Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.

Digital Object Identifier 10.1109/ACCESS.2019.DOI

Misbehavior Detection in EphemeralNetworks: A Local Voting Game inPresence of UncertaintyALI BEHFARNIA, 1 (Student Member, IEEE), ALI ESLAMI, 1 (Member, IEEE)1Department of Electrical Engineering and Computer Science,Wichita State University, Wichita, KS, USA

Corresponding author: Ali Behfarnia (e-mail: axbehfarnia@shockers.wichita.edu).

This material is based upon work supported by the National Science Foundation under Award No. OIA-1656006 and matching supportfrom the State of Kansas through the Kansas Board of Regents.

ABSTRACT Emerging short-lived (ephemeral) connections between wireless mobile devices have raisedconcerns over the security of ephemeral networks. An important security challenge in these networks isto identify misbehaving nodes, especially in places where a centrally managed station is absent. Totackle this problem, a local voting-based scheme (game) in which neighboring nodes quickly decidewhether to discredit an accused (target) node in mobile networks has been introduced in the literature.However, nodes’ beliefs and reactions significantly affect the outcome of target node identification in thecollaboration. In this paper, a plain Bayesian game between a benign node and a target node in one stageof a local voting-based scheme is proposed in order to capture uncertainties of nodes for target nodeidentification. In this context, the expected utilities (payoffs) of players in the game are defined accordingto uncertainties of nodes regarding their monitoring systems, the type of target node and participants,and the outcome of the cooperation. Meanwhile, incentives are offered in payoffs in order to promotecooperation in the network. To discourage nodes from abusing incentives, a variable-benefit approachthat rewards each player according to the value of their contribution to the game is introduced. Then,possible equilibrium points between a benign node and a malicious node are derived using a pure-strategyBayesian Nash equilibrium (BNE) and a mixed-strategy BNE, ensuring that no node is able to improveits payoffs by changing its strategy. Finally, the behavior of malicious and benign nodes is studied viasimulations. Specifically, it is shown how the aforementioned uncertainties and the designed incentivesimpact the strategies of the players and, consequently, the correct target-node identification.

INDEX TERMS Misbehavior detection, local voting-based scheme, game theory, uncertainty, ephemeralnetworks. a

aThis work was presented in part at 2019 Allerton Conference on Communication, Control, and Computing [1].

I. INTRODUCTIONA. MOTIVATION

The proliferation of temporal peer-to-peer communicationbetween wireless devices gives rise to the formation ofshort-lived (ephemeral) networks due to the unpredictableexistence of mobile nodes. Ephemeral networks are perva-sive over a variety of applications, such as vehicular ad hocnetworks, mobile social networks, wireless sensor networks,etc. [2]–[5]. These networks are attractive to maliciousnodes so they join and manipulate sensed data, therebyinfluencing network performance [6]. For example, in thecase of vehicular ad hoc networks (VANETs), a malicious

vehicle can inject false information to its neighbors andtrigger a serious problem on the road. In addition to datamanipulation or sending false information, a malicious nodecan compromise the vehicle’s routing efficiency by notforwarding the packets that it received in the network.Therefore, an important security improvement in ephemeralnetworks is to reveal the type of nodes, either benign ormalicious, especially in places where centrally managedstations are absent. In such transitory distributed networks,quick cooperation among neighboring nodes can provideeffective solutions toward improving network performance[7], [8]. However, nodes are usually selfish and reluctant to

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cooperate, simply because they must use their resources. Inaddition, each node has some inherent uncertainties in a col-laboration, including the type of participants, the accuracyof its own components (e.g., detection system), the valueof its contribution, and attainable outcomes, all of whichaffect the node’s decision about whether to participate. Inthis respect, it is crucial to provide incentives according todifferent reactions of nodes under uncertainty in order toachieve malicious-node detection.

B. RELATED WORKA variety of precise and effective schemes have beenproposed to detect misbehaving nodes. The authors in [9][10] have provided a comprehensive literature review onmisbehavior detection in cyber-physical systems (CPSs) andintelligent transportation systems, which covered a varietyof solutions for transient networks wherein nodes havedifferent limitations. In particular, Liu et al. [11] studiedthe interactions between an attacker and a defender in orderto detect misbehaving nodes in ad hoc networks. Theyconsidered scenarios where a malicious node can eitherattack or not attack a benign node, while the defender maybe in a monitoring or non-monitoring state. The interactionsbetween players were analyzed by game theory, which is apowerful tool to obtain the best strategies of independentdecision makers [12]. Although this paper clearly describedthe individual interactions and uncertainties between twoagents, it did not consider the identification of a misbehavingnode for all nodes in the network. Another approach formisbehavior detection is to use reputation systems for whicha history of credits is built for nodes based on their pastbehavior in the network. However, this method needs adatabase that can be created over time, and nodes may haveto continuously monitor their neighbors [13]. Consideringshort time connections between nodes, reputation systemsdo not appear to be a proper approach in ephemeral net-works [14].

In other work, a local revocation process is introduced toconsider the dynamic nature of ephemeral networks [14]–[21]. In the revocation process, a benign node as an initiatoris assumed to detect (or become suspicious of) a maliciousnode and broadcasts its identification (ID) as a target node oran accused node. Then, other benign neighboring nodes runa local voting-based scheme to decide whether to discreditthe target node. Scholars in [14]–[16] studied the localrevocation process as a sequential voting game in which abenign node can choose one of three strategies with regard to(w.r.t.) a target node: voting, abstaining, or self-sacrificing.A benign node chooses between a voting strategy or anabstaining strategy based on its economic considerationsin the game. Also, the benign node might use a self-sacrificing strategy to declare the invalidity of its currentidentity as well as the identity of the target node. Somelimitations of the proposed revocation process in vehicularad hoc networks (VANETs) have been pointed out by Liuet al. [16]. For example, they underlined the assumption

of complete information for nodes in the game and provedthat the identification of the target node may not be possiblewithout considering the rate of false positives and the rateof false negatives.

To address existing problems and introducing new ap-proaches for misbehavior detection in mobile networks,Abass et al. [17] introduced an evolutionary game whereinall benign nodes cooperate in the voting game, focusing onunsuccessful revocation and over-reacted revocation deci-sions. Scholars [18], [19] have developed a weighted votinggame based on clustering architecture to effectively solvethe problem of false accusation. Masdari [20] proposeda collaborative false accusation approach to stop wrongaccusations in the network. Diakonikolas and Pavlou [22]emphasized the inverse power index problem in designingweighted voting games and proved that the problem is com-putationally intractable for a broad family of semi-values.In another work, Subba et al. [23] proposed an intrusiondetection system (IDS) that employs the concept of electionleaders and a hybrid IDS, with the aim of avoiding thecontinuous monitoring of nodes in mobile ad hoc networks(MANETs). These authors then extended their work in[24] by providing a multi-layer game theory to addressthe problem of dynamic network topologies in VANETs.While these efforts are effective at minimizing the volumeof IDS traffic, they do not address the uncertainty of nodesalongside incentives in local voting games. Others [25] havestudied the impact of incentives on misbehavior detection inunmanned aerial vehicle (UAV)-assisted VANETs. Silva etal. [26] proposed a voting scheme to generate a large setof novel strategies from available expert-based ones. Theproposed scheme selects the best strategies while the modelof opponents are considered during the game. However, thiswork does not focus on identifying a malicious node in anephemeral network.

C. NOVELTIES AND CONTRIBUTIONSSome differences distinguish this paper from other sig-nificant contributions in the literature. First, we design alocal voting game wherein the type of target node can beeither malicious or benign. This is in contrary to severalpapers [15], [17], [20], in which authors presume that abenign initiator broadcasts the ID of a malicious node asthe target node. The reason of our assumption is that everynode, including malicious nodes, can accuse others and allbenign nodes do not necessarily need to know each otherin the network. In our design, nodes vote regarding thetype of target node based on the accuracy and cost oftheir monitoring systems as well as a pre-defined beliefabout the portion of malicious nodes in the network. Theseconsiderations provide a framework in which nodes lookat their resources and potential hostile environments beforetaking part in the voting game. Second, we consider thatbenign nodes are uncertain about the strategy of maliciousnodes in the network. This is because a malicious nodemight intentionally not attack a benign node in order to

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obtain its support during the local voting game. This pointis omitted in the design of the local voting game in [14]–[20]. In particular, these works mainly focused on targetnode identification based on the rate of participation ratherthan considering the malicious node strategy of avoidingbeing accused. Third, we design incentives to encourageonly knowledgeable nodes (i.e., nodes that have alreadymonitored the target node) in the game. Otherwise, theincentives lead to many random votes in the game, whichmight spoil the result of cooperation. This point is mostlyoverlooked in the above literature, and [27] that studied therole of nodes’ incentives in order to contribute to ephemeralsocial networks. Fourth, we consider that both benign nodesand malicious nodes can take part in the voting game. Thisimplies that a benign node cannot rely solely on others’votes, owing to misleading votes from malicious nodes. Theincomplete information of nodes in voting was not studiedin the above literature, including [14], [17], [20]. Fifth, thecost of a group participating in the game (a.k.a., social cost)should be designed based on nodes’ contributions and theiruncertainties about the results. For example, a cooperativenode should be punished less than an abstaining node whenthe collaboration becomes unsuccessful. Finally, a potentdesign should provide more rewards for decisive votes.Although some authors studied weighted voting games [19],[28], they mainly based their designs on clustering headsrather than the value of a vote in the middle of a localvoting games.

We implement the points above in analyzing misbehaviordetection using the local voting-based scheme in the pres-ence of uncertainty. Our main contributions in this papercan be summarized as follows:

• We provide a game layout between a benign player anda target node in one stage of a local voting game usinga static Bayesian game in order to detect misbehavingnodes in an ephemeral network. In this regard, we de-sign expected utilities (payoffs) that capture uncertainties(explained above) regarding detection systems, types ofparticipants, type of target node, and outcome of the game.We offer incentives in node payoffs in order to promotecooperation in the network. Furthermore, we introduce avariable benefit for cooperative nodes in which rewardsare adjusted according to the value of contributions inthe game. This scheme prevents nodes from abusingincentives by pointless participation.

