forwarding actions for timeslot t, which is based on the observation of node v. If the node does not encounter any nodes within node-class ψj , the second term at the right hand

t side of Eq. (1) is 0. Fv,j approaches to 0 or 1 if one node-class keeps the same packet-forwarding behavior such as selfish or cooperative.

Remark 1. One phenomenon may happen that even the destination has successfully decoded an original packet, some of its replicas may still be stored in some intermediate nodes. Due to the fact that transmitting multiple copies will cause too much but inefficient network resource consumption, the destination will broadcast a message to other nodes and inform them to discard these copies for preventing these copies from spreading in the network. V. E VOLUTIONARY G AME AMONG VARIOUS N ODE - CLASSES A. Evolutionary Game among Nodes with different Nodeclasses Since a intermediate node earns some rewards if its forwarded packet reaches the destination as the first copy, its payoff is also influenced by other nodes’ forwarding actions. We assume that the node does not completely know others’ current actions (strategy uncertainty) due to the channel noise and the dynamic environment. Hence, the network node has to try different strategies in different rounds of play and learns its optimal strategy by using the methodology of understandingby-building about the changing strategic environment based on its local information. Based on the analysis above, we build an evolutionary game to explore the player’s optimal forwarding strategy among different node-classes. Note that the cost parameters and payoff functions are referred to as private knowledge among the game players, and that one has no idea of these knowledge about others in this paper. When a packet is generated by the source, the competition takes place at this timeslot. Each node who has received the packet has two strategies: either to participate in the cooperative packet forwarding f , or to denial the packet forwarding d. Each strategy corresponds to a certain utility of the node. Let Gd = [N, P, U K (·)] denote the evolutionary game, where N is the number of network nodes, P is the strategy set and U K (·) is the set of utility functions for network players. The utility of each player depends on its own resources, social ties and also on the choices of other players’ strategies. Formally, the evolutionary game Gd is expressed as follows. • Players: Players are the populations in the network, where the same node-class’s nodes make up one population, ψi ∈ Ψ. • Strategies: Strategy set for each node in ψi is Pi = [f, d]i . Then, strategy set for players in total is P ⊇ {P1 , P2 , · · · , PK }. • Payoffs: Player’s payoff is determined by multiple factors, including the cost of forwarding the packet, the reward obtained by forwarding/dropping the packet. B. Utility Function Formation In the multi-copy packet forwarding mechanism for MSN, the reward received by the node depends explicitly on the

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number of nodes adopting the forwarding behaviors. The more the nodes participate in the cooperative packet forwarding, the higher probability for packet to be successfully delivered to the destination, but indeed the less chance for the node to receive the rewards. In the meanwhile, the node needs to consume its bandwidth and also has the risk to expose its location privacy, while forwarding the received packet. Additionally, all network nodes are assumed to have more bandwidth reserved for receiving more packets destined to itself. The node-class ψi ’ node who has participated in forwarding this packet will gain the reward R if it has completed a successful packet forwarding as the first copy and 0 if it failed. When the node has dropped the packet, it will receive nothing. Additionally, the node’s friendship with APs ρi,S will affect the efficiency of currency usage χi which is defined in Section III, and further influence the node’s revenue. As we assume that the contact process between nodes follows Poisson process, the possibility that the node-class ψi ’ node contacts the source within one timeslot is 1 − eρi,S τ and τ is the duration of one timeslot. In this paper, we use the meeting probability above to measure the node-class ψi ’ efficiency of currency usage, i.e. χi = 1 − eρi,S τ . Therefore, the payoff of the node-class ψi ’ node for a timeslot is  i if forward and success,  χi R − Ctr i −Ctr if forward but fail, (3) Ui =  0 if drop, i is the cost for one packet transmission. The probwhere Ctr i ability P rsucc that the packet is delivered to the destination via the node-class ψi ’ node successfully (as the first copy) is related to the packet delivery probability of node-class ψi and network copy degree C(pfi , pf−i ) which denotes the average number of packet’s copies in the network. •

