2. Related Work
Numerous studies have investigated methods to promote cooperation in VANETs. Among the most popular are incentive mechanisms and game theory approaches. Incentive mechanisms are designed to motivate vehicles to cooperate with one another. Shevade et al. proposed an incentive-aware routing protocol that uses a Tit-for-Tat (TFT) mechanism to maximize performance in Delay Tolerant Networks (DTNs) [4
]. Zhu et al. developed a multilayer credit-based incentive system to encourage cooperation among DTN bundle nodes [5
]. Lu et al. tackled the free-riding problem in DTNs by introducing an incentive protocol that offers rewards to participating nodes to promote fairness [6
]. Patel et al. dealt with free-riding in VANETs by creating a honeypot that traps selfish vehicles in the network [7
]. Dubey et al. developed an incentive-based protocol based on coalition game theory, in which vehicles have an incentive to forward data as members of a coalition to improve reliability and fairness in VANETs [8
]. Liu et al. designed an incentive system that uses the reciprocal altruistic factor to reconstruct the utility function of vehicles [9
]. They demonstrated that their system can reduce message transmission delay by up to 30% compared to other cooperative transmission mechanisms. Additionally, Liu et al. proposed an incentive mechanism from the perspective of behavioral economics that incorporates the anchoring effect and loss aversion on offloading in the Internet of Things [10
]. They introduced the reference factor and price-break discount factor based on the anchoring effect and time pressure and regret value on loss aversion to encourage nodes to participate in data offloading. Rehman et al. developed a mechanism with both incentives and punishments for VANET [11
]. In this mechanism, vehicles with higher weight and cooperation are elected as Heads during the election process. Vehicles that participate in the election can increase their reputation by cooperating in forwarding data, while those who exhibit repeated selfish behavior are punished.
In addition, game theory has been utilized to address the issue of free-riding in wireless network routing [12
]. Game theory approaches can be broadly classified into two categories, pairwise and collective games. Pairwise games assume that interactions between players are limited to two players only, where the effect of a player’s strategy is only on one other player. On the other hand, collective games assume that interactions are collective, meaning that the strategy of one player can impact several other players’ strategies. The collective game category includes public goods games (PGG) and coalition games.
Previous studies have investigated the problem of cooperation in VANETs using game theory, and have utilized a model with pairwise interactions [13
]. This means that the effect of each player, or vehicle, is assumed to be limited to only one other companion player. However, this assumption may not be appropriate for the effect of forwarded messages, since each player can actually affect multiple other vehicles simultaneously. Therefore, other studies have considered the effect of each player on multiple other vehicles simultaneously, which is a more suitable assumption for the problem at hand. While models with pairwise interactions are commonly used in game theory, their results cannot always be extended to games with collective interactions, and may lead to misleading conclusions [20
The cited studies have contributed to the investigation of cooperation in VANETs through game theoretical models. Zhang et al. [13
] proposed a cooperative information dissemination scheme for vehicular networks using a Markov decision process. Mao et al. [15
] proposed a cooperative content sharing mechanism using evolutionary game theory. Altman et al. [16
] considered the problem of finding stable cooperative strategies in VANETs using a repeated game approach. Naserian et al. [17
] proposed a game-theoretic model for cooperation in VANETs that considers the impact of obstacles on communication. Saeed et al. [19
] proposed a cooperative transmission mechanism that considers the impact of obstacles and interference on transmission rates.
The problem of promoting cooperation in social dilemmas involving groups of interacting individuals is traditionally investigated using the Public Goods Game (PGG), which assumes that individuals can make contributions to a public good. In VANETs, the data packet can be considered as the public good. Shivshankar and Jamalipour were the first to propose a game theoretic strategy, called Tit-for-Tat, to promote cooperation among vehicles in VANETs based on PGG. Their approach measured the cooperation level of the vehicles based on time and the number of message forwards, and they examined the effect of different parameters on the cooperation level to evaluate the performance of the proposed game [21
In a different study, Zhang et al. used PGG to investigate cooperative behavior among vehicles [13
]. They showed that the cooperation level in VANETs is proportional to the synergy factor. Shivshankar and Jamalipour proposed a PGG framework to analyze the effects of networking properties on the dynamics of cooperation in VANETs. In another work, Ding et al. designed a PGG for VANETs with high node density. In the proposed game, vehicles could dynamically adjust their grouping strategy according to the real situation around their strategies. Additionally, they defined a hub vehicle that exchanges data packets with all neighbors to prevent isolated vehicles. The authors demonstrated that increasing the multiplication factor in the PGG would promote the cooperation level, which is an expected result for VANETs [22
In VANETs, messages sent by vehicles and road side units can impact multiple vehicles. Thus, it is more appropriate to model the effect of event messages as a multiplayer game with collective interactions rather than pairwise. However, current PGG-based models in VANETs have limitations. Researchers usually assume that each vehicle has fixed neighbors or can perfectly connect to all other players, which is not realistic in VANETs. Additionally, the topology of vehicles does not change throughout the game, which is also unrealistic. In VANETs, a vehicle benefits from diffusible messages produced by all vehicles in its group, which randomly form at each generation. Therefore, the vehicle’s payoff should be calculated by weighting the payoffs obtained in the randomly composed groups, weighted by the probability that such groups occur. Moreover, in VANETs, the PGG is played in populations of varying sizes. Small population size (or density) results in small group sizes and vice versa for a larger population size.