• We derive possible equilibrium points between a benignplayer and a target node in the game using a pure-strategyBayesian Nash equilibrium (BNE) and a mixed-strategyBNE, ensuring that no node can improve its utility bychanging its strategy. The best strategies can be adoptedby malicious nodes and benign nodes w.r.t. different gameparameters.

• We provide extensive numerical results to verify theanalysis and investigate the impact of cooperation param-eters and uncertainties on the identification of malicious

nodes. Our results confirm the influence of the designedincentives, hence participation rate, on the strategies ofmalicious and benign nodes in an ephemeral network.We observe, in particular, that if the participation incen-tives go beyond a certain limit, then correct target-nodeidentification will be decreased, in spite of the growingparticipation rate.

D. PAPER ORGANIZATIONThe remainder of this paper is organized as follows. SectionII describes assumptions, the local voting game, and theobjectives of this paper. Section III formulates the game,including defining parameters, payoff design, and a variablebenefit scheme. Section IV applies Bayesian game analysisto derive equilibrium points in the proposed model. SectionV is devoted to extensive numerical results. Section VIconcludes the paper.

II. ASSUMPTIONS AND PROBLEM DESCRIPTIONA. NETWORK MODELWe study misbehavior detection in a network where nodeshave short-lived connections, and a centrally managed sta-tion is absent. We use a vehicular ad hoc network (VANET)as a typical example of an ephemeral network to explainour approach. We assume that nodes (e.g., vehicles) arepowerful enough to have wireless communication amongthemselves. We consider a contention-based medium, e.g.,IEEE 802.11p in a VANET, which can represent the natureof wireless channel access [14]. We further assume that abase station or a certificate authority has already establishedthe credential of the nodes; hence, each node has a uniqueID.

We assume that there are two types of nodes in thenetwork: malicious and benign. Malicious nodes may at-tack benign nodes by disseminating false information. Forexample, a malicious car might inject faulty data to thesensors of the car that follows it, in order to manipulatean optimal space between them [29]. On the other hand,a benign node is equipped with a monitoring system todetect abnormal or counterfeit signals. For example, anautonomous vehicle can use a set of anti-spoofing techniquesto detect fake global positioning system (GPS) signals [30].However, benign nodes do not necessarily need to monitorall of their neighbors, due to the cost of monitoring over allshort-lived connections.

B. LOCAL VOTING GAMENodes can participate in a local voting-based scheme (game)in order to determine the identity of a node in the network.The voting game starts when an initiator broadcasts the IDof a target node. Then, neighboring nodes choose either tovote or not to vote (abstain) on type of the target node. Eachnode calculates its costs and benefits in order to choose astrategy. Nodes can broadcast their decisions sequentially,and each node’s decision is made in one stage of the game.We assume that the belief of a node w.r.t. the target node is

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independently inferred and does not change (e.g., by othervotes) during the game. This is because nodes are uncertainabout the correctness of other votes. In other words, a nodemay not know the types of all previous nodes that havealready voted. Also, the node is unaware of the target node’sstrategy w.r.t. previous nodes. Hence, we assume that a nodevotes based on its own detection system. It is worth notingthat monitoring all neighboring nodes and their interactionsmight be too costly for a node in an ephemeral network.The target node is identified when the number of votes inone type (either malicious or benign) reaches a pre-definednumber. This number is denoted by nth and is studied insection V-B . If correct (wrong) votes reach nth, then wewill have correct (wrong) target node identification. If nth isnot reached during the game, then we will have undecidedtarget node identification.

Malicious nodes and benign nodes can choose somestrategies in the game. A malicious node can select to attackor not to attack a benign node, while it is unaware of beingmonitored. After a target node is determined, a benign nodechecks whether it has already monitored the target node. Ifit has not monitored the target node, it will abstain fromvoting, simply because it does not have any informationabout the node. But, if the benign node has monitored thetarget node, it will calculate its payoffs. If its voting payoffoutweighs its abstaining payoff, then the benign node willvote; otherwise, it will abstain. On the other hand, maliciousnodes vote against a benign target node and for a malicioustarget node. If there is no possibility for malicious nodesto change the result of the game in their favor, then theyabstain from voting. We do not consider strategic maliciousnodes that can optimize their types of votes to collect somecredits or to send multiple wrong votes (e.g., Sybil attack[31]).

C. PROBLEM DEFINITION

Fig. 1 shows an example of a local voting game in a VANETthat helps us explain the problem. As can be seen, nodes0, 2, and 5 are malicious, and the other nodes are benign. Weassume that node 0 is the target node and nth = 2. We alsoassume that node 1 casts a correct vote, and node 2 castsa wrong vote. Now, node 3 should reveal its strategy. Here,we study the possible reactions of node 3 (as an exampleof a benign node) in the game where identification of thetarget node is not yet finalized. If node 3 has not monitorednode 0, then it simply abstains. Otherwise, if it has alreadymonitored node 0, it can select between voting or abstaining.To choose its strategy, node 3 faces some uncertainties thatgreatly impact its decision.

The first uncertainty of node 3 is about its own monitoringsystem, which has a predefined detection rate and falsealarm rate. For example, the monitoring system of node3 might recognize node 0 as benign, even though it ismalicious. Next, node 0 might choose not to attack node 3in order to produce a wrong perception. Hence, node 3, evenif not attacked by node 0, would still be uncertain about the

?

3

4

5Initiator

1

2

0

FIGURE 1: Example of local voting game in VANETs.

type of node 0. In addition, node 3 might not have monitorednodes 1 and 2; hence, it cannot vote based on previous votes.Moreover, node 3 may not have monitored all remainingnodes in the voting game, so it cannot count on their correctvotes. This implies that the node is also uncertain about theoutcome of the game, which potentially affects its decision.Nevertheless, incentives should be offered to encouragenode 3 (and others) in cooperation. This could also avoidthe formation of free-riders (i.e., nodes that benefit fromthe results without making any contributions [32]) in thenetwork. It is worth noting that, depending on the stage ofthe game, a node’s strategy could make a different level ofimpact on the results. For instance, if node 3 abstains, thenthe vote of node 4 is absolutely necessary for correct targetidentification (nth = 2). Hence, incentives should be offeredw.r.t. the value of a contribution.

On the other hand, malicious nodes are aware of anexisting voting game in the network. That is, a maliciousnode knows that it might become a target node. Theobjective of a malicious node is to maximize the level ofits aggressiveness in the network without being identified.However, it is uncertain about being monitored by a benignnode, and the strategy of a benign node in the game (i.e.,voting or abstaining). In contrast, a benign node knows thatsome of its neighbors may be malicious. The objective of amonitoring benign node is to choose a strategy with the aimof target node identification. However, a benign node hassome limitations in its detection system. Also, it is uncertainabout the strategies of malicious nodes and, therefore, isuncertain about the type of the target node. Taking thesepoints into consideration, our goal is the following:• To design payoffs for a benign node w.r.t. the explained

uncertainties and the value of its contribution in the game,• To determine the best strategies for malicious nodes and

benign nodes.We address the first problem in section IV by considering

the following: (i) the vote of a benign node that could beeither correct or incorrect; (ii) the probability of correcttarget node identification in each stage, which is mainlybased on votes that have already been cast; and (iii) theimpact of a benign node’s strategy on correct, wrong, andundecided target node identification. We address the secondproblem in section V. In particular, we develop one stage ofthe voting game as a Bayesian game to study the reactions ofa benign node w.r.t. a target node. This helps us understandthe best strategies of both types of nodes in the network.

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To find the best strategies of players in the game, onecould employ the concept of subgame perfect equilibrium,wherein nodes have complete information in a sequentialgame. However, as explained above, complete informationdoes not seem to be a realistic assumption for nodes in anephemeral network. Hence, we do not apply this conceptto our model. Another solution is to use the concept ofperfect Bayesian equilibrium (PBE) in dynamic Bayesiangames to capture the incomplete information of nodes. Inthis context, a node needs to update its belief regarding itsopponent’s type according to the game evolution. However,we believe that the belief of a benign player regarding thetype of target node should not change by other votes duringthe game because of existing uncertainties about the typesof neighboring nodes, their interactions, and the cost ofmonitoring over all neighbors in an ephemeral network. Inprinciple, we assume that nodes are relying on their owndetection systems to vote on the type of the target node. Inthis regard, we can not properly apply the concept of PBEto our model.

On the other hand, a Bayesian game is general enough tocapture many scenarios between attackers and defenders inone stage of the game [32]. In particular, we can consider apotential attacker as a target node that chooses to attackor not to attack a benign node, and the benign playerthat can vote or abstain in one stage of the game. Oneadvantage of applying Bayesian games to our model is thatnodes independently infer the type of a target node using adetection system, and they do not need to know the type ofall neighbors and their interactions with each other. In otherwords, a node can efficiently select its strategy according tothe BNE that maximizes its expected utility with a set ofparameters. However, one drawback of using BNE is thata benign node requires a sensible prior belief regarding thetype of neighboring nodes. We let µ denote the prior beliefof benign nodes about the portion of malicious nodes ina network (parameters are described in section III-A). Inpractice, a node should be able to adjust the values of µaccording to its knowledge of an environment. Nodes couldassign a high value for µ in potentially hostile places suchas crowded areas in mega cities, and a low value for µ insafe places such as rural areas.

III. PROBLEM FORMULATIONIn this section, we first introduce a set of parameters that arerequired to define the underlying game. Next, we providesome definitions to facilitate the explanation of our model.Then, we design payoffs that include individual beliefsand available voting information in the game. Finally, wepropose a variable-benefit scheme that rewards participantsaccording to the value of their contributions.

