# Shapley's Value as a Resource Optimization Strategy for Digital Radio Transmission over IBOC FM
## 2 Materials and Methods  
### Game Theory and Shapley Value  
Game theory, conceptualized by [PERSON] in 1928, offers a mathematical framework for assessing individual decisions within competitive contexts of gain or loss compared to decisions made by other competitors. This competitive scenario is referred to as a \"Game\", and the individuals who are part of this scenario are called \"Players\". Game theory provides three models for representing real scenarios: extensive, strategic, and coalition. While the first two are applicable to non-cooperative games, emphasizing individual player interests, the coalition model pertains to cooperative games. In cooperative games, players collaborate to achieve common goals, increasing the likelihood of higher gains than individual actions. Cooperative game analysis avoids the need to scrutinize player strategies, focusing instead on coalition utility and payment vectors [9].  
In cooperative game theory, a common question arises concerning the fair distribution of a good's net value among players, particularly in cases of insufficient resources. This scenario, known as the \"bankruptcy\" problem, is typically analyzed either as a transferable utility game or through axiomatic methods [10; 11]. This paper adopts the transferable utility approach.  
Definition 1 ([12]): A Transferable Utility (TU) cooperative game is a pair (N, v), where N is the set of players {1, 2, \(\ldots\), n} and v: \(2N\rightarrow\mathbb{R}\) is the characteristic function, with v(\(\emptyset\)) = 0. Each coalition \(S\subset N\) associates with (S), representing the assured payoff for its players, irrespective of others' actions.  
Definition 2 ([13]): A bankruptcy game, represented as (N, d, E), comprises creditors (N), claims vector (d), and net value (E). A cooperative game (N, v) is defined from each bankruptcy problem. The value of coalition S is the property not claimed by non-S members, calculated as follows:  
\[v(S)=max\{0,\ E-d(N\smallsetminus S)\}\ \forall S\subset N \tag{1}\]  
In TU cooperative game theory, a major challenge lies in the equitable allocation of the total gain among players. The Shapley value, introduced by [PERSON] in 1953, tackles this issue by accounting for each player's marginal contributions to all potential coalitions, satisfying established axioms [14; 15]. The expression corresponding to the Shapley value is as follows:  
\[\varphi_{i}(v)=\sum_{S\subseteq N\
i S}\frac{(s-1)!(n-s)!}{n!}[v(S)-v(S-\{i\})] \ \text{where}\ n=|N|\ y\ s=|S| \tag{2}\]  
The Shapley value guarantees efficient, symmetric, passive, and additive resource allocations. [PERSON] (1982) demonstrated the correspondence between the recursive partitioning process in a bankruptcy problem and Shapley's value. For the solution to be adequate, it is necessary for the vector of payoffs to comply with the efficiency principle, as outlined in [16]:  
\[\sum_{i\in N}\varphi_{i}(v)=v(N) \tag{3}\]  
\[\varphi_{i}(v)\geq(v\{i\})\ \forall i\in N \tag{4}\]  
In the context of equitable bandwidth distribution in the digital radio spectrum for IBOC FM, represented as a TU game (N, v), the Shapley value emerges as a strategy for fair allocation among nodes under saturation conditions. This scenario involves N nodes as players, where \(\lambda_{i}\) represents the bandwidth demand of each node \(i\) (\(d_{i}\)) and \(E=BW_{T}\).