Packet delivery probability: The packet delivery probability for node-class ψi represents the likelihood that node-class ψi ’ node can deliver the packet to its destination directly or indirectly. Let DCi,D denote the probability that node-class ψi ’ node delivers the packet to destination directly during one timeslot. Similarly, DCi,D = 1 − e−ρi,D τ . Additionally, ICi,D is the indirect packet delivery probability of the node-class ψi for one timeslot, i.e. the probability that the packet is delivered to the destination finally after having experienced a series of nodes’ cooperative forwarding. Based on the proposed forwarding protocol, ICi,D depends on network nodes’ packet forwarding strategies together with some system parameters (social ties and network scale). For instant, we consider a three-hop relaying scenario where the nodeclass ψi ’ node receives the packet from the source, then encounters and forwards the packet to the node-class ψk ’ node who has higher forwarding capability. Then, the node-class ψk ’ node meets and forwards the packets to the destination within one timeslot. In such a case, motivated by [24], the node-class ψi ’ indirect packet ∑ Ii,k,D pfk ∑ delivery probability is ICi,D = k Ii,h,D where h

∫x dx 0 ρi,k e−ρi,k ς ρk,D e−ρk,D (τ −ς) dς = 1 + 0 ρk,D e−ρk,D τ −ρi,k e−ρi,k τ denotes the node-class ψi ’ indiρi,k −ρk,D rectly packet delivery probability via the node-class ψk ’ I node, and as well ∑ i,k,D is the probability that the h Ii,h,D Ii,k,D =

∫τ

packet is delivered to the destination via a ψi ’ node and a ψk ’ node. Then, the packet forwarding probability of the node-class ψi ’ node is Φi (pf−i , ρ) = (1 − e−ρi,D τ ) + e−ρi,D τ ICi,D . • Network copy degree: Each packet is associated with a network copy degree, which indicates the average number of copies delivered to the destination within a specific timeslot. The network copy degree depends on the network nodes’ forwarding strategies. Based on the proposed forwarding protocol, the source forwards the packet copies to κ distinct nodes it encounters. The initial number of packet copies is κ. Then, if it decides to forward the packet, the carrier keeps the packet copy until it meets the destination or the other node who has higher forwarding capability. Due to the precondition that the packet has been delivered to the destination via the node-class ψi ’ node directly or indirectly, the network copy degree at this timeslot is C(pfi , pf−i , ρ) = ∑ ρj,S rj pfi Φj (pf−j ,ρ) ρ rj ∑ 1+(κ−1) j where ∑ j,S , h ∈ K, h ρh,S rh j ρh,S rh is the probability that the node-class ψh ’ node is one of the source’s κ encountered nodes. The more copies result in the lower probability for a copy to be successfully delivered to the destination before other copies, while the larger Φi (pf−i , ρ) leads to the higher i successful probability. Hence, P rsucc is inversely proportional f f to C(pi , p−i ) while it is proportional to Φi (pf−i , ρ). Then, the i is calculated as probability P rsucc Φi (pf−i , ρ)

Φi (pf−i , ρ) ∑ ρj,Srj pfi Φj (pf−j ,ρ) . C(pfi , pf−i , ρ) ∑ 1 + (κ − 1) j h ρh,S rh (4) i depends on the privacy sensiThe node’s relative cost Ctr tivity and its available bandwidth resource. • Privacy sensitivity: For node-class ψi , its privacy sensii tivity is denoted as Pri , 0 ≤ Pri ≤ 1. In addition, Pco denotes the leakage amount of location privacy for one forwarding packet. Because the same node-class has the i same mobile property, Pco is related to the portion of f cooperators pi in node-class ψi , which is also denoted i as Pco (pfi ). • Node’s resource: The bandwidth consumed by node-class i ψi is denoted as Bco while Bi is its bandwidth capacity. Note that, the node within node-class ψi will spend location i i privacy Pco and bandwidth resource Bco on forwarding one packet. Different node-classes, which have different privacy sensitivities and bandwidth capacity, will pay different attentions to resources consumption. For instance, the node-class who has the higher privacy sensitivity and shorter bandwidth will more cherish its resource consumption, and be less willing i to forward the packet. Accordingly, node’s relative cost Ctr i P rsucc =