A. PARAMETERSTo describe our model, we need to define a set of givenparameters to design player’s payoffs in the game. We listthese parameters in Table 1. To begin, we assume that a

TABLE 1: List of parameters in alphabetical order.

Meaning↵

�

pk

w

cm

ca

cv

b

cgm

cgb

nv1

nv2

nth

SymbolsProbability of detection (true positive)

Probability of false alarm (false positive)

Value of an asset

Cost of attack

Cost of voting

n Total number of nodes

Probability of monitoringPm

nl

nr

µ

Cost of monitoring of an asset

Prior probability of node being malicious

Cost of group for incorrect identification ofbenign target node

malicious target nodeCost of group for incorrect identification of

Number of nodes left at kth stage

Number of required votes at kth stageto identify target node

Number of required votes to identify target node

Number of correct votes for target node

Number of incorrect votes for target node

Benefit of correct strategy

Probability of attack for malicious PLT

Probability of voting for monitoring PLBs

q

� Probability of a remaining node stays in the neighborhood

�b Punishment of incorrect strategy

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k � 1<latexit sha1_base64="pbiTt/4MnVrclk5DenVh3Gy5iys=">AAACLnicdVDLSgMxFM34rOOr6tJNsBQUtczU2sdCKIjgsoJ9QFtLJk1tmExmSDKFMswXufFXdCGoiFs/w0xbQUUvhHvuuedyc48TMCqVZT0bc/MLi0vLqRVzdW19YzO9td2QfigwqWOf+aLlIEkY5aSuqGKkFQiCPIeRpuOeJ/3miAhJfX6txgHpeuiW0wHFSGmql77I8l40suMjmOR8bGYhh8dw3z7UtYgPkjoBOrs3kRrGXwQ8mwhd0z22zV46Y+XsfKl4UoBWzioVi+VTDQqV03LFgnbOmkQGzKLWSz92+j4OPcIVZkjKtm0FqhshoShmJDY7oSQBwi66JW0NOfKI7EaTc2OY1UwfDnyhH1dwwn6fiJAn5dhztNJDaih/9xLyr147VINyN6I8CBXheLpoEDKofJh4B/tUEKzYWAOEBdV/hXiIBMJKO5yY8HUp/B808jlbe3VVyFQLMztSYBfsgX1ggxKogktQA3WAwR14AC/g1bg3now3430qnTNmMzvgRxgfnwZ9o5k=</latexit><latexit sha1_base64="pbiTt/4MnVrclk5DenVh3Gy5iys=">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</latexit><latexit sha1_base64="pbiTt/4MnVrclk5DenVh3Gy5iys=">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</latexit><latexit sha1_base64="pbiTt/4MnVrclk5DenVh3Gy5iys=">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</latexit>

nl = n � k<latexit sha1_base64="JqEs9meZkLoy9aSV7vMZ00z8l+k=">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</latexit><latexit sha1_base64="JqEs9meZkLoy9aSV7vMZ00z8l+k=">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</latexit><latexit sha1_base64="JqEs9meZkLoy9aSV7vMZ00z8l+k=">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</latexit><latexit sha1_base64="JqEs9meZkLoy9aSV7vMZ00z8l+k=">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</latexit>

nr

kth node

FIGURE 2: kth stage of the game, where nv1, nv2, andnr denote the number of correct, incorrect, and remaining

votes, respectively, and nl refers to the total number ofnodes left to vote.

benign node holds an asset with a security value of w,where w > 0. This parameter is considered in order tocount for the value of a vulnerable asset in benign nodesthat can be exploited by malicious nodes. . A maliciousnode could compromise the asset by paying the cost of anattack, denoted by ca. In contrast, a benign node protects itsasset by monitoring for attacks, with probability Pm. Thismonitoring costs cm for the node, and all costs are positive.It is rational to assume that w > ca and w > cm. Otherwise,the attacker and the benign node lose their motivation toattack and to protect the asset, respectively. A benign nodeassigns a prior probability of µ for its neighbors to bemalicious. We measure the performance of the monitoringsystem by considering the following: (i) α, which representsthe detection rate (i.e., true positive rate), and (ii) β, whichdenotes the false alarm (i.e., false positive rate) for detectingabnormalities. It is sensible to expect that α > 0.5 > β.

It is assumed that n nodes are in a neighboring area. Eachbenign node can vote by paying cv as the cost of voting.The benefit of a correct strategy and the punishment of anincorrect strategy for a benign node are denoted by b and

VOLUME 4, 2016 5

A. Behfarnia et al.: Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

−b, respectively. To generalize the analysis, we design thegame at one stage, say the kth stage, in which the typeof a target node has not yet been determined. Until thisstage, it can be assumed that nv1 votes in one type (e.g.malicious) and nv2 votes in another type (e.g. benign) havealready been cast for the type of target node. We let nrdenote the number of remaining votes required to identifythe target node. Therefore, nr = nth − nv1 or nr = nth −nv2, depending on the belief of the kth node on the type oftarget node. Also, there are nl nodes left in the game. Fig. 2helps us understand these parameters. We use pk to denotethe probability of correct target node identification at the kth

stage. It is assumed that the cost of the group (neighboringnodes) for the incorrect identification of a malicious targetnode and a benign target node are cgm and cgb, respectively.

A malicious node can choose to either attack or not attacka benign node based on its information in the game. Inthis regard, the probability of attack is considered and it isdenoted by q in the analysis. Likewise, a node can chooseto either vote or not vote. This is captured by defining theprobability of voting and is denoted by s in the analysis. Itis worth noting that a node considers different parameters toselect its strategy, including the accuracy of the monitoringsystem, costs, benefits, punishments, and the probabilitiesintroduced here. Equipped with these parameters, we areable to design the expected payoffs for players in the game.

B. GAME DEFINITION AND NOTATIONSHere, we provide some definitions in order to facilitate thedescription of our model.

Definition 1: A Bayesian game G is defined by a tuple(N,A,Θ, µ, U), where N is a set of players, A is a set ofactions, Θ is a set of players’ types, µ is a common prior,and U is a set of utilities. These are defined as follows:

• N = (PLT, PLB), where PLT is a target node and PLBis a benign player.

• A = (aPLT , aPLB), where aPLT ∈{

attack, notattack

}, and aPLB ∈

{vote, abstain

}.

• Θ = (θPLT , θPLB), where θPLT ∈{

malicious,benign

}, and θPLB ∈

{benign

}.

• Prior belief: PLB assigns µ for PLT being malicious,and PLB’s type is common knowledge.

• U = (u, v), where u refers to PLB’s payoff, and vrefers to PLT’s payoff.

Fig. 3 shows the strategic form of the game G, whererows and columns indicate the actions of a target node anda benign player, respectively. As can be seen, each windowincludes pairs of payoffs (uz , vz), where 1 ≤ z ≤ 9, andthe subscript z refers to the actions of both PLB and PLTin one scenario. For example, (u1, v1) at top left window inFig. 3 corresponds to voting PLB and attacking PLT. As canbe seen in Fig. 3, we need to define uz and vz to providethe layout of the game.

Definition 2: Each player’s payoff can be defined as thesummation of an individual payoff and a group payoff as

Attack

Not

Vote Abstain

Attack

Vote Abstain

NotAttack

Malicious node, µ

Benign node, 1 � µ

⇣Pm u2 + (1 � Pm) u3,

Pm v2 + (1 � Pm) v3

⌘

Pm v5 + (1 � Pm) v6

⌘⇣Pm u5 + (1 � Pm) u6,

⇣Pm u8 + (1 � Pm) u9,

Pm v8 + (1 � Pm) v9

⌘

�u4, v4

�

�u1, v1

�

�u7, v7

�

PLB

PLT

FIGURE 3: Players’ payoffs in the game relative to a benignplayer (PLB) and malicious or benign target node (PLT).

follows:

uz = uz,i + uz,g, (1)vz = vz,i + vz,g, (2)

where uz,i and vz,i denote individual payoffs, and uz,gand vz,g denote group payoffs. The individual payoff onlyconsiders interactions between two players, while the grouppayoff accounts for the impact of a player’s strategy on allmembers in the neighborhood.

It is worth noting that the group payoffs capture theimpact of a node’s strategy on the security of all nodesin the network. This comes from a node by either voting orabstaining in the voting game. Group payoff also provides aframework to capture incentives for promoting cooperationin the game. In particular, the collaboration of nodes formalicious node identification would reward them, whileabstaining from the collaboration could punish them in thegame. These rewards and penalties are included in the grouppayoff of a node, because they relate to actions that impactall nodes in the network. The following section describesthe design of both individual and group payoffs in detail.

C. PAYOFF DESIGNIn what follows, we describe individual payoffs and grouppayoffs and find all uz and vz shown in Fig. 3. Individualpayoffs account for the costs and benefit of monitoring andnon-monitoring, whereas group payoffs account for the costsand benefits of voting or abstaining from the game. Theincentives of the game, including benefits and punishments,are considered in the group payoffs. This means that anode measures its group payoff before choosing its strategy.Initially, we focus on obtaining individual payoffs, i.e., uz,i.Then, we explain group payoffs, i.e., uz,g . Finally, we usedefinition 2 to add individual and group payoffs to obtain

6 VOLUME 4, 2016

A. Behfarnia et al.: Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

all uz and vz .

1) Individual payoffs

To study individual interactions between two nodes, notethat a malicious node could choose either to attack or not toattack a benign node. Also, the benign node could be eitherin a monitoring state or a non-monitoring state. Hence, wehave four possible scenarios between a malicious node anda benign node:

• A monitoring PLB faces an attacking PLT.• A non-monitoring PLB faces an attacking PLT.• A monitoring PLB faces a non-attacking PLT.• A non-monitoring PLB faces a non-attacking PLT.

What follows is a study of the payoffs for these scenariosto evaluate related uz,i and vz,i.

