Routing and network performance are the main constraints to be considered while designing VANETs. As malicious nodes are termed to degrade the performance of VANETs, dynamic Bayesian signaling game is used to reveal vehicles with normal and abnormal activities. In this approach, cooperative game theory is chosen. Figure 1. gives the detailed flow diagram of the proposed system. The game consists of two players Vehicle 1 (V1) Vehicle 2 (V2) where V1 acts as the Sender and V2 as the Receiver. Here V1, V2 are not aware of each other's type. A = [\({a}_{1},{a}_{2}\dots {a}_{j}\)] is the set of messages from which V1 chooses a message and sends it to V2. Finally V2 fixes the required action from the action space B = {cooperate(C), decline(D)}. Perfect Bayesian equilibrium (PBE) is a strategy in which V1 chooses the message and V2 chooses the action. The node’s strategy is determined by the payoff calculation and belief update mechanism. It can be either pure, mixed, or PBE. In the case of pure strategy, the vehicle’s type cannot be altered or changed. In PBNE ( Perfect Bayesian Nash Equilibrium) based on the type of the other players, the strategy profile and the beliefs are specified for each player. In this approach, some of the sender and relay vehicles exhibits abnormal activities and are termed as malicious. This information about the malicious nodes is reported to the neighbor vehicles.
A. Algorithm to choose feasible action for players.
Input: Two players [Sender (V1), Receiver (V2)]
Output: Project the malicious activity
Start:
Initiate Vehicle (V1) and Vehicle (V2)
Define the strategy profile of the players.
Determine the vehicle’s type {Regular or Malicious}
Apply Bayes rule and analysis the belief for the vehicles.
Apply Bayes rule and analysis the belief for the vehicles.
Intend the optimal payoff for V1 and V2.
Find the operable action{C or D}
if not operable then
Convey to other vehicles/nodes as malicious
else
Choose to Cooperate or Decline (C or D)
end if
Stop
The sender V1 has type δ = {Regular, Malicious}. The receiver V2 trusts its own type as probability of p(δ) is 1.The sender V1, observes the information about its own type and decides to choose an action. Similarly, V2 checks the action chosen by V1 and chooses a reaction. Here, \({ x}_{i}\left(c,d,{\delta }\right)\)denotes the payoff of the sender vehicle V1, \({p}_{1}\)(δ) denotes the probability distribution for the sender’s strategy over the action Cooperate (C) and \({\rho }_{2}\) (\({y}_{1}\)) denotes the probability distribution over the action Decline (D).
Sender’s payoff is calculated in Eq. (1)
\({x}_{1}\) (\({\rho }_{1},{\rho }_{2},\theta\)) =\({\sum }_{{y}_{1}}^{l}o{\sum }_{{y}_{2}}{\rho }_{1 }\left({y}_{1}\right{\delta }){\rho }_{2}\)(\({y}_{2}\left{y}_{1}\right){x}_{1}\)(\({y}_{1},{y}_{2},\)δ) (1)
Receiver’s payoff is calculated in Eq. (2)\(\)
\({ x}_{2}\) (\({\rho }_{1},{\rho }_{2},\theta\)) = \(\sum _{{\delta }}^{p}p\left({\delta }\right)\) \({\sum }_{{y}_{1}}^{l}o{\sum }_{{y}_{2}}{\rho }_{1 }\left({y}_{1}\right{\delta }){\rho }_{2}\)(\({y}_{2}\left{y}_{1}\right){x}_{1}\)(\({y}_{1},{y}_{2},\)δ) (2)
where \({y}_{1}\)and \({y}_{2}\) are the actions chosen by V1 and V2.
\({P}_{V1 }\) : Ɐ\({\delta }\), \({ x}_{1}\)(\({\rho }_{1}^{*},{\rho }_{2}, \theta )\) ≥ \({ x}_{1}\)(\({\rho }_{1}^{*},{\rho }_{2}, \theta )\), (3)
\({P}_{V2}\) : Ɐ, \({y}_{1}{x}_{2}\)(\({\rho }_{1}^{*},{\rho }_{2}, ɸ)\) ≥ \({ x}_{2}\)(\({\rho }_{1}^{*},{\rho }_{2},{ɸ}^{*})\), (4)