=

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is inversely proportional to its bandwidth capacity Bi while i it is proportional to privacy sensitivity Pri . Moreover, Ctr is also related to node’s resource consumption for one packet i i forwarding, i.e. Bco and Pco . Then, the relative cost is defined as follows i i Pri (w1 Bco + w2 Pco (pfi )) , (5) Bi where w1 and w2 represent the weights for bandwidth and location privacy respectively, and w1 + w2 = 1. The utility of node is generally a function of multiple attributes representing the different dimensions of problem, namely packet reward, resource capacity, the cost for one packet forwarding, social tie, as well as other nodes’ actions, etc. Then, the average payoff for the node-class ψi ’ node at one timeslot, who forwards the received packet, is expressed as i Ctr =

i i i ¯ f = P rsucc U (χi R − Ctr ) − (1 − P rsucc )Ctr i i i = P rsucc χi R − Ctr

¯ f (t) is the average payoff where η is a positive scale factor, U i ¯ of node adopting strategy f and Ui (t) is the average payoff of node-class ψi at timeslot t. Since the sum of the probabilities that a node chooses strategy f and d is equal to one in the replicators dynamic of the evolutionary game for one node-class, we can simply analyze Eq. (8) for node-class ψi . In our system, based on ¯ f and U ¯ d described in Eqs. (6) and node’s average payoffs U i i (7) respectively, the average payoff for node-class ψi at time t is ¯i (t) = pf (t)U ¯ f (t) + (1 − pf (t))U ¯id (t) U i i i f f ¯ (t), = p (t)U i

i

(10)

and replicator dynamic equation is ( ) ¯ f (t) − U ¯i (t) , p˙fi (t) = ηpfi (t) U i ¯ f (t). = ηpfi (t)(1 − pfi (t))U i

(11)