Scenario I: A monitoring PLB faces an attacking PLT.Here, the monitoring PLB pays −cm as the cost of monitor-ing. However, it gains (2α−1)w from its detection system.This is because the expected gain of the PLB relates to thetrue positive rate (α) and the false negative rate (1− α) ofits detection system, i.e., αw − (1 − α)w. Therefore, theindividual payoff of a monitoring PLB in this scenario isequal to −cm + (2α− 1)w. This payoff corresponds to u1,iand u2,i, where the PLT attacks a monitoring PLB. Hence,

u1,i = u2,i = −cm + (2α− 1)w. (3)

On the other hand, the PLT pays −ca as the cost of theattack. The loss of the PLT can be assumed as the negativegain of the PLB’s individual payoff [11], i.e. −(2α− 1)w.Therefore, the individual payoff for an attacking maliciousPLT in this scenario equals −ca − (2α − 1)w. This payoffcorresponds to v1,i and v2,i. Hence,

v1,i = v2,i = −ca − (2α− 1)w. (4)

Scenario II: A non-monitoring PLB faces an attackingPLT. Here, the non-monitoring PLB does not pay the costof monitoring but loses its asset as a result of being in anon-monitoring state and attacking the PLT. Therefore, theindividual payoff of a non-monitoring PLB in this scenariois equal to −w. This payoff corresponds to u3,i. Hence,

u3,i = −w (5)

On the other hand, the PLT pays −ca as the cost ofthe attack, and its gain is equal to the loss of the non-monitoring PLB, i.e. +w. Therefore, the individual payofffor an attacking malicious PLT in this scenario equals−ca + w. This payoff corresponds to v3,i. Hence,

v3,i = −ca + w. (6)

Scenario III: A monitoring PLB faces a non-attackingPLT. Here, a monitoring PLB pays −cm as the cost ofmonitoring. However, since the PLT is in a non-attackingstate, any possible detection comes from its false alarm rate,i.e., −βw. Therefore, the individual payoff of a monitoring

PLB in this scenario is equal to −cm − βw. This payoffcorresponds to u4,i, u5,i, u7,i, and u8,i. Hence,

u4,i = u5,i = u7,i = u8,i = −cm − βw. (7)

The non-attacking PLT does not gain or lose in the interac-tion. Thus,

v4,i = v5,i = v7,i = v8,i = 0. (8)

Scenario IV: A non-monitoring PLB faces a non-attackingPLT. Here, since there has been neither an attack from thePLT nor monitoring from the PLB, we have 0 payoff. Thispayoff corresponds to u6,i, u9,i, v6,i, and v9,i. Hence,

u6,i = v6,i = u9,i = v9,i = 0. (9)

Having uz,i and vz,i in hand, we continue defining uz,gand vz,g to be able to use definition 2 and obtain all payoffs.

2) Group payoffs

To study group payoffs, note that a monitoring PLB canchoose between voting and abstaining on the type of PLTin the game. A non-monitoring PLB always chooses toabstain from voting because it does not have any informationabout PLT. In this respect, we can study group payoffs byconsidering four scenarios:

• Monitoring PLB faces an attacking malicious PLT.• Monitoring PLB faces a non-attacking malicious PLT.• Monitoring PLB faces a (non-attacking) benign PLT.• PLB is in a non-monitoring state.

In the first three scenarios, we study the voting and ab-staining strategies of a monitoring PLB w.r.t. the probabilityof correct target node identification (pk) in the game. This isbecause the target node is not yet identified in the network;hence, the PLB should consider the probability of correct,wrong, or undecided target node identification before choos-ing its strategy. In this regard, we use Fig. 4(a) to show thedependency between a monitoring PLB’s strategy and pk.As shown, the left column corresponds to the player’s votingstrategy and the right column corresponds to its abstainingstrategy. Also, the top row (labeled pk) refers to the casethat the target node is correctly identified in the game, andthe lower row (labeled 1 − pk) refers to the case that thetarget node is not correctly identified in the game. In thisfigure, X, Y, Z, and W denote payoffs for different possiblecases. For example, X in the top left window in Fig. 4(a)represents the case where the PLB votes and target nodeis correctly identified in the game. We will design X, Y,Z, and W for different scenarios between the PLT and amonitoring PLB. Using Fig. 4(a), we define group payoffsfor the voting and abstaining strategies of the PLB.

Definition 3: Considering two possible outcomes fortarget node identification, denoted by pk and 1− pk in Fig.4(a), we define two group payoffs for possible strategies of

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A. Behfarnia et al.: Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

Vote Abstain

pkb � cv 0

�cv

�cv � cgm

�(1 � pk)b

pkb

�cgm

(a) (b)

�cv � cgb

(c)

pkb � cv 0

�(1 � pk)b�cgm

�(1 � pk)b�cgb

�cv � cgm

Vote AbstainVote Abstain

pk

1 � pk

Vote Abstain

pk

1 � pk

FIGURE 4: Group payoffs: (a) for a monitoring benignplayer in general, which is then broken down to the

following scenarios: (b) malicious target node has attackeda monitoring benign node, (c) malicious target node hasnot attacked a monitoring benign node, and (d) benign

target node versus a monitoring benign node.

a monitoring PLB at the kth stage:

ug (vote) = pk (X) + (1− pk)(Z), (10)ug (abstain) = pk (Y ) + (1− pk)(W ), (11)

where the payoff of each strategy is weighted by thecorresponding probabilities.

In the last scenario, since a non-monitoring PLB alwaysabstains from voting regardless of the PLT’s strategy, itsgroup payoff depends on other nodes’ actions in the game.In what follows, we study payoffs for all the above scenariosto evaluate related uz,g and vz,g .

As can be seen by Eqs. (10)-(11), pk plays an importantrole in the payoffs. In this respect, we obtain pk to evaluatethe voting and abstaining strategies of players. It is note-worthy that the value of pk increases when the PLB votescorrectly. We use δ to denote this improvement in pk.

Lemma 1. pk and δ can be written as follows:

pk =

nl∑

i=nr

(nli

) (ps)i (

1− ps)nl−i

, (12)

δ =

(nl

nr − 1

) (ps)nr−1 (

1− ps)nl−(nr−1)

, (13)

where ps , λ(1 − µ)αPm represents the probability ofcorrect target identification by a remaining node in thegame.

This lemma and theorems are proved in the appendix.

Scenario I: The monitoring PLB faces the attackingmalicious PLT, which relates to the first row of Fig. 3.The group payoffs in this scenario correspond to u1,g ,u2,g , v1,g , and v2,g . Fig. 4(b) shows the voting payoff (i.e.,u1,g) and the abstaining payoff (i.e., u2,g) for a monitoring

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1pk

-4

-3

-2

-1

0

1

Gro

up P

ayof

fs

VotingAbstaining

FIGURE 5: Group payoffs for monitoring benign noderelative to pk.

PLB w.r.t. pk. As can be seen in Fig. 4(b), −cv in theleft column (i.e., vote) represents the cost of voting. Also,−cgm in the lower row (i.e., 1 − pk) denotes the cost ofincorrect identification for a malicious target node. Thereward for voting in correct target identification (top leftwindow) and the punishment of abstaining in incorrect targetidentification (bottom right window) are represented by bpkand −b(1− pk), respectively. These are proportional to pkbecause the player’s expected outcome is entangled with theprobability of correct target node identification (pk) in themiddle of the game. The reward and the punishment areconsidered (as incentives) to encourage nodes in coopera-tion. Using Fig. 4(b) along with equations (10) and (11) indefinition 3, we have

u1,g = pk ×(pkb− cv

)+(1− pk

)×(− cv − cgm

),

⇒ u1,g = p2kb− cv − (1− pk)cgm, (14)

u2,g = pk ×(0)

+(1− pk

)×(− (1− pk)b− cgm

),

⇒ u2,g = −(1− pk)2b− (1− pk)cgm. (15)

We also define group payoffs for the PLT in this scenarioby (1 − pk)cgm, which indicates the inverse proportionalrelationship between pk and the gain of the malicious PLT.Hence, we have

v1,g = v2,g = (1− pk)cgm. (16)

In order to better understand the impact of a node’s grouppayoff on selecting its strategy, we provide a plot in Fig. 5that shows voting payoffs (i.e., u1,g) and abstaining payoffs(i.e., u2,g) of PLB w.r.t. different pks in this scenario. Here,it is assumed that cv = 1, b = 1.5, and cgm = 2. As canbe seen, depending on the value of pk, voting payoffs canoutweigh abstaining payoffs, and vice versa. For example,voting payoffs are dominant for pk < 0.2, which meansthat the player votes in this interval of pk. In fact, votingis an attempt by PLB to increase pk and avoid an incorrectoutcome of the game. The main motivation of the player,however, comes from the game’s punishment. That is, thecost of voting is lower than the punishment of the gamewhen the malicious target node is not correctly identified. Inother words, if pk → 0, then −cv > −(1− pk)b (see lower

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row of Fig. 4(a)). Therefore, the player votes not only toincrease pk but also to avoid punishment in the game. Thesame reasoning can be used for other intervals of pk, i.e.,pk > 0.8 and 0.2 < pk < 0.8, to determine the motivationsof the player for voting and abstaining in the game.

Scenario II: The monitoring PLB faces the non-attackingmalicious PLT, which relates to the second row of Fig.3. The group payoffs in this scenario correspond to u4,g ,u5,g , v4,g , and v5,g . Fig. 4(c) shows the voting payoff (i.e.,u4,g) and the abstaining payoff (i.e., u5,g) for a monitoringPLB w.r.t. pk. Here, a monitoring PLB might unintentionallysupport a malicious PLT by its vote because the maliciousPLT is in a non-attacking state. In this regard, we definea penalty by −(1 − pk)b for voting in an incorrect targetidentification (bottom left window), and a reward by pkbfor abstaining in a correct target identification (top rightwindow). This prevents the PLB from blind voting (solelyto gain the benefit of collaboration) when it did not senseabnormalities from a node. Using Fig. 4(c) along withequations (10) and (11) in definition 3, we have

u4,g = pk(− cv

)+(1− pk

)(− (1− pk)b− cv − cgm

),

⇒ u4,g = −(1− pk)2b− cv − (1− pk)cgm, (17)

u5,g = pk ×(pkb)

+(1− pk

)×(− cgm

),

⇒ u5,g = p2kb− (1− pk)cgm. (18)

Since the malicious PLT is in a non-attacking mode, wedefine v4,g = v5,g = 0, which implies that the non-attackingPLT does not gain or lose in this scenario.