\({P}_{Q}\) : \({ɸ}^{*}\)(\({\delta }{y}_{1}\)) = \(\frac{p\left({\delta }\right){{\rho }}_{1}^{\text{*}}\left({\text{y}}_{1}{\delta }\right)}{\begin{array}{c}\sum p({{\delta }}^{\text{'}}\left){\rho }_{1}^{*}\right({y}_{1}\left{{\delta }}^{\text{'}}\right)\\ l\end{array}}\) (5)
where \({ɸ}^{l}\)is uncertainty of nodes and \({P}_{V1 }\), \({P}_{V2}\) are the perfect Bayesian Equilibrium for the sender and receiver. The belief of type δ is given by \({P}_{Q}\). In case of mixed strategy, the stranger can have two types: {Regular, Malicious}. The probability for stranger vehicles to be determined as malicious and its action space is given as {Attack, Normal}. \({P}_{V1 }\) always behaves normally as it's probability is determined to be regular. {Doubt, Trust} are the two actions the neighbor nodes may perform on the stranger. When a 'Doubt' arises, the neighbor's help is asked to find the trustworthiness of the stranger.
B. Payoff Articulation
Generally, payoff ate the outcomes of the players in the game. It is usually a number. The brief procedure is as follows:
 Consider a stranger to be a regular vehicle, ‘\(g\)’ amount of payoff will be obtained by the target if it trusts, where\(g>0\).
 Consider a stranger to be a malicious vehicle; an amount of harm ‘ \(h\) ’ is caused to the target, if the target is attacked successfully. \(0<h<1\)
 Consider the stranger being doubted by the stranger, then it costs 1.
 Consider the stranger to be a malicious vehicle but pretends to be a normal one. In this case, the cost is more for the target to doubt, but the target may threaten the stranger more frequently.
 For invaluable trust, the payoff lost by the target is ‘h’ amount.
As in dynamic Bayesian signaling game, decline is a strategy dominated by cooperate, (D) is the best result.
C. Pure Strategy:
In pure strategy, the vehicle chooses the strategy those outcome the highly beneficial payoffs. This strategy provides at most profit to the players. The Nash Equilibrium is an action profile in which a vehicle is restricted to increase or decrease its payoffs. Consider the strategies {Attack, Doubt} and {Normal, Trust}. The receiver response to the sender as Doubt if the sender vehicle is aimed to show malicious activity. Similarly, the receiver vehicle response to the sender vehicle as Trust if the sender vehicle is aimed to show normal activities.
D. Mixed Strategy:
In the case of mixed strategy, the vehicle chooses more than one action based on the probability value in the strategic game. Here one of the two players plays a randomized strategy.
Definition 1: A mixed strategy is a neutral strategy for players to choose \(n\) number of actions dependent over the probability distribution, where\(n=({n}_{1}+{n}_{2}\dots \dots {n}_{k}\)) = 1 .
E. Perfect Bayesian Equilibrium:
PBE, a strategy profile whose payoff is dependent on its own belief and other players belief, regarding to update beliefs based on Bayes rule. The strategy is given by \(\rho =\left({\rho }_{1}{\rho }_{2}\right)\) and the fact set is given by \(X=\left\{x1,x2,x3\right\}\). Equations (6), (7) and (8) calculates the probability distribution on\(X.\)
$$ɸ\left(x1\right)=\alpha q,$$
6
$$ɸ\left(x2\right)=\alpha (1q),$$
7
$$ɸ\left(x3\right)=1\alpha .$$
8
The payoff of the receiver and the differential coefficient is calculated in the equations (9) and (10).
\(ɸ\) (\(\rho )=\left(3hg\right)\alpha qt+\left(gh\right)\alpha q+\left(g\alpha h\alpha 1\right)t+g\left(1\alpha \right). (9)\)
\(\frac{\partial ɸ}{\partial q}\) = \(\left(3hg\right)\alpha t+\left(gh\right)\alpha .\) (10)
Finally it is concluded as if q increases, the expected payoff the receiver(V1) will increase .
\(t> \frac{gh}{(g3h)}\) , considering the Eq. (9) < 0.