pfi ,

From the above replicator dynamics of we observe that the number of nodes who participate to forward the + = . packets increases if the forwarding fitness is above the average Bi fitness, and vice versa. In the evolutionary game, each player 1 + (κ (6) dynamically adjusts his/her action by observing the utilities under different behaviors. It is an effective approach for i is related to node-classes converging to a stable equilibria after having where successful forwarding probability P rsucc the nodes’ current strategies, and varies with the timeslot. experienced a period of strategic interactions, and such a stable Similarly, the average payoff for the node-class ψi ’ node at a equilibria is called the evolutionary stable strategy (ESS). timeslot, whose strategy is d, is ESS [25]: ESS is the solution of the evolutionary game. ¯id = 0. U (7) When the ESS is reached, the proportion of the active nodes in each population will be stable and no player will try to C. Analysis of Evolutionary Stable Strategies change its strategy to get a higher utility. 1) Evolutionary Stable Strategies: Note that any one of the According to the replicator dynamic equation of packet players is uncertain about other players’ actions and utilities. ¯ f = 0 or pf = 0 or pf = 1 forwarding in Eq. (11), U i i i To improve his/her own utility, each player will try different is the solution to the replicator dynamics. Furthermore, if strategies in different rounds of play and then learns its optimal packet forwarding can lead to a higher utility than the average strategy from the interactions. During this process, the portion level, the portion pfi will increase. Note that, in order to take of nodes using a certain pure strategy may vary with the action for new arriving packet, each node of node-class ψi timeslot in one node-class. ¯i and U ¯ f . Similar needs to learn about the average utilities U i Replicator Dynamics: Replicator dynamics is used to phenomenon can be found for the evolutions of pf−i . model such population evolutions. In strategy evolution, the node updates its strategy to get a higher payoff by imitating Theorem 1. For the packet forwarding evolutionary game the better strategy. Note that the higher payoff of a node’s Gd = [N, P, U K (·)], there exists a unique ESS for each nodestrategy has, the more possibility the strategy to be imitated class under particular conditions of the specific reward R. by the other nodes within one node-class. Thus, the payoff of a Specially, when K = 2, κ > 1, r1 = r2 , ρ1,D > ρ2,D and strategy can be interpreted as its fitness, and the strategy with ρ1,S > ρ2,S , we have Eq. (12) at the top of the next page where higher fitness has more chance to reproduce in each node- Υ2 = 1 − m2 + m2 ϱ2 , Ξ1 = κ(1 − ϖ)m1 + 1, m2 = e−ρ2,D τ , ρ class. Thus, along with the nodes’ strategies updating, the m1 = 1 − e−ρ1,D τ , ϖ = ρ2,D2,D +ρ1,D , ϱ2 = IC2,1,D and node-class state pi , i.e. the ratio of active nodes to total nodes (pf , pf ) is the solution to the equations 12 22 in population ψi , also varies across two adjoint timeslots.  Rχ1 m1  − c1 pf12 = 0, Let us define the varying rate of the node-class state as 1+κ[ϖpf22 (1−m2 +m2 ϱ2 pf12 )+(1−ϖ)pf12 m1 ] the population dynamics [p˙ fi , p˙di ]. According to the replicator Rχ2 (1−m2 +m2 ϱ2 pf12 )  − c2 pf22 = 0. f f f 1+κ[ϖp (1−m 2 +m2 ϱ2 p12 )+(1−ϖ)p12 m1 ] dynamics, the dynamics of nodes’ number for specific strategy 22 in node-class ψi at timeslot t can be modeled as, Proof: Please refer to Appendix A. ( ) The existence of ESS also can be proved as the similar way ¯ f (t) − U ¯i (t) , p˙fi (t) = ηpfi (t) U (8) i when K > 2. Because of the complexity of the analysis and ) ( d ¯i (t) − U ¯i (t) , (9) the limitation of paper, we take K = 2 as an example to prove p˙di (t) = ηpdi (t) U i χi RΦi (pf−i , ρ) P i (w1 Bco − r ∑ f ρj,Sr Φj (p ,ρ) − 1) j ∑ ρj h,S rh−j h

i (pfi )) w2 P rco

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  (1, 1)   √ Rχ2 f∗ f∗ (1, κϖc + (p1 , p2 ) = 2    (pf , pf ) 12

c2 (1+κ[ϖΥ2 +(1−ϖ)m1 ) , χ 1 m1 c1 (1+κ[ϖΥ2 +(1−ϖ)m1 ) c2 Ξ1 κc2 ϖ , χ + χ2 Υ2 < R < χ 1 m1 2 2 )+c2 c1 m1 Ξ1 0 < R < ϖ(c1 ) κ(1−m2 )(2mχ21ϱm2 +1−m 1

if R > Ξ1 1 4κ2 ϖ 2 (Υ2 )2

−

Ξ1 2κϖΥ2 )

22

if if

the existence of ESS in Theorem 1. Note that, the network is an intrinsical three-hop relaying scenario1 according to our protocol when K = 2. Remark 2.∗ From Theorem 1, we can observe that the stable strategy pfi is a function of the node’s individual selfishness which is affected by its limited resources (bandwidth, local privacy) and the node’s social selfishness which is influenced by its social ties. Thus, Eq (12) is the specific expression of the function F(Sindi , Ssoci ). i i 2) Distributed Learning Algorithm for ESS: In the above evolutionary game with multiple populations, we have shown that the ESS is solvable. However, solving the equilibrium requires the knowledge of utility function and the exact actions adopted by the other nodes which results in huge communication overhead. Moreover, others’ exact actions cannot be obtained by the node in this paper. Therefore, a distributed learning algorithm that gradually converges to the ESS without any information exchange for node is preferred. According to the Wright-Fisher model [26], the probability of individual node taking forwarding strategy f is proportional to the sum of utilities for the nodes who participate in the cooperative packet forwarding process. Moreover, based on the assumption that the number of nodes within populations is sufficiently large, the portion of nodes who participate in packet forwarding is equal to the probability of one individual node forwarding the packet. Therefore, the Wright-Fisher model is equivalent to the replicator dynamic equations when the number of network nodes N is sufficiently large. In such a case, the packet forwarding probability at timeslot t + 1, p˜fi (t + 1), is calculated by ˜ f (t) ¯ p˜f (t)U i p˜fi (t + 1) = i . (13) ˜ ¯ Ui (t) From Eq. (13), we find that the potential relay node learns and makes its forwarding decision based on its local information independently. Specifically, in order to update p˜fi (t + 1), each node-class ψi ’ node has to learn about the average forwarding ˜¯ f (t) and the average utility U ˜ ¯i (t) of the node-class. In utility U i this paper, the node estimates its average forwarding payoff ˜¯ f (t) and average payoff U ˜ ¯i (t) by following method. Note U i that, node’s local information contains its historical payoff and observed payoff information. Historical payoff information: We assume that each node˜ ¯ f (t) of the class ψi ’ node has a prior historical perception U i payoff performance for cooperative forwarding action. 1 The