Scenario III: The monitoring PLB faces a (non-attacking)benign PLT, which relates to the third row of Fig. 3. Thegroup payoffs in this scenario correspond to u7,g , u8,g , v7,g ,and v8,g . Fig. 4(d) shows the voting payoff (i.e., u7,g) andthe abstaining payoff (i.e., u8,g) for a monitoring PLB w.r.t.pk. The design of payoffs in Fig. 4(d) is quite similar toscenario I in Fig. 4(b), where cgm is replaced by cgb. Hence,the reasoning for this scenario follows the same lines asscenario I. Using Fig. 4(c) along with equations (10) and(11) in definition 3, we have

u7,g = pk ×(pkb− cv

)+(1− pk

)×(− cv − cgb

),

⇒ u7,g = p2kb− cv − (1− pk)cgb, (19)

u8,g = pk ×(0)

+(1− pk

)×(− (1− pk)b− cgb

),

⇒ u8,g = −(1− pk)2b− (1− pk)cgb. (20)

Since benign PLT is in a non-attacking mode, we havev7,g = v8,g = 0.

Scenario IV: The PLB is in a non-monitoring state. In thiscase, the PLB abstains from voting, regardless of the PLT’sstrategy. The group payoffs in this scenario correspondto u3,g , u6,g , u9,g , v3,g , v6,g , and v9,g . To define u3,gand u6,g , where the PLT is malicious, we know that thenon-monitoring PLB completely relies on other nodes fortarget node identification. If pk = 1, then the node isnot harmed, but if pk = 0, then it gets −cgm as thecost of the incorrect malicious PLT identification. Thus,

we define u3,g = u6,g = −(1 − pk)cgm, where the nodedoes not impact the group decision while it is affected byother decisions. We can apply a similar reasoning for u9,g ,with the only difference being that cgm is replaced by cgb,because the PLT is benign, i.e., u9,g = −(1−pk)cgb. On theother hand, we define v3,g = (1−pk)cgm, which reflects theinverse proportional relationship between pk and the gain ofthe attacking malicious PLT. In the case of v6,g and v9,g ,the benign PLT does not attack; hence, there is no gain orloss, i.e., v6,g = v9,g = 0.

3) Total payoffs

By designing individual payoffs and group payoffs foreach scenario in the game, we can use definition 2 to obtainall payoffs. For example, u1 is the summation of u1,i (i.e.,equation (3)) and u1,g (i.e., equation (14)). Thus, usingequation (1) in definition 2, we have

u1 = −cm + (2α− 1)w + p2kb− cv − (1− pk)cgm, (21)

We obtain the remaining payoffs in the same fashion, asfollows:

u2 = −cm + (2α− 1)w − (1− pk)2b− (1− pk)cgm,(22)

u3 = −w − (1− pk)cgm, (23)

u4 = −cm − βw − (1− pk)2b− cv − (1− pk)cgm, (24)

u5 = −cm − βw + p2kb− (1− pk)cgm, (25)u6 = −(1− pk)cgm, (26)

u7 = −cm − βw + p2kb− cv − (1− pk)cgb, (27)

u8 = −(1− pk)2b− (1− pk)cgb − cm − βw, (28)u9 = −(1− pk)cgb, (29)v1 = v2 = −ca − (2α− 1)w + (1− pk)cgm, (30)v3 = −ca + w + (1− pk)cgm, (31)v4 = v5 = v6 = v7 = v8 = v9 = 0. (32)

D. VARIABLE-BENEFIT SCHEMESo far, we have assumed that the benefit of a correct strategy(b) is constant, irrespective of its impact on the outcome ofthe game. In principle, a variable benefit could be designedto be commensurate with the impact of kth player’s strategyon the target node identification. In this regard, we choosepk and µ as two important arguments that directly affect thevalue of a strategy in the game. We categorize these benefitsinto two cases. The first is the benefit when the PLB hasdetected an abnormality in the PLT. This implies that themalicious PLT has attacked the PLB, or that the abnormalitysimply comes from a false alarm. We denote the benefit fora correct strategy in this case by b1. Here, group payoffs arethe same as in Fig. 4(a), wherein b = b1. The second case iswhen a monitoring PLB has not detected any abnormalitiesfrom the PLT, i.e., whether the PLT is malicious or benign.We denote the benefit for a correct strategy in this case by

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A. Behfarnia et al.: Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

0 0.5 1pk

1

1.5

2

Ben

efit

(b1)

01

0.4

5B

enef

it (b

2)

0.3pk

0.5 0.2

10

0.10 0(a) (b)

FIGURE 6: Benefits with regard to probability ofsuccessful target identification in kth stage (pk) and

portion of malicious nodes (µ) in network.

b2. Payoff tables for this case are the same as those in Fig.4(b) and Fig. 4(c), wherein b = b2. By making the PLBindifferent, b1 and b2 can be derived.

Lemma 2. b1 and b2 for the kth stage of the game can beobtained as follows:

b1 =cv

(2p2k − 2pk + 1), (33)

b2 =cv

(1− 2µ)(2p2k − 2pk + 1). (34)

Assuming cv = 1, the value of b1 and b2 versus pk and µare shown in Fig. 6. As can be seen in Fig. 6(a), whenpk = 0.5, the highest b1 occurs, and when pk = 0 orpk = 1, the lowest b1 occurs, underlining the fact that thehighest benefit is rewarded for a PLB that faces the highestuncertainty of pk (similar to gambling!). On the other hand,as shown in Fig. 6(b), when µ (the portion of maliciousnodes) grows, the value of benefits also increases. This isnot surprising because as µ increases, benign nodes shouldbe more motivated for participation to reveal the identity ofa target node before malicious nodes determine the game’sresult with their votes.

Here, we study the impact of b1 on group payoffs. Thestudy of group payoffs with b2 can be done in the samefashion. Fig. 7 shows group payoffs with b = b1, where eachsubplot corresponds to a window in Fig. 4(b). For example,the top left subplot in Fig. 7 refers to pkb− cv in Fig. 4(b),where b = b1. In Fig. 7, we assume that cv = 1 and cgm =4. As can be seen, payoffs in the upper row dominate overthose in the lower row. This is because the upper payoffsshow successful target node identification, while the lowerpayoffs indicate the unsuccessful counterpart. To understandthis better, let us study the trend of graphs for a few pks.First, let pk = 0.25, which yields b1 = 1.6 from eq. (33).Under this assumption, if the game is to successfully identifythe target node (upper row in Fig. 7), then PL2 collects ahigher payoff by abstaining (0 > −0.6). However, if thegame becomes unsuccessful (lower row), then PL2 shouldhave voted in the game (−5 > −5.2). Next, assume thatpk = 0.75. Interestingly, we obtain the same benefit (i.e.,b1 = 1.6). In this case, however, the strategy of PL2 will

0 0.5 1-1

-0.5

0

0.5

Payo

ff

0 0.5 1-2

0

2

Payo

ff

0 0.5 1Probability

-5.5

-5

-4.5

-4

Payo

ff

0 0.5 1Probability

-6

-5

-4

Payo

ff

Voting AbstainingX: 0.5Y: 0

(pk) (pk)

X: 0.5Y: -5

FIGURE 7: Expected group payoffs for scenario I withvariable benefits.

be the opposite. That is, PL2 is willing to vote if it thinksthe game will be successful (0.2 > 0), while it abstainsif it thinks the game will be unsuccessful (−.4.4 > −5).This example helps us summarize the following intuitionsfor the case of adaptive benefit: (i) There is no pure strategyfor PL2 during the game w.r.t. a specific pk or b; and (ii)pk = 0.5 is the only point where PL2 becomes completelyindifferent between voting and abstaining, regardless of theresult of the game (see pk = 0.5 in Fig. 7). This is the placewhere our design offers the highest benefit (see Fig. 6) inorder to encourage PL2 to participate, hence, increase pk.

IV. EQUILIBRIUM ANALYSISThe objective of the players is to maximize their payoffs inthe game. In this regard, we obtain possible equilibriumpoints using a Bayesian game to better understand thebehavior of the players. In particular, we obtain the beststrategies of benign players to identify a malicious node,while we find the maximum level of aggressiveness formalicious nodes without being identified. In this respect, weuse the interactions between a PLB and a PLT, as illustratedin Fig. 3. Let us quickly summarize the possible strategiesof the players. A non-monitoring PLB has one pure strategy:abstaining. A monitoring PLB has two strategies: votingor abstaining. A benign PLT has one pure strategy: to notattack. Finally, a malicious PLT could choose two strategies:to attack or not to attack. Depending on game parameters, apure-strategy BNE may or may not exist. This is addressedin the following theorem.

Theorem 1. Given µ and Pm, if cgm ≥ ca + (2α− 1)w+δcgm, and b > 2 cv , then the game has one pure-strategyBNE. A malicious node attacks and a monitoring benignnode votes in the game.

Remark 1: When the damage caused by incorrect mali-cious node identification (cgm) is higher than the summationof PLT’s cost of attack (ca), PLT’s individual loss againsta monitoring PLB, i.e., (2α − 1)w, (see eq. (4)), and theimpact of PLB’s vote in the game (δcgm), then a maliciousnode attacks a benign node. When the benefit of voting fora monitoring PLB is more than twice of its cost, then the

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benign node chooses to vote.The pure-strategy BNE, as seen above, holds only under

certain conditions on the parameters. Our game is a finitestrategic-form game. Hence, a mixed-strategy BNE can beapplied to gain a broader perspective for the game analysis.In this regard, to determine each player’s indifference strat-egy, we define q as the probability of attack for a maliciousPLT, and s as the probability of voting for a monitoringPLB.