E. Calculating the payoff
Payoff of the regular and malicious vehicles constitutes the player's vantage point. This payoff depends on the player's action and the action of its neighbors. Figure (2) portraits the payoff calculation of each vehicle. Based on the expected payoff value, corresponding action is chosen by the sender and the receiver. Decline (D) is categorized for a node that is rejected for participating in packet forwarding. Cooperate(C) means that the vehicle can cooperate for packet forwarding.
Here, the sender receives a certain amount of payoff if it attacks the receiver. In the case of Decline (D), the regular vehicle has zero gain. If the receiver finds the sender to be malicious, it reports to the neighbors and fetches the gain Signaling Malicious(SM). Here PM (Prefer to Send) and SM are the profit gained to report and cooperate. (PSPS, C) and (SMSM, D) are the Nash Equilibrium for Perfect Bayesian Game. In the first case, (SMSM, D)  the sender avoids to send the message and checks in for decline. In the second case, (PSPS, C)  the sender sends the message and as the final step the receiver chooses to cooperate or decline. Hence based on the vehicle's type payoff is calculated.
F. Belief Update Mechanism
Malicious vehicles can raise the overhead and this leads to performance degradation while communicating. Belief updating needs to be carried out to avoid this issue. It is given by
p(θa) =\(\frac{p\left({\theta }^{\text{'}}\right)\rho \left(a\right\theta )}{p\left(\theta \right)\rho \left(a\right\theta )}\)
where ρ denotes the action and a is the message that the sender sends. The trust opinion is determined using Bayes rule. Decline(D) is categorized for a vehicle that is rejected for participating in packet forwarding. Cooperate(C) means the vehicle can cooperate for packet forwarding. Malicious and regular vehicles are determined by the belief update process. The following algorithm finds the belief strategy pairs for the PBE strategy.
INPUT: Incomplete information of the Sender (V1) and Receiver (V2).
OUTPUT: Belief strategy pairs for PBE.
MAIN:
Find the \(Type\) \({t}_{n}\) for \({V1}_{n}\) from \(type\) = {T,……\({T}_{L}\)} based on the distribution p(\({t}_{n}\)), where p(\({t}_{n}\)) > 0\({Ɐ}_{i}\) and \({\sum }_{L}\) p(\({t}_{n}\)) = 1. \({V1}_{n}\) sends message \({A}_{{sg}_{j}}\)from A ={\({a}_{1}\dots ..{a}_{j}\)} depending on the observed \({T}_{n}\). \({V2}_{n}\) chooses a reaction \({V2B}_{k }\)from B ={\({V2B}_{1}\)…..\({V2B}_{k}\)} depending on the observed\({A}_{{sg}_{j}}\)
Calculation of payoffs for Sender (\(POV1\)) and Receiver (\(POV2\)):
\({POV1}_{n}\) (\({t}_{n},{A}_{{sg}_{j}},{V2B}_{k}\)), \({POV2}_{n}\)(\({t}_{i},{a}_{j},{x}_{k}\))
\({V2}_{n}\) has a belief ɸ(\({t}_{n}\)\({A}_{{sg}_{j}}\))
for each \({A}_{{sg}_{j}}\) є A, V2B * \({A}_{{sg}_{j}}\); do
max \({a}_{j}\) є A \({V1}_{n}\)(\({t}_{n},{A}_{{sg}_{i}},{a}_{k}\))
end for
for each \({t}_{n}\) є \(Type\), \({A}_{sg}\)* \({t}_{n}\) do
max \({A}_{{sg}_{j}}\) є \(A \sum {t}_{n}\) є\(Type\)
end for
Pure strategies:
ɸ(\({t}_{n}\)\({A}_{{sg}_{j}}\)) = \(\frac{p\left({t}_{n}\right)}{\sum {t}_{n} є Type}\) =\(p\left({t}_{n}\right)\)
Belief strategy pairs (PBE):
[\({A}_{sg}\)* \({t}_{n}\) ,V2B * \({A}_{{sg}_{j}} \text{ɸ}\left({t}_{n}\right{A}_{{sg}_{j}})\)]
Table 2. Parameters for Simulation
PARAMETER

VALUE

Simulation area

1000 m× 500 m

Simulation Time

1000 s

Density of vehicles

5080 vehicles

Transmission range

50 m

Mobility model

Freeway model

Speed of Vehicle

[2060] m/s

Packet size

512 bytes

Pause Time

400 s
