# Energy-Efficient BWP Configuration for Multi-Slice Users
## 4 Distributed Approach: Congestion Game  
Users connected to multiple slices strive to choose between the multi-numerology (MN) and single-numerology (SN) BWP configurations in order to curb their energy consumption without sacrificing their QoS. Thus, the problem can be modeled as a non-cooperative game where the players are autonomous UEs competing over limited radio resources, which are the PRBs available in the BWPs. The corresponding non-cooperative game \(\mathcal{J}\) is presented as follows:  
* \(\mathcal{P}\) is the set of players which are none other than the UEs connected to multiple slices.
* The set of strategies is \(\mathcal{S}=\{MN,SN\}\) where \(MN\) and \(SN\) designate the multi-numerology and single-numerology BWP configurations, respectively. Additionally, \(v_{u}\) is the strategy vector of UE \(u\) which is composed of binary variables \(v_{u,c}\). The latter is equal to 1 when UE \(u\) chooses the BWP configuration \(c\). Therefore, the strategy profile is \(v=(v_{u})_{u\in\mathcal{P}}\in\mathcal{S}\). Also, the space of all profiles is denoted by \(\mathcal{S}=S_{1}\times S_{2}\ldots\times S_{U}\).
* \(\{C_{1},C_{2},\ldots,C_{U}\}\) denotes the set of cost functions based on Equation (4) which assess the players' gain after their strategy selection.  
In \(\mathcal{J}\), the Nash Equilibrium (NE) is sought. In fact, the NE is an equilibrium state where every player \(u\) would select an optimal strategy in response to other players' strategies and where no further benefit is acquired after deviating from this strategy unilaterally.  
The game \(\mathcal{J}\) is guaranteed to have a mixed NE, since it is a finite game. With mixed NEs, players select their optimal strategy following a probability distribution which may be inconvenient. However, thanks to the Finite Improvement Path property which \(\mathcal{J}\) possesses similarly to the work in [6], the existence of a Pure NE (PNE) is guaranteed [18]. Thus, we prove this property below.  
Proposition 1: _The game \(\mathcal{J}\) holds the Finite Improvement Path property._  
Proof: The cost function of \(\mathcal{J}\) is player specific and non-decreasing in the number of players which select the same strategy. This is because the sojourn time \(\frac{V_{u,c}}{Thr_{u,s,hyp}}\) is non-decreasing in the number of players that selected the same strategy as the throughput is inversely proportional to the number of users selecting the same BWP configuration. Thus, \(\mathcal{J}\) is an unweighted congestion game. Moreover, this type of game has the FIP property only when two strategies can be selected according to [19]. \(\mathcal{J}\) verifies this condition, since only two strategies may be chosen \(MN\) and \(SN\).  
Subsequently, games holding the FIP property converge to the PNE using simple Best-Response dynamics, as stated by [20], where each player in turn will choose the strategy minimizing its cost function in response to other players' strategies until convergence, when the chosen strategy of each player is the same as in the previous round presented in the below Algorithm 1.  
In the next section, the same problem is tackled in a centralized approach.