Theorem 2. Given µ and Pm, the game defined in sectionIII has a mixed-strategy BNE, which is as follows:

• Malicious node attacks with a probability of q∗, whichis

q∗ =q1 + q2 + ...+ qn

n, (35)

where qk, k = 1, ..., n, is the probability of attack forthe kth node in the game

qk =Ak

Bk, (36)

Ak = µ(1 + Pm

)(2p2k − 2pk + 1

)b+

(1− Pm

)

×(cm + βw

)+ cv − p2kb− Pm

(1− pk

)2b,

Bk = µ(1 + Pm

)(2p2k − 2pk + 1

)b

+ µ(1− Pm

)(2α+ β)w.

• Monitoring benign node votes with probability of s∗,which is equal to

s∗ =ca + (2αPm − 1)w − cgm

(1− Pm)[−ca + (1− 2α)w + cgm]− δcgm.

(37)

Note that the mixed-strategy provides general equilibriumpoints w.r.t. different parameters. In a special case, if allnodes monitor their neighbors, i.e., Pm = 1, then an upperbound for the benefit and a lower bound for the detectionrate can be derived using eqs. (36) and (37), respectively.

Corollary 1. In Theorem 2, if Pm = 1, then

b <cv

(1− 2µ)(2p2k − 2pk + 1), (38)

α >w − ca + cgm(1− δ)

2w. (39)

From eq. (38), it can be seen that as µ→ 12 , the upper bound

increases. This allows network designers to select highervalues of benefit in environments where the probability of amalicious PLT is higher. On the other hand, eq. (39) impliesthat a monitoring system must have a minimum true positiverate in order to make a malicious node indifferent in thegame.

Corollary 2. In Theorem 2, if Pm = 1, and benefits aredesigned using b1 and b2 in Eqs. (33) and (34), then theprobability of attack by a malicious node will be zero, i.e.,

1 1.5 2 2.5 3 3.5 4 4.5 5Benefit (b)

0

20

40

60

80

100

Perc

enta

ge

Undecided Identification Correct Identinfication Wrong Identification

Region IVRegion III

Region II

Region I

FIGURE 8: Game outcomes versus benefit variations.

q∗ = 0.

While the result of this corollary is the most desirableoutcome for every game designer, we should note thatachieving it might require a strong assumption (Pm = 1)and a complex system for a beneficial design, because itrequires all monitoring nodes in an ephemeral network thatadapt their benefits in each stage according to pk.

In the next section, we evaluate the performance of thegame in order to obtain a better picture of the above analysis.

V. NUMERICAL RESULTSTo evaluate our analysis, we assume that 40 nodes can runthe game in an area that is 625 m × 625 m (normal density≈ 100 nodes

km2 in [33]). Since the analysis is probabilistic, werun 100 iterations for each simulation. Then, we take anaverage of the results with 95% confidence interval. Thedefault game parameters are as follows:

• Monitoring system parameters: α = 0.95, β = 0.05,• Probabilities: Pm = 0.75, µ = 0.2, and q = 0.4,• Costs and benefits: cgb = cgm = 4, w = 4, b = 3,cm = ca = cv = 1.

If we change these parameters to better explain a scenario,then we will explicitly mention it. Nevertheless, we describethe theoretical results in three subsections. Initially, we studythe impact of incentives (in particular, b) on correct, wrong,and undecided target node identification. Then, we focuson the behavior of malicious nodes w.r.t. their portion andaggressiveness in the network. This section also includes acomparison among different nths. Finally, we compare ourwork with scenarios where the uncertainties discussed inthis paper have not been considered, e.g., [15], [17], [20].In all these cases, we evaluate and compare the percentageof target node identification for different sets of givenparameters that lead to different equilibria for the game.

A. IMPACT OF INCENTIVESFig. 8 illustrates the percentage of target node identificationversus b. Here, it is assumed that q = 0.7. As shown,this figure can be categorized into four different regions. Inregion I, the percentage of undecided target identificationoutweighs correct and wrong identifications for a simplereason: the benefit is not large enough to persuade nodes

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A. Behfarnia et al.: Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Probability of Attack (q)

0

20

40

60

80

100

Perc

enta

ge Undecided Ident. = 0.2 Correct Ident. = 0.2 Wrong Ident. = 0.2 Undecided Ident. = 0.1 Correct Ident. = 0.1 Wrong Ident. = 0.1

FIGURE 9: Impact of portion of malicious nodes (µ) andprobability of attack (q) on identification results.

to participate in the game. Region II, however, illustrates adrastic reduction of undecided identification. This indicatesthat voting payoffs become larger in comparison to abstain-ing payoffs. In addition, correct identification dominatesover wrong identification, which is the result of the follow-ing: (i) benign nodes with high monitoring and detectionrates (i.e., Pm = 0.75 and α = 0.95), and (ii) maliciousnodes with a high level of aggressiveness (i.e., q = 0.7).Region III shows a slight increase in correct identificationand a decrease in undecided identification because of lowerpayoffs for abstaining from the game. The increase of wrongidentification over undecided identification is remarkable inregion IV. Wrong votes in this region mainly come fromhighly encouraged benign nodes that have not been attackedby a malicious target node. In other words, since votingpayoffs are significantly larger than abstaining payoffs (i.e.,u4 > u5 and u7 > u8), a benign node votes in favor ofa non-attacking target node. This observation reveals thatpersuading every node to vote by applying the leverage ofbenefit does not necessarily lead to a better outcome. Takingall regions into consideration, region III indicates the bestoption for the benefit design.

B. IMPACT OF MALICIOUS NODES AND NTH

Fig. 9 shows the percentage of target identification w.r.t.the portion of malicious nodes and their probability ofattack (q) in the network. As shown, when q increases,correct identification generally increases, which confirmsthat aggressive attackers can be more easily identified.However, wrong identification is reduced after a certainvalue of q; for example, q = 0.1 for µ = 0.1. When thenumber of malicious nodes increases in the network, thisdecreasing trend starts at higher values of q; for instance,q = 0.4 for µ = 0.2. This reveals that malicious nodesbecome more aggressive when their number increases inthe network.

Fig. 10 depicts identification results w.r.t. the variationof nth and µ. As nth grows larger, a turning point occurs,whereby undecided identification prevails over correct andwrong identifications. The reason is that a higher number ofnodes must participate to make an outcome of identification.

10 15 20 25 30 35 40nth

0

20

40

60

80

100

Perc

enta

ge

Undecided Ident., = 0.10Correct Ident., = 0.10Wrong Ident., = 0.10Undecided Ident., = 0.20Correct Ident., = 0.20Wrong Ident., = 0.20

FIGURE 10: Impact of required votes, nth, on targetidentification.

0.8 0.85 0.9 0.95 1(a) Detection rate ( )

0

20

40

60

80

100

Perc

enta

ge o

f co

rrec

t ide

ntifi

catio

n Pm = 0.95Pm = 0.75

0.1 0.15 0.2 0.25 0.3 0.35 0.4(b) Fraction of malicious nodes ( )

0

20

40

60

80

100

with uncertainty without uncertainty

FIGURE 11: Impact of game uncertainties relative: (a)detection rate, and (b) correct identification rate.

For example, consider plots for µ = 0.2, where nth = 23is the turning point. This number comes from the fact thatout of 40 neighboring nodes, on average, 8 of them aremalicious nodes (40 × 0.2). Also, from the remaining 32benign nodes, those that are in the monitoring state, i.e.,24 (32 × 0.75), might vote depending on their payoffs. Ifnth = 23, then almost all such nodes must participate toavoid undecided target identification. A designer can adjustthe nth w.r.t. a range of µs to obtain an acceptable correctidentification rate.

C. COMPARISONIn this section, we compare our work with the scenarioswhere some of the explained uncertainties have not beenconsidered (e.g. [15], [17], [20]). It is worth mentioningthat the comparison is limited to highlighting uncertaintiesin those scenarios. This is because the nature of theirgames and objectives are slightly different. However, thiscomparison provides us with insight into the effect ofincomplete information at nodes on the outcome of a localvoting game.

Fig. 11(a) shows the impact of the true positive detectionrate (α) on the correct target identification. As can be seen,it is essential for nodes to have high values of α in orderto gain high correct target identification. The values of αbecome more important when fewer benign nodes monitortheir neighbors (i.e., smaller Pm). Fig. 11(b) indicates acomparison between a design with and without uncertainties

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A. Behfarnia et al.: Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

in the local voting game. In particular, we assume that adesign without uncertainty has the following parameters:α = 1, β = 0, and q = 1. As shown, the difference betweengraphs is the growth of µ. This is because a player withoutuncertainty considers a non-attacking malicious node asa benign node and votes for it. On the other hand, theproposed design prevents benign nodes from voting whenthey are unsure about the strategy of malicious nodes. Inboth scenarios, when µ goes beyond a threshold, here 0.3,correct target identification is significantly reduced. Thiscomes from higher payoffs for abstaining in comparison tovoting. Interpreted differently, benign nodes are unwilling tocooperate in a game in which a high portion of participantsare malicious.

VI. CONCLUSIONIn this paper, we have provided a game-theoretic approachto identify malicious nodes in ephemeral networks, wherecentral stations are not available. In particular, we havestudied the strategies of nodes in a local voting-based gameusing a Bayesian game, in which nodes have incompleteinformation about the accuracy of their monitoring systems,the type of neighbors (benign or malicious), and the outcomeof the game. By offering incentives in expected utilities, wehave provided encouragements for game participation withthe aim of improving correct node identification. We havederived possible Bayesian Nash equilibrium (BNE) pointsand mixed-strategy BNE points to study the best strategiesof players in the game. Simulation results have shown theimpact of different parameters, such as participation benefitsand detection rate, on the identification of malicious nodes.