packet is delivered to the destination via one or two relays.

andm1 >

(m2 ϱ2 +1−m2 )χ2 c1 . c2 χ1

(12)

Observed payoff information: The node v within node-class o ˜ f (t) with local ψi , i ∈ K, estimates its observed payoff U i t t t information Fv,t , Fv,t ⊇ {Fv,1 , Fv,2 , · · · , Fv,K } obtained in Section III about other nodes’ actions. This procedure is repeated round after round, which generates a discrete time stochastic process. Such a stochastic process is called the learning process. At timeslot t, the ˜¯ f (t) will be used by node v estimated forwarding payoff U i to choose action f or d. The discrete time stochastic process represents the evolution of nodes’ portion and can be written in the following equivalent form ( ) f fo f f t t˜ ˜ ˜ ¯ ¯ ¯ Ui (t) = δ U i (t − 1) + (1 − δ ) γ1 Ui (t − 1) + γ2 U i (t) , (14) where γ1 , γ2 , δ t are weights, 0 ≤ δ t ≤ 1 and γ1 + γ2 = 1. If node’s action at time t − 1 is dropping (ai (t − 1) = d), carrier v cannot obtain Uif (t − 1), i.e. Uif (t − 1) = 0, then setting ˜¯ f (t). And also, we have γ1 = 0 to compute U i ) ( ˜¯ (t) = δ t U ˜¯ (t − 1) + (1 − δ t ) γ U (t − 1) + γ U ˜¯ o U i i 1 i 2 i (t) , (15) ˜¯ o (t) can be obtained according to Eqs. (6) and (7) where U i ˜¯ (t) based on local observed information Fv,t . The payoffs U i f ˜ ¯ and Ui (t) for timeslot t are then obtained and used to update the forwarding action at this timeslot (Eq. (13)). The details for the distributed evolutionary learning algorithm are outlined in Algorithm 2. Through the proposed algorithm, the players will try different strategies at each timeslot, calculate the utilities changes of some strategies and update their strategies to achieve the ESS eventually. The convergence of Algorithm 2 can be guaranteed by Theorem 1. The learning process proposed in Algorithm 2 coincides with the reality of MSNs, and leads to a tradeoff between exploration and exploitation. Specifically, on one hand, the greedy method is introduced, with the goal of selecting the optimal strategy according to the current learning information (exploration, steps 6-8). On the other hand, since there exists a random number γ ∈ [0, 1], the player exploits the promising strategy with a probability of γthr (exploitation, step 4). Then, the learning process can prevent the node from falling into a locally optimal solution. Algorithm 2 also has the following attractive features: 1) It is genuinely distributed, as strategy updating decision is local to potential relays; 2) It depends uniquely on the nodes’ local observations to estimate the average payoffs of the node-classes.

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Delivery ratio

1 0.8 0.6 Proposed ICMPF Incentive forwarding Selfish epidemic

0.4 0.2 0

5

10 15 20 25 Distance away from the source(m)

Fig. 2.

