# Stochastic Game Analysis of Cooperation and Selfishness in a Random Access Mechanism
## 6 Numerical Results
Figure 3: Retransmission policy for cooperative users and selfish users. From the above results, it seems that the aggressive behavior of selfish users is beneficial for the system, even if it is done at the expense of the performance of cooperative users. However, it is not always the case, as shown in Figure 4b where the number of selfish users dominates over the number of cooperative users. Figure 4b shows that selfish users always get better throughput, not only because of their dominant number but also because of their aggressive behavior. However, when \(p_{a}\geq 0.5\), the system throughput drops considerably due to the collisions caused by the selfish users. Figure 5 shows the number of backlogged users as a function of the arrival rate. In the first game scenario Figure 5a, the cooperative users get heavily penalized by the aggressive behavior of selfish users since the number of their backlogged packets grows more rapidly as the traffic load increases. As for the selfish users, \(\approx\)30% of them get backlogged when \(p_{a}=0.53\). Note that when \(p_{a}\geq 0.88\), cooperative users refrain from transmitting their backlogged packets, refer to Figure 3, as a result they all get backlogged, see Figure 5a. Indeed, due to the high arrival rate, cooperative users stay silent to allow the newly arrived packets to access the channel without any disturbance. On the other hand, selfish users take advantage of the situation and access the channel with the highest probability, i.e., \(q_{nc}^{*}=1\), refer to Figure 3. Even in the presence of two selfish users, they reach the maximum throughput thanks to the ZigZag decoding technique that decodes all their collided packets. Figure 4: Normalized throughput for different retransmission policies. (**a**) \(M=3\), \(N=2\), and (**b**) \(M=2\), \(N=3\). Figure 5: Number of backlogged users for different retransmission policies. (**a**) \(M=3\), \(N=2\), and (**b**) \(M=2\), \(N=3\). In the second game scenario Figure 4(b), selfish users are penalized by their aggressive behavior more than cooperative users because they collide very often, especially in high load conditions. Figure 6 shows the access delay as a function of the arrival rate. The access delay represents the time in slots elapsed from the packet generating time and its successful transmission. The results show that the delay increases with the arrival rate. Furthermore, cooperative users exhibit the worst delay due to their low retransmission policy. For instance, under heavy load conditions, the delay reaches \(9.99\times 10^{7}\) and \(10^{8}\) slots in the first and second game scenarios, respectively. However, in the first game scenario, the delay of selfish users under the same load conditions is limited to 1 slot. That is to say, the selfish users are able to successfully transmit all their packets at the first attempt. On the other hand, selfish users experience a considerably high delay of \(5\times 10^{5}\) slots in the second game scenario due to the increasing number of selfish users. Figure 7 shows the delay of backlogged packets as a function of the arrival rate. Backlogged packets are the pending packets that went through a collision, and the ability to deliver backlogged packets is crucial for real-time traffic. Similar to the previous results, cooperative users have the worst delay. The first game scenario shows that backlogged packets get delayed more as the traffic load increases. However, as the arrival rate gets closer to 1, the backlogged packets of cooperative users become very large. In contrast, selfish users do not report any delay when \(p_{a}\geq 0.88\) since they do not possess any backlogged packet as shown in Figure 4(a). On the other hand, both cooperative and selfish users suffer a longer delay when the load increases in the second game scenario (see Figure 6(b)). However, selfish users experience slightly higher backlogged delay than cooperative users due to their number and exaggerated aggressive behavior.