To address these limitations, this study models vehicles as players that form randomly composed groups at each round of the game based on the population size. This game is called ecological-aware PGG. The study investigates, for the first time to the best of our knowledge, the effect of rate of departure and density of vehicles on the cooperation dynamics in VANETs based on ecological-aware PGG.
4. Dynamical Analysis
Equilibrium points in replicator Equation (8
) are obtained by setting them to zero. In our system, this corresponds to an internal steady state between cooperators and defectors if system works infinity. In this section, we investigate the stability of the equilibrium points in parametric form and represent individuals behavior by several scenarios under different conditions. Some of these scenarios represent normal behavior and others indicate rare cases that theoretically can take place.
) Consider a system without any cooperators. In such situation (
), according to Equation (6
), the average payoff of the defector will be
, hence the rate of change of defector frequencies is
. This results in decreasing defector frequency in the population and, consequently, it goes extinct.
) In the absence of defectors in the population (
), according to Equation (6
), the average payoff of cooperators is given by:
and based on the maximum value of the function
three equilibria can be readily observed in the system. The maximum of the function
is actually attained at the point
and is given by:
then population goes to extinct (i.e.,
), while for
a bifurcation arise in the system (Figure 1
) that results in one stable and one unstable interior equilibrium at higher and lower x
, respectively. However, another equilibrium emerged exactly at the maximum point
that represents an unstable point.
shows population dynamics in the absence of defectors for others parameters.
) The system can enable cooperators to survive even when defectors free-ride on contributions of the cooperators. In order to investigate this scenario, a new variable
is introduced. By using Equation (8
), the changing rate of
are given by:
By introducing variable
the dynamic system in Equation (8
) which represented by a 3-simplex reduced to a dynamics that shown in a rectangle determined by
. This sheds some new light on the vehicle dynamic from viewpoint of the evolutionary game theory. Figure 3
shows an example of vehicles dynamics for such system. As shown in this figure, natural selection favors cooperator vehicles and, eventually, all populations select this strategy. The red rectangle in this figure represents the valid region of the population dynamics induced by Equations (12
) and (13
). It is worth pointing out that the dynamics on the rectangle boundaries can be analyzed clearly. The boundaries
represent the scenarios i
, respectively. Definitely, on the boundary of
, z will be equal to 1. However, the dynamics on the boundary
has 1, 2, or 3 equilibrium states: extinction of defector (i.e., cooperator dominance), extinction of the cooperators (i.e., defector dominance), and coexistence of the cooperators and defectors.
Up to now, we have discussed about the monomorphic equilibrium points of the game which defectors or cooperators strictly dominate. Now, conditions under which a polymorphic equilibrium can arise are investigated. Since in the polymorphic equilibrium
, so in accordance with [26
] the equilibrium
are obtained by setting
. Analyzing the parametric form of
is very complicated and maybe impossible. Hence, we analyze different conditions for
using numerical simulations and under various environment conditions. According to the numerical simulation we claim that for
at most has one root, and for
at most has two roots in the interval
. Figure 4
shows the roots of the function for
. Consequently, there are at most two equilibrium points in the system. Figure 5
shows all dynamics of the system regards to the number of equilibria.
An equilibrium point in the system can be stable, unstable, and saddle. An equilibrium point X
is stable if for every neighborhood U
the system starts from another neighborhood
, eventually the system goes to U
. An unstable point is an equilibrium if it is not stable. A saddle point is an unstable equilibrium where it is stable at least in one direction. In general, if equilibrium points exist, their stability depend on values of the parameter. Our simulations show that the dynamical system at most can have one polymorphic stable equilibrium point (Figure 5
). As can be seen in Figure 5
a, the boundary
is attracting for sufficiently small population densities (approximately for
). Hence, for the small population, vehicles remain unable to recover from exploitation and finally become extinct. Additionally, for large population densities, cooperation arises in the system but there is no stable point. This means that, by changing the number of defectors and cooperators, the final state maybe changed. Figure 5
b shows the system has one polymorphic stable equilibrium. In this example, population goes to stable point (
) for z
. It is easily seen that for high population densities cooperation arises, while for medium and small densities relative defector frequency increases in the population over time and finally approaches to the stable point (
). The important point to note here is that although in high densities cooperation arises, there is no any equilibrium in this area. Figure 5
c is similar to Figure 5
b except that it includes a polymorphic saddle point. For a given
and low c
, if population density increases, the members of the randomly composed group increase until eventually
. This decreases the return from the public good (i.e., forwarded messages) and results in increasing defectors in the system. However, decreasing population density result in smaller randomly composed group until in the long run
holds, and cooperator vehicles arise.
One of the advantage of the proposed model lies in the fact that cooperation level of vehicles can be determined by setting model’s parameters. Figure 6
suffice to show it. As illustrated in this figure, a high death rate, that is equal to birth rate in the proposed model, increases the chance of cooperation in large population densities while low death rate, by contrast, increases the cooperator frequency in the small population densities. This is for the simple reason that the payoff of cooperators and defectors directly influence growth by increasing d
proportional to x
(see Equation (8
The size of randomly composed group N
is another determinant element in promoting cooperation. Indeed, in our model, N
determines the average number of vehicles that are able to receive the message from the focal vehicle. Generally, low (high) value of N
increases (decreases) the return from the public good and leads to promote (demote) defection between the vehicles. Figure 7
shows the effect of N
on the level of cooperation of vehicles.
If the costs of contribution c
are sufficiently small the cooperation arises between vehicles. Figure 8
represents the effect of c
on the level of cooperation between the vehicles. As shown, increasing c
leads to continuous loss of cooperation strength in the population insofar as
c) results in the lowest cooperativeness.