30

Packet delivery ratio

5

10 Delivery overhead

Algorithm 2 Distributed Evolutionary Learning for ESS Input: Node randomly initializes its strategy; Set the threshold value of γthr ∗ Output: The node packet-forwarding probability pfi ; 1: for each timeslot t do 2: For the node-class ψi ’ nodes, generate a random number 0 ≤ γ ≤ 1; 3: if γ ≤ γthr then 4: Node randomly chooses a forwarding strategy a(t); 5: else 6: Node estimates its node-class’s average payoffs ˜¯ f (t) and U ˜ ¯i (t) according to Eq. (14) and Eq. (15), U i respectively; 7: t = t + 1; 8: Update the probability of cooperation forwarding, pfi (t) using Eq. (13); 9: end if 10: end for

Proposed ICMPF Incentive forwarding Selfish Epidemic

0

10

VI. P ERFORMANCE E VALUATION In this section, we integrate our scheme with a number of experiments based on MATLAB. Our evaluations have two objectives. One is to measure the efficiency of the proposed ICMPF protocol in the MSNs with selfish nodes. The other is to verify that our evolutionary game theoretic scheme indeed prevents some nodes from being selfish and obtains the stable strategies for various node-classes. A. Experiment Setup We consider an MSN with 480 mobile nodes randomly distributed in a certain area of (480 × 480)m2 . All the nodes have the same power level and the same maximal transmission range of 50m. In our simulation, node movements are slotted into time intervals, during which different reference locations in the network area are associated with nodes. Then, within each timeslot, a node can either move in a restricted area (its reference location) or freely in the whole scenario [27]. Moreover, the reference locations for the different nodeclasses’ nodes are dissimilar, which are based on their social properties. Whereas, the reference locations for the same nodeclass’s nodes are equal. To show the effectiveness of proposed ICMPF protocol, we compare it with two existing schemes: selfish epidemic [28] and incentive forwarding [29]. We utilize the following criteria to evaluate the packet forwarding performance. • Delivery Ratio: it is the proportion of packets that have been successfully delivered to destinations out of the total packets generated in the source within a given timeslot. • Delivery Overhead: it is a measure of the average number of forwarders used for the successfully delivered packets. B. Simulation Results 1) Effectiveness of the proposed ICMPF protocol: As shown in Figs. 2 and 3, we observe that our ICMPF protocol

5

10 15 20 25 Distance away from source(m)

Fig. 3.

30

Delivery overhead

performs better than the other two packet forwarding schemes jointly in term of delivery ratio and delivery overhead. Although the selfish epidemic scheme has the higher delivery ratio, it also has the highest delivery overhead owing to its blind replication strategy. And also, the delivery overhead of incentive forwarding is less than that of the ICMPF protocol, but its delivery ratio is much slower than that of the ICMPF protocol. Our result clearly shows that the ICMPF protocol can effectively achieve good delivery ratio without relative large delivery overhead. We claim that two reasons can explain it. Firstly, the ICMPF protocol employs a forwarding decision for nodes and further maximizes the forwarding profit of each communication. Thus, the number of packet’s copies in the network is minimized by aware of the resource levels and other nodes’ actions. Secondly, the ICMPF protocol measures the node’ selfishness of the neighbors when choosing the nexthop relays. In this way, our scheme can guarantee that the packet moves along the more reliable paths including nodes with higher forwarding willingness and improve the delivery ratio in the network. 2) ESS of evolutionary game: Fig. 4 shows the percent of cooperation nodes versus the timeslot when there are three node-classes in the system. We observe that the slop of curve for a particular node-class becomes smooth and eventually tends to a certain value. This implies that the network nodes reach the equilibrium forwarding strategy after having experienced several strategy-updating. Figs. 5 and 6 show the simulation results for different parameters: learning rate δ t = δ and strategic learning error σn . We can observe from Fig. 5 that the larger δ results in worse convergence.

