# Shapley's Value as a Resource Optimization Strategy for Digital Radio Transmission over IBOC FM
## 4 Discussion
The analysis conducted aims to determine which of the two optimization methods provided the most suitable adjustment to the requirements of each node, a criterion discernible through the technique resulting in the lowest disparity between the requested and allocated bandwidth for each node in an IBOC FM network. Accordingly, the following hypotheses are proposed [27]:  
\[H_{o}:\ \mu_{x}\leq\mu_{y}\rightarrow\mu_{x}-\mu_{y}\leq 0\rightarrow\mu_{z}\leq 0 \tag{13}\]  
\[H_{a}:\ \mu_{x}>\mu_{y}\rightarrow\mu_{x}-\mu_{y}>0\rightarrow\mu_{z}>0\]  
Here, \(\mu_{y}\) and \(\mu_{x}\) represent the means for the difference between requested bandwidth and assigned bandwidth through the PL optimization methods and the Shapley value, respectively. Hypothesis \(H_{o}\) suggests a significant difference between means, with \(\mu_{x}\) being lower than \(\mu_{y}\), indicating that the Shapley value leads to a more effective optimization process. Hypothesis \(Ha\) suggests the opposite. Additionally, a new variable \(Z\) is introduced to refine the hypotheses. To evaluate these hypotheses, a paired \(t\)-test is employed, commonly used for assessing the statistical validity of differences between two random samples [28]. The process involves the following steps:  
Step 1: Define a new random variable \(Z=X-Y\) and calculate the mean (\(Z\)) and standard deviation (\(S_{z}\)) for \(Z\). The resulting values are 0.0008 and 2.7544, respectively. Step 2: Calculate the test statistic (\(d\)) using the expression:  
\[d=\frac{\overline{Z}}{S_{z}}\sqrt{n}=\frac{0.0008}{2.7544}\sqrt{12}=1.006\times 1 0^{-3} \tag{14}\]  
where \(d\) is the statistic value, and \(n\) is the number of samples for both proposed models. Step 3: Establishing the acceptance range for \(H_{o}\) as {\(t\): \(t<T_{(\alpha;n-1)}\)} at a 5% significance level (\(\alpha=0.05\)) with \(n-1\) degrees of freedom; for this case, \(T(0.05;11)=1.7959\). Defining the acceptance interval for \(H_{o}\) as \((-\infty,1.67959)\). Evaluation of the statistic \(d\) reveals that it falls within the acceptance interval, leading to the non-rejection of \(H_{o}\). Consequently, it can be concluded with 95% confidence that the Shapley value stands as an excellent alternative for executing high-quality optimization processes. One potential disadvantage of the Shapley value is its computational cost, as calculating it may require evaluating all possible node coalitions, which can be complex in systems with a large number of agents. On the other hand, linear programming may be less flexible in situations where node relationships are complex or changing, as it requires a precise formulation of the problem and may not fully capture nonlinear interactions between nodes. Both the Shapley value and linear programming offer valid approaches to resource optimization in IBOC transmission systems in FM, each with its own advantages and disadvantages. The choice between them will depend on the specific characteristics of the problem, such as the complexity of interactions between nodes, the availability of precise information, and computational efficiency requirements. The use of the Shapley value can be considered a starting point of great importance for the optimization of resources in digital broadcasting over IBOC FM, through the use of techniques related to the theory of cooperative games. However, to advance in this field, it is necessary to evaluate in future work other cooperative techniques such as Nucleolus, Max-Min Fairness, and power indices, which can offer complementary perspectives and even the possibility of obtaining more equitable solutions in the allocation of digital broadcasting resources, considering that most of the studies that have been carried out by other researchers are mainly related to making better use of bandwidth through the development of advanced audio compression techniques and the use of more efficient modulation techniques.