VII. APPENDIXA. PROOF OF LEMMA 1Proof. To obtain pk, note that nv1 and nv2 votes havealready been cast until the kth stage, while there are nlnodes left in the game. To derive a closed form for pk, notethe following: (i) If nr > nl, then pk = 0, which means thatthe number of left nodes is less than the number of requiredvotes to identify the PLT; (ii) if nr = 0, then pk = 1,which implies that the PLT has been already identified; (iii)pk directly depends on nl and their type; and (iv) if nris reduced, then pk will be increased. Taking these pointsinto account, pk can be written in the form of eq. (12),where ps represents the probability of correct target nodeidentification. For instance, assume n = 10, k = 7 (i.e.,nl = 3), ps = 1/3, and nth = 4. Under such assumptions,if nv1 = 0 (i.e., nr = 4), then equation (12) yields pk = 0because of the first condition. If nv1 = 4 (i.e., nr = 0),then eq. (12) yields pk = 1 because of the second condition.Also, substituting nr = 1 and nr = 3, respectively, yieldspk = 0.7 and pk ≈ 0.04, which confirm the last condition.We can define ps = λ(1− µ)αPm, where λ represents theprobability of a remaining node to be in the network and1−µ is the probability of the remaining node to be benign.

Since δ is defined as the difference that a correct vote can

make in pk, we have

δ = pk(voting)− pk(abstaining),

⇒ δ =

nl∑i=nr−1

(nl

i

) (ps)i (

1− ps)nl−i −

nl∑i=nr

(nl

i

) (ps)i (

1− ps)nl−i

, (40)

which yields eq. (13).

B. PROOF OF LEMMA 2Proof. A node must remain indifferent between voting andabstaining for all pks and µs. That is,

Eu(voting)z = Eu(abstaining)z,

z , {attack, not attack}, (41)

where Eu(.) denotes expected utility function, and z is thestrategy of PLT. Applying eq. (41) for b1 (Fig. 4(a)), andassuming small values of β, we obtain

pk(pkb1 − cv) + (1− pk)(−cv − cgm)

= (1− pk)[−(1− pk)b1 − cgm]. (42)

Simplifying eq. (42) yields eq. (33). Eq. (34) can beobtained in the same fashion using Figs. 4(b)-(c).

C. PROOF OF THEOREM 1Proof. We first derive combined payoffs for two types ofPLT (malicious and benign) and the PLB. Then, usingstrictly dominated strategies, we prove the theorem w.r.t.the conditions.

Fig. 12 shows the combination of payoffs. The first andthe second element of each column (e.g., {A,NA}) indicatethe strategy of a malicious PLT and a benign PLT. Forinstance, {A,NA} represents the attacking (A) and thenon-attacking (NA) strategies from PLT. In addition, φisand νis denote expected payoffs for the PLT and the PLB,respectively. For instance, ν1 is obtained as follows:

ν1 = µPmu1 + (1− µ)Pmu7, (43)

where u1 and u7 are obtained in eqs. (21) and (27),respectively. Based on the values of φis, if cgm ≥ ca+(2α−1)w + δcgm, then the left column in Fig. 12 ({A,NA})strictly dominates the right column ({NA,NA}). On theother hand, we need to show that ν1 > ν3 to obtain pure-strategy BNE. Substituting the equations of ν1 and ν3 yields

(1− Pm)µw + (1− Pm)(1− µ)(1− pk)cgb + (1− Pm)×µ(1− pk)cgm + Pm[(2p2k − 2pk + 1) b− cv] > 0 (44)

As can be seen, the first three items in eq. (44) are equalto or greater than zero. Hence, it suffices to say that the lastterm is positive. This happens only if b > cv

(2p2k−2pk+1)

. Thisinequality, however, must hold for all pks, which means thatthe right-hand side of the inequality must be maximized.This yields pk = 0.5, which implies b > 2 cv .

VOLUME 4, 2016 13

A. Behfarnia et al.: Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

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A, NA<latexit sha1_base64="8E2baxfPBxhzqauxfrzQy8YBKYM=">AAACGnicbVA9SwNBEN3z2/MrammzGA4sQriTgBYWBhsLkQgmCrlw7G0mZnFv79idE8OR32HjX7GxUMRObPw3bmIKvx4MPN6bYWZenElh0Pc/nKnpmdm5+YVFd2l5ZXWttL7RMmmuOTR5KlN9GTMDUihookAJl5kGlsQSLuLro5F/cQPaiFSd4yCDTsKulOgJztBKUSnwQpVHtUpYoWHWF1HN9Rongeu1UgTXq8cGmVBuiHCLRb1CT+vDqFT2q/4Y9C8JJqRMJmhEpbewm/I8AYVcMmPagZ9hp2AaBZcwdMPcQMb4NbuCtqWKJWA6xfi1IfWs0qW9VNtSSMfq94mCJcYMkth2Jgz75rc3Ev/z2jn29juFUFmOoPjXol4uKaZ0lBPtCg0c5cASxrWwt1LeZ5pxtGm6NoTg98t/SWu3GvjV4KxWPjyYxLFAtsg22SEB2SOH5Jg0SJNwckceyBN5du6dR+fFef1qnXImM5vkB5z3T6NAnYM=</latexit><latexit sha1_base64="8E2baxfPBxhzqauxfrzQy8YBKYM=">AAACGnicbVA9SwNBEN3z2/MrammzGA4sQriTgBYWBhsLkQgmCrlw7G0mZnFv79idE8OR32HjX7GxUMRObPw3bmIKvx4MPN6bYWZenElh0Pc/nKnpmdm5+YVFd2l5ZXWttL7RMmmuOTR5KlN9GTMDUihookAJl5kGlsQSLuLro5F/cQPaiFSd4yCDTsKulOgJztBKUSnwQpVHtUpYoWHWF1HN9Rongeu1UgTXq8cGmVBuiHCLRb1CT+vDqFT2q/4Y9C8JJqRMJmhEpbewm/I8AYVcMmPagZ9hp2AaBZcwdMPcQMb4NbuCtqWKJWA6xfi1IfWs0qW9VNtSSMfq94mCJcYMkth2Jgz75rc3Ev/z2jn29juFUFmOoPjXol4uKaZ0lBPtCg0c5cASxrWwt1LeZ5pxtGm6NoTg98t/SWu3GvjV4KxWPjyYxLFAtsg22SEB2SOH5Jg0SJNwckceyBN5du6dR+fFef1qnXImM5vkB5z3T6NAnYM=</latexit><latexit sha1_base64="8E2baxfPBxhzqauxfrzQy8YBKYM=">AAACGnicbVA9SwNBEN3z2/MrammzGA4sQriTgBYWBhsLkQgmCrlw7G0mZnFv79idE8OR32HjX7GxUMRObPw3bmIKvx4MPN6bYWZenElh0Pc/nKnpmdm5+YVFd2l5ZXWttL7RMmmuOTR5KlN9GTMDUihookAJl5kGlsQSLuLro5F/cQPaiFSd4yCDTsKulOgJztBKUSnwQpVHtUpYoWHWF1HN9Rongeu1UgTXq8cGmVBuiHCLRb1CT+vDqFT2q/4Y9C8JJqRMJmhEpbewm/I8AYVcMmPagZ9hp2AaBZcwdMPcQMb4NbuCtqWKJWA6xfi1IfWs0qW9VNtSSMfq94mCJcYMkth2Jgz75rc3Ev/z2jn29juFUFmOoPjXol4uKaZ0lBPtCg0c5cASxrWwt1LeZ5pxtGm6NoTg98t/SWu3GvjV4KxWPjyYxLFAtsg22SEB2SOH5Jg0SJNwckceyBN5du6dR+fFef1qnXImM5vkB5z3T6NAnYM=</latexit><latexit sha1_base64="8E2baxfPBxhzqauxfrzQy8YBKYM=">AAACGnicbVA9SwNBEN3z2/MrammzGA4sQriTgBYWBhsLkQgmCrlw7G0mZnFv79idE8OR32HjX7GxUMRObPw3bmIKvx4MPN6bYWZenElh0Pc/nKnpmdm5+YVFd2l5ZXWttL7RMmmuOTR5KlN9GTMDUihookAJl5kGlsQSLuLro5F/cQPaiFSd4yCDTsKulOgJztBKUSnwQpVHtUpYoWHWF1HN9Rongeu1UgTXq8cGmVBuiHCLRb1CT+vDqFT2q/4Y9C8JJqRMJmhEpbewm/I8AYVcMmPagZ9hp2AaBZcwdMPcQMb4NbuCtqWKJWA6xfi1IfWs0qW9VNtSSMfq94mCJcYMkth2Jgz75rc3Ev/z2jn29juFUFmOoPjXol4uKaZ0lBPtCg0c5cASxrWwt1LeZ5pxtGm6NoTg98t/SWu3GvjV4KxWPjyYxLFAtsg22SEB2SOH5Jg0SJNwckceyBN5du6dR+fFef1qnXImM5vkB5z3T6NAnYM=</latexit>

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PLB

PLT

FIGURE 12: Expected payoffs for combined types of PLTand the PLB.

D. PROOF OF THEOREM 2Proof. To obtain q∗, we first equalize the expected utilitiesfor voting and abstaining to obtain qk. Then, we take anaverage over all possible values of the pks to get eq. (35).In this way

Eu[voting

]= Eu

[abstaining

](45)

where,

Eu[voting

]= µqu1 + µ

(1− q

)u4 +

(1− µ

)u7,

(46)

Eu[abstaining

]= µ q Pmu2 + µ q

(1− Pm

)u3 + µ

(1−

q)Pmu5 + µ

(1− q

)(1− Pm

)u6 +

(1−

µ)Pmu8 +

(1− µ

)(1− Pm

)u9. (47)

Substituting eqs. (21), (24), and (27) into eq. (46), and eqs.(22), (23), (25), (26), (28), and (29) into eq. (47), and thensubstituting eqs. (46) and (47) in eq. (45) yields eq. (36).Since the malicious PLT might attack the neighboring nodesregardless of their stage in the game, we take an averageover all values of qks, which yields eq. (35).