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to guide nodes’ forwarding behaviors, and we derived the evolutionary stable strategy (ESS). Furthermore, we designed a distributed learning algorithm for nodes to achieve the desired ESS.

node−class ψ3

0.8

node−class ψ

0.7

node−class ψ1

i

Percent of cooperation nodes (pf)

1 0.9

2

0.6 0.5

A PPENDIX A P ROOF OF T HEOREM 1

0.4 0.3 0.2 0.1

0

200

400

600

800

1000

Slot

Fig. 4.

Steady packet-forwarding probability when K = 3.

1

Percent of cooperation nodes

0.9

δ=0.8 δ=0.65 δ=0.3

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

200

400

600

800

1000

Slot

Fig. 5. Convergence performance when K = 3, impact of δ to the packetforwarding probability achieved by node-class ψ1 .

This is because the exploration lasts for a longer time during learning procedure even if the forwarding strategy achieving optimal transmission is already visited. The curves of Fig. 6 illustrate that our evolutionary models used here are quite robust against error-probability for our learning algorithm when various strategic learning error settings exist. The higher error-probability (σn = 2%) of strategy learning also can make sure the convergence of algorithm 3 while expending longer time to achieve steady forwarding strategy.

Proof: Considering replicator dynamic equation Eq. (11) as a nonlinear dynamic system, we can examine whether the equilibrium is ESS through analyzing the Jacobian matrix J of the replicator dynamic equations. The asymptotical stability requires the conditions that Det(J) > 0 and that Tr(J) < 0 [26]. Substituting these equilibria into stability conditions separately, we can examine whether they are ESSs. For the numerical analysis, we consider a three-hop network scenario where the packet is delivered to the destination via one or two relays. That is to say, when it receives the packet from another node with lower forwarding capability, the carrier can only forward the packet to the destination if it decides to forward the packet in our protocol. In particular, regarding the rewarding policy adopted by the source nodes, we assume that upon successful delivery of a packet, the relay receives a positive reward R if and only if the packet is delivered to the destination as the first copy via it directly or indirectly. Simply, we take K = 2 as an example. Note that the network is an intrinsical three-hop relaying scenario according to our protocol. Moreover, when K = 2, α = 1, κ > 1 and r1 = r2 , we have p˙fi = 0, i = 1, 2. i Now, the probability of the packet successful delivery P rsucc for a cooperative node as the first copy is: m1 1 = P rsucc , f 1 + κ[ϖp2 (1 − m2 + m2 ϱ2 pf1 ) + (1 − ϖ)pf1 m1 ]

VII. C ONCLUSIONS In this paper, we proposed the ICMPF protocol and analyzed different selfish behaviors for different node-classes based on a game-theoretic packet forwarding mechanism to support a satisfied transmission performance when various selfishness of different node-classes were considered. To model the complicated interactions among nodes with dynamic strategies in our protocol, an evolutionary game framework was exploited

0.9

σn=0

0.8

σn=0.5%

0.7

σn=2%

1 − m2 + m2 ϱ2 pf1 1 + κ[ϖpf2 (1 − m2 + m2 ϱpf1 ) + (1 − ϖ)pf1 m1 ]

,

where m2 = e−ρ2,D τ , m1 = 1 − e−ρ1,D τ , ϖ = ρ2,D2,D +ρ1,D and ϱ2 = IC2,1,D . Then, the average gain for node-class ψi is ρ

¯f = U 1

¯f = U 2

1

Percent of cooperation nodes

2 P rsucc =

Rχ1 m1 1+

κ[ϖpf2 (1

− m2 +

m2 ϱ2 pf1 )

+ (1 −

ϖ)pf1 m1 ]

Rχ2 (1 − m2 + m2 ϱ2 pf1 ) 1 + κ[ϖpf2 (1 − m2 + m2 ϱ2 pf1 ) + (1 − ϖ)pf1 m1 ]

−c1 pf1 ,

−c2 pf2 ,

where χ1 < χ2 . According to Eq. (11), we can get seven possible equilibria: (0, 0), (0, 1), (1, 0), (0, 1), (pf11 , 1), (1, pf21 ), (pf12 , pf22 ), where pf11 satisfies

0.6 0.5 0.4

Rχ1 m1 )

0.3

1 + κ[ϖ(1 − m2 + m2 ϱ2 pf11 ) + (1 − ϖ)m1 pf11 ]

0.2 0.1

0

200

400

600

800

1000

Slot

Fig. 6. Convergence performance when K = 3, impact of σn to the packetforwarding probability achieved by node-class ψ2 .

