# Stochastic Game Analysis of Cooperation and Selfishness in a Random Access Mechanism
## 6 Numerical Results
The mathematical and numerical models in this study provide the basis for the performance analysis and protocol design of future generation wireless networks and protocols. With the ever increasing number of interconnected wireless devices, there is an increasing need of developing more robust protocols capable of limiting the impact of interference over the network performance. Furthermore, the lack of cooperation of even a single user may result in the malfunctioning of the overall network. Figure 6: Delay of transmitted packets for different retransmission policies. (**a**) \(M=3\), \(N=2\), and (**b**) \(M=2\), \(N=3\). In order to illustrate the potential use of our models in the context of the research and development efforts of future telecommunications systems, let us consider a network of \(M=10\) and \(N=2\) users. Table 1 shows the results as a function of the arrival probability \(p_{a}\). In the table, \(TH_{c}^{i}\) and \(TH_{nc}^{i}\) indicate the throughput of a cooperative user and a non-cooperative user, respectively. As seen from the table, as the arrival probability increases, selfish users get more aggressive and attempt accessing the shared wireless channel using the highest probability \(q_{nc}^{*}=0.9999\). In contrast, the cooperative users lower their retransmission probability to avoid collisions for allowing everyone to use the network resources. Under very heavy load conditions (i.e., \(p_{a}\geq 0.9\)), the cooperative users reduce drastically their retransmission probability \(q_{c}^{*}\) while the two selfish users can successfully transmit all their packets thanks to the use of the ZigZag scheme. In fact, the impact of the ZigZag scheme can be particularly seen on all the performance metrics under very high load conditions. The last two rows of the table show that the delay experienced by the traffic of the cooperative users experience increases by several orders of magnitude as all the cooperative users get their packets backlogged, \(S_{c}=10\). On the contrary, the non-cooperative users benefit from the use of the ZigZag scheme by equally sharing the channel \(TH_{nc}^{j}=0.49\) and are able to transmit practically all their packets at their first attempt, \(S_{nc}=1.99\times 10^{-3}\). The results also show that the collision probability resulting on packet losses reduces drastically to \(P_{col}=9.99\times 10^{-4}\) as the packets of the two non-cooperative users rarely collide with the packets of the cooperative users. Remember that by implementing the ZigZag code, the receiver may properly recover the information of up to two colliding packets. Even though the use of the ZigZag proves effective in improving the performance of the network, it also raises some concerns on the dangers of the misuse of such features. As seen from our results, non-cooperative users may take advantage of such a decoding scheme. Therefore, further studies are required to explore the use of such scheme even in the presence of non-cooperative users. One possible line of research will be a dynamic implementation of such code by enabling and disabling its operation as a means to discourage non-cooperative users. Figure 7: Delay of backlogged packets for different retransmission policies. (**a**) \(M=3\), \(N=2\), and (**b**) \(M=2\), \(N=3\).