To calculate s∗, we can equalize the expected utilities ofattack and not attack from the PLT, hence obtaining

µ s v1 + Pm µ (1− s) v2 + (1− Pm)µ v3 = 0. (48)

Plugging eqs. (30) and (31) back into eq. (48) yields eq.(37).

REFERENCES

[1] A. Behfarnia and A. Eslami, “Local voting games for misbehavior de-tection in vanets in presence of uncertainty,” in 57th Annual AllertonConference on Communication, Control, and Computing (Allerton), Sep.2019.

[2] A. Rahim, X. Kong, F. Xia, Z. Ning, N. Ullah, J. Wang, and S. K. Das,“Vehicular social networks: A survey,” Pervasive and Mobile Computing,vol. 43, pp. 96–113, 2018.

[3] L. Yin, Y. Guo, F. Li, Y. Sun, J. Qian, and A. Vasilakos, “A game-theoreticapproach to advertisement dissemination in ephemeral networks,” WorldWide Web, vol. 21, no. 2, pp. 241–260, 2018.

[4] T. Qiu, B. Chen, A. K. Sangaiah, J. Ma, and R. Huang, “A survey of mo-bile social networks: Applications, social characteristics, and challenges,”IEEE Systems Journal, vol. 12, no. 4, pp. 3932–3947, 2018.

[5] E. C. Akrida, L. Gasieniec, G. B. Mertzios, and P. G. Spirakis, “Ephemeralnetworks with random availability of links: The case of fast networks,”Journal of Parallel and Distributed Computing, vol. 87, pp. 109–120, 2016.

[6] R. van der Heijden, S. Dietzel, and F. Kargl, “Misbehavior detection invehicular ad-hoc networks,” 1st GI/ITG KuVS Fachgespräch Inter-VehicleCommunication. University of Innsbruck, pp. 23–25, 2013.

[7] S. Shivshankar and A. Jamalipour, “An evolutionary game theory-basedapproach to cooperation in vanets under different network conditions,”IEEE Transactions on Vehicular Technology, vol. 64, no. 5, 2015.

[8] H. Hasrouny, A. E. Samhat, C. Bassil, and A. Laouiti, “Misbehaviordetection and efficient revocation within vanet,” Journal of InformationSecurity and Applications, vol. 46, pp. 193–209, 2019.

[9] R. W. van der Heijden, S. Dietzel, T. Leinmüller, and F. Kargl, “Survey onmisbehavior detection in cooperative intelligent transportation systems,”IEEE Communications Surveys & Tutorials, 2018.

[10] G. Loukas, E. Karapistoli, E. Panaousis, P. Sarigiannidis, A. Bezemskij,and T. Vuong, “A taxonomy and survey of cyber-physical intrusion detec-tion approaches for vehicles,” Ad Hoc Networks, vol. 84, pp. 124–147,2019.

[11] Y. Liu, C. Comaniciu, and H. Man, “Modelling misbehaviour in ad hocnetworks: A game theoretic approach for intrusion detection,” Journal ofSecurity and Networks, vol. 1, no. 3-4, pp. 243–254, 2006.

[12] M. H. Manshaei, Q. Zhu, T. Alpcan, T. Bacsar, and J.-P. Hubaux, “Gametheory meets network security and privacy,” ACM Computing Surveys(CSUR), vol. 45, no. 3, p. 25, 2013.

[13] F. Hendrikx, K. Bubendorfer, and R. Chard, “Reputation systems: Asurvey and taxonomy,” Journal of Parallel and Distributed Computing,vol. 75, pp. 184–197, 2015.

[14] M. Raya, M. H. Manshaei, M. Félegyházi, and J.-P. Hubaux, “Revocationgames in ephemeral networks,” in Proceedings of the 15th ACM Confer-ence on Computer and Communications Security, Alexandria, VA, USA,2008, pp. 199–210.

[15] I. Bilogrevic, M. H. Manshaei, M. Raya, and J.-P. Hubaux, “Optimalrevocations in ephemeral networks: A game-theoretic framework,” inProceedings of the 8th International Symposium on Modeling and Opti-mization in Mobile Ad Hoc and Wireless Networks (WiOpt), Avignon,France, 2010, pp. 21–30.

[16] B. Liu, J. T. Chiang, and Y.-C. Hu, “Limits on revocation in vanets,” in 8thInternational Conference on Applied Cryptography and Network Security,Beijing, China, 2010, pp. 38–52.

[17] A. A. A. Abass, N. B. Mandayam, and Z. Gajic, “An evolutionary gamemodel for threat revocation in ephemeral networks,” in 51st AnnualConference on Information Sciences and Systems (CISS), Baltimore, MD,USA, 2017, pp. 1–5.

[18] S. I. Lekha and R. Kathiroli, “Trust based certificate revocation of mali-cious nodes in manet,” in International Conference on Advanced Commu-nications, Control and Computing Technologies, 2014, pp. 1185–1189.

[19] S. Kim, “Effective certificate revocation scheme based on weighted votinggame approach,” IET Information Security, vol. 10, no. 4, pp. 180–187,2016.

[20] M. Masdari, “Towards secure localized certificate revocation in mobile ad-hoc networks,” IETE Technical Review, vol. 34, no. 5, pp. 561–571, 2017.

[21] M. Arshad, Z. Ullah, N. Ahmad, M. Khalid, H. Criuckshank, and Y. Cao,“A survey of local/cooperative-based malicious information detectiontechniques in vanets,” EURASIP Journal on Wireless Communicationsand Networking, vol. 2018, no. 1, p. 62, 2018.

[22] I. Diakonikolas and C. Pavlou, “On the complexity of the inverse semi-value problem for weighted voting games,” in Proceedings of the AAAIConference on Artificial Intelligence, vol. 33, 2019, pp. 1869–1876.

[23] B. Subba, S. Biswas, and S. Karmakar, “Intrusion detection in mobilead-hoc networks: Bayesian game formulation,” Engineering Science andTechnology, an International Journal, vol. 19, no. 2, pp. 782–799, 2016.

[24] ——, “A game theory based multi layered intrusion detection frameworkfor vanet,” Future Generation Computer Systems, vol. 82, pp. 12–28, 2018.

[25] C. A. Kerrache, A. Lakas, N. Lagraa, and E. Barka, “Uav-assisted tech-

14 VOLUME 4, 2016

A. Behfarnia et al.: Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

nique for the detection of malicious and selfish nodes in vanets,” VehicularCommunications, vol. 11, pp. 1–11, 2018.

[26] C. R. Silva, R. O. Moraes, L. H. Lelis, and K. Gal, “Strategy generationfor multi-unit real-time games via voting,” IEEE Transactions on Games,2018.

[27] R. Esmaeilyfard, F. Hendessi, M. H. Manshaei, and J. Grossklags, “Agame-theoretic model for users’ participation in ephemeral social vehic-ular networks,” International Journal of Communication Systems, vol. 32,no. 12, p. 3998, 2019.

[28] F. Hof, W. Kern, S. Kurz, K. Pashkovich, and D. Paulusma, “Simple gamesversus weighted voting games: bounding the critical threshold value,”Available at SSRN 3270445, 2018.

[29] A. Ferdowsi, U. Challita, W. Saad, and N. B. Mandayam, “Robust deepreinforcement learning for security and safety in autonomous vehicle sys-tems,” in 2018 21st International Conference on Intelligent TransportationSystems (ITSC). IEEE, 2018, pp. 307–312.

[30] A. Behfarnia and A. Eslami, “Risk assessment of autonomous vehiclesusing bayesian defense graphs,” in IEEE 88th Vehicular TechnologyConference (VTC-Fall), Aug 2018, pp. 1–5.

[31] B. Yu, C.-Z. Xu, and B. Xiao, “Detecting sybil attacks in vanets,” Journalof Parallel and Distributed Computing, vol. 73, no. 6, pp. 746–756, 2013.

[32] D. Fudenberg and J. Tirole, Game Theory. MIT Press, 1991.[33] J. A. Sanguesa, F. Naranjo, V. Torres-Sanz, M. Fogue, P. Garrido, and F. J.

Martinez, “On the study of vehicle density in intelligent transportationsystems,” Mobile Information Systems, vol. 2016, 2016.

ALI BEHFARNIA (S’16) received the B.S. degreeand the M.S. degree in electrical engineeringfrom University of Tabriz and Iran University ofScience and Technology (IUST), respectively. Heis currently a Ph.D. candidate in the Departmentof Electrical Engineering and Computer Scienceat Wichita State University, Kansas, USA. He is arecipient of the Bright Future Award from Wichita

State Ventures, and the Donald D. Sbarra Endowed Fellowship in 2016.His research interests include resilient cyber-physical system, error controlcoding, game theory, and communications over wireless networks.

ALI ESLAMI (S’07-M’13) received the PhD inelectrical and computer engineering from theUniversity of Massachusetts, Amherst, in 2013.He is an assistant professor of electrical engi-neering and computer science at Wichita StateUniversity, Wichita, Kansas. He is the recipient ofWichita State’s Young Faculty Risk Taker Awardfor 2016-2017 academic year. From August 2014

to June 2015, he worked as a visiting research scholar of informationinitiative at Duke (iiD). He held the position of postdoctoral research fellowwith Texas A&M University, College Station, Texas, from March 2013 toApril 2015. He is a member of the IEEE Communications and ComputerSocieties, and has served as a reviewer for numerous IEEE journals. He hasalso served on several NSF review panels, and as session chair in severalIEEE conferences and workshops. His current research interests includenano-communications, applications of coding theory in biology, resilientdesign of cyber-physical systems, fault-tolerant quantum computing, andbig-data storage systems.

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