−c1 pf11 = 0,

pf21 satisfies Rχ2 (1 − m2 + m2 ϱ2 ) 1+

κ[ϖpf21 (1

− m2 + m2 ϱ2 ) + (1 − ϖ)m2 ]

− c1 pf21 = 0,

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and (pf12 , pf22 ) is the solution to the following equations  Rχ1 m1  − c1 pf12 = 0, 1+κ[ϖpf (1−m +m ϱ pf )+(1−ϖ)pf m ] 

2

22

2 2 12

12

1

Rχ2 (1−m2 +m2 ϱ2 pf12 ) f 1+κ[ϖp22 (1−m2 +m2 ϱ2 pf12 )+(1−ϖ)pf12 m1 ]

− c2 pf22 = 0.

In the following, we will check which equilibria is the ESS. For the equilibrium (0, 0), i.e. the node-class ψ1 ’ nodes and node-class ψ2 ’ nodes choose to drop the packets, we have lim J11 {pf →0,pf →0} 1

2

Rχ1 m1 κ[ϖpf2 (1

1+ = Rχ1 m1 ,

− m2 + m2 ϱ1 pf1 ) + (1 − ϖ)pf1 m1 ]

− c1 pf1

and lim J22 {pf →0,pf →0} = Rχ2 (1 − m2 ). In such a case, 1 2 equilibrium (0, 0) is not the ESS. Similarly, the equilibrium (0, 1), i.e. the node-class ψ1 ’ nodes choose to drop the packets while the node-class ψ2 ’ nodes choose to cooperatively forward the packets, is not an ESS; The equilibrium (1, 0), i.e. the node-class ψ2 ’ nodes choose to drop the packets while the node-class ψ1 ’ nodes choose to cooperatively forward the packets is not an ESS; The equilibrium (1, 1), i.e. all node-class ψ2 ’ nodes and node-class ψ1 ’ nodes choose to forward the packets, is an ESS if and only if J11 {pf =1,pf =1} < 0 and J22 {pf =1,pf =1} < 0, 2 2 1 1 when J12 {pf =1,pf =1} = 0 and J21 {pf =1,pf =1} = 0. That 1

2

1

2

2 ϱ2 )+(1−ϖ)m1 ) is R > c1 (1+κ[ϖ(1−m2χ+m ; The equilibrium 1 m1 f (1, p21 ), i.e. the node-class ψ2 ’ nodes forward the packets with the probability of pf12 while the node-class ψ1 ’ nodes choose to cooperatively forward the packets, is an ESS if κcχ22ϖ + c1 (1+κ[ϖ(1−m2 +m2 ϱ2 )+(1−ϖ)m1 ) c2 (κ(1−ϖ)m1 +1) ; χ2 (1−m2 +m2 ϱ2 ) < R < χ 1 m1 f The equilibrium (p11 , 1), i.e. the node-class ψ1 ’ nodes forward the packets with the probability of pf11 while the nodeclass ψ2 ’ nodes choose to cooperatively forward the packets, is not an ESS; The equilibrium (pf12 , pf22 ), i.e. the nodeclass ψ2 ’ nodes forward the packets with the probability of pf22 while the node-class ψ1 ’ nodes choose to forward the packets with the probability of pf12 , is an ESS if and only 2 2 )+c1 c2 m1 (κm1 (1−ϖ)+1) if 0 < R < ϖ(c1 ) κ(1−m2 )(2m2 ϱ2 +1−m χ1 m1 2 )c1 χ2 and as well m1 > (m2 ϱ2 +1−m . c2 χ1

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