# Stochastic Game Analysis of Cooperation and Selfishness in a Random Access Mechanism
## 6 Numerical Results  
In this section, we present the numerical results, and we discuss the main findings of this study. The overall system throughput is taken as the utility function of cooperative users \(U_{g}:=TH_{c}\), whereas the utility function of a single selfish user is the individual throughput. We investigate and compare two game scenarios. First, we consider a network where the number of cooperative users exceeds the number of selfish users, \(M=3\) and \(N=2\). Then, in the second game scenario, we consider the opposite case where the number of selfish is \(N=3\) and the number of cooperative users is \(M=2\). In each scenario, we compare the performance of cooperative users and selfish users. Furthermore, we provide the performance of the overall system in each case. To maintain a fair channel condition, we set for all users the same arrival rate, i.e., \(p_{a}=p_{c}=p_{nc}\). Finally, we take \(\epsilon=10^{-4}\). Figure 3 shows the game equilibrium \((\Psi^{*}_{c},\Psi n\epsilon^{*})\) as a function of the arrival rate \(p_{a}\), where \(\Psi^{*}_{c}=(q^{*}_{c},1-q^{*}_{c})\) and \(\Psi n\epsilon^{*}=(q^{*}_{nc},1-q^{*}_{nc})\). The retransmission probabilities \(q^{*}_{c}\) and \(q^{*}_{nc}\) represent the optimal retransmission probability of cooperative and selfish users, respectively. The results show that selfish users access the wireless channel more aggressively than cooperative users. Cooperative users use a high retransmission probability in light load conditions since the collision rate is meager. However, as the arrival rate increases, they lower their retransmission rate to reduce network congestion. In contrast, selfish users transmit with high probability than cooperative users. Moreover, they increase the transmission rate even in the case of high load conditions. Even though this aggressive behavior is not suitable for any user, selfish users cannot unilaterally deviate from their retransmission strategy due to the rationality concept. Assuming that all others will cooperate, a selfish user could improve his utility function by choosing to cooperate. However, if he cooperates while the others defect, he will be heavily penalized. Thus, in such a situation, selfish users prefer to defect since they do not know the strategy chosen by others. This conflict situation is known in game theory as the prisoner dilemma. On the other hand, cooperative users assume that everyone is cooperating. Thus, they do not have any information about the existence of selfish users. Therefore, they play a cooperative game, assuming that everyone behaves similarly. In the following results, we present the performance metrics at the equilibrium, i.e., when the cooperative and selfish users use the retransmission probability \(q_{e}^{*}\) and \(q_{nc}^{*}\), respectively. Figure 4 shows the throughput of cooperative and selfish users as well as the global system throughput. Furthermore, we compare two scenarios: (a) when the number of cooperative users exceeds the number of selfish users and (b) when the number of selfish users is greater than the number of cooperative users. Figure 3(a) shows that in light load conditions, cooperative users have a higher throughput than selfish users. This is because both users use a high retransmission probability. Even though the retransmission probability of selfish users is slightly higher, its impact is not significant in light load conditions. Thus, due to the dominant number of cooperative users, their throughput is slightly higher than that of selfish users. However, in high load conditions, selfish users get significantly better throughput since they transmit with higher probability and also because cooperative users lower their transmission probability as seen from Figure 3. In very high load conditions \(p_{a}\geq 0.88\), cooperative users stop the retransmission of backlogged packets while selfish users transmit at the highest probability. This aggressive behavior allows selfish users to dominate the overall system throughput.