# Marginal Contribution-Based Distributed Subchannel Allocation in Small Cell Networks
## 3 Game Theoretic Framework and Utility Function Design
### Utility Function Design with Limited Information Feedback  
We consider a potential function, \(\mathbf{P}^{\text{MC}}:\mathcal{S}\rightarrow\mathbb{R}\), with the objective to maximize total capacity of the SCN for transmission of link \(i\in\mathcal{M}\) on subchannel \(k\), such that,  
\[\mathbf{P}^{\text{MC}}(\mathcal{S})=\sum_{k}W_{k}(\mathbf{s}) \tag{9}\]  
where \(W_{k}(\mathbf{s})=\sum_{i:s_{i}=k}C_{i,k}\) is the total welfare (in term of capacity) from all links that are using subchannel \(k\). Hence, the potential function \(\mathbf{P}^{\text{MC}}(\mathcal{S})\) essentially reflects overall capacity in Equation (2).  
From Equations (8) and (9), we can express the utility of link \(i\) when it is using subchannel \(k^{\prime}\), i.e., \(s_{i}=k^{\prime}\), while the other links are using strategies \(\mathbf{s}_{-i}\) as,\[u_{i}^{\text{MC}}(s_{i}=k^{\prime},s_{-i}) =P^{\text{MC}}(s_{i}=k^{\prime},s_{-i})-P^{\text{MC}}(s_{-i})\] \[=\sum_{k}W_{k}(s)-\sum_{k}W_{k}(s_{-i})\] \[=\left(W_{k^{\prime}}(s)+\sum_{k\
eq k^{\prime}}W_{k}(s_{-i}) \right)-\left(W_{k^{\prime}}(s_{-i})+\sum_{k\
eq k^{\prime}}W_{k}(s_{-i})\right)\] \[=W_{k^{\prime}}(s_{i},s_{-i})-W_{k^{\prime}}(s_{-i}) \tag{10}\]  
where \(W_{k^{\prime}}(s_{i},s_{-i})\) and \(W_{k^{\prime}}(s_{-i})\) are the total welfare on subchannel \(k^{\prime}\) with presence and absence of link \(i\) in the game, respectively. Here while learning the utility on subchannel \(k^{\prime}\) to improve its own performance, the link \(i\) is aware of its influence to other links that are using subchannel \(k^{\prime}\). From Equation (10), since link \(i\) is using only subchannel \(k^{\prime}\), the total welfare on all other subchannels, \(k\
eq k^{\prime}\), is the same as that without link \(i\), i.e., we have for any subchannel \(k\
eq k^{\prime}\), \(W_{k}(s_{i}=k^{\prime},s_{-i})=W_{k}(s_{-i})\) and hence the total welfare from all other subchannels (except \(k^{\prime}\)) is the same with and without link \(i\). With this observation, the utility of link \(i\) is expressed simply as \(u_{i}(s_{i}=k^{\prime},s_{-i})=W_{k^{\prime}}(s_{i},s_{-i})-W_{k^{\prime}}(s_{ -i})\), which is the additional welfare on subchannel \(k^{\prime}\) when link \(i\) is in the game, compared to that when link \(i\) is not in the game, assuming all other links stay using the same strategies \(s_{-i}\). Hence, the computational complexities to compute the utility are reduced in this game, since link \(i\) would require information feedback only from those links that are also using subchannel \(k^{\prime}\), i.e., for which the strategy is \(s_{-i}=k^{\prime}\).  
The game with utility function in Equation (10) is an EPG for the potential function in Equation (9) as shown in Theorem 1.  
Theorem 1: _The game \(\mathcal{G}=(\mathcal{M},\{\mathcal{S}\},\{u_{i}^{\text{MC}}(s)\})\) is an EPG and has at least one pure strategy NE._  
Proof of Theorem 1.: As shown in Section 3.2.2. of [8], when we set MC as a utility function, the game with this utility function is proven to be an EPG where the strategy profile that maximizes the potential function is the pure strategy NE.  
Corollary 1: _For a particular initial subchannel profile and sequence of play, the pure strategy NE of the game \(\mathcal{G}=(\mathcal{M},\{\mathcal{S}\},\{u_{i}^{\text{MC}}(s)\})\) is unique._  
Proof of Corollary 1.: Consider an initial subchannel profile, \((s_{1},\ldots,s_{i},\ldots,s_{M})\) where \(\forall i\in\mathcal{M}\), and there is a strategy update sequence of players, say \((m)_{m\in\mathcal{M}}\).  
Since the game \(\mathcal{G}=(\mathcal{M},\{\mathcal{S}\},\{u_{i}^{\text{MC}}(s)\})\) is an EPG, the potential function \(P^{\text{MC}}(\mathcal{S})\) is guaranteed to converge to a maxima and that maxima in turn corresponds to a pure strategy NE [17]. Hence, a pure strategy NE is unique, depending on the initial subchannel profile and the sequence of play for the strategy update by players.