# Game-Based Resource Allocation Mechanism in B5G HetNets with Incomplete Information
## 3 Distributed Resource Allocation Algorithm
### Follower Game  
Given the bandwidth price declared by networks, mobile users compete with each other for the limited resources to maximize their payoffs. In the follower subgame with incomplete information, the channel gain is the private information regarded as the type of the mobile user, while its probability distribution is public information. As a result, we modelled the competition among users with there Bayesian game, where:  
* Players are \(n\) mobile users in the set of \(N=\{i|i=1,2,\ldots,n\}\),
* The action set of user \(i\) is the set of the requested bandwidth \(\mathbf{q}_{i}=\left\{q_{ij},\mathbf{q}_{-ij}\right\}\), consisting of the bandwidth requesting from network \(j\) and the other networks. Let \(\mathbf{q}=\left(\mathbf{q}_{i^{\prime}},\mathbf{q}_{-i}\right)\) denote the vector of actions of all mobile users, where \(\mathbf{q}_{-i}=\left(\mathbf{q}_{1},\mathbf{q}_{2},\ldots,\mathbf{q}_{i-1}, \mathbf{q}_{i+1},\ldots,\mathbf{q}_{n}\right)\) denotes the vector of users, except user \(i\). * The type of user \(i\) is the channel gain \(h_{ij}\) between the user and its connected network \(j\). Let \(\mathbf{h}_{i}=\left\{h_{ij},h_{-ij}\right\}\) be the type set of user \(i\) and \(\mathbf{h}=\left(\mathbf{h}_{i},\mathbf{h}_{-i}\right)\) be the vector of types of all mobile users, where \(\mathbf{h}_{-i}=\left(\mathbf{h}_{1},\mathbf{h}_{2},\ldots,\mathbf{h}_{i-1}, \mathbf{h}_{i+1},\ldots,\mathbf{h}_{n}\right)\) denotes the vector of users, except user \(i\). We assumed the types of all users were continuous and had the same probability distribution, and the probability density function was \(f_{i}(\mathbf{h}_{i})\). * Strategy of user \(i\) is a mapping from the type set to the action set, which is \(\mathbf{s}_{i}\!:\!\mathbf{h}_{i}\!\to\!\mathbf{q}_{i}\). \(\mathbf{s}_{i}(\mathbf{h}_{i})=[s_{i1}(h_{i1}),\ldots s_{ij}(h_{ij}),\ldots s_ {in}(h_{in})]\) represents the strategy distribution given in the channel gain set \(\mathbf{h}_{i}\), where \(s_{ij}(h_{ij})\) is equal to \(q_{ij}\). To simplify the presentation, we will use \(s_{ij}(h_{ij})\) and \(q_{ij}\) interchangeably in the following sections. Let \(\mathbf{s}_{-i}=\left(\mathbf{s}_{1},\mathbf{s}_{2},\ldots,\mathbf{s}_{i-1}, \mathbf{s}_{i+1},\ldots\mathbf{s}_{n}\right)\) represent the vector of strategies made by other players, except user \(i\). * Payoff of user \(i\) is the function of its action \(q_{ij}\) and its type \(h_{ij}\), given the actions and types of others, which is defined as: \[u_{i}^{F}(\mathbf{q}_{i^{\prime}},\mathbf{q}_{-i^{\prime}},\mathbf{h}_{i}, \mathbf{h}_{-i})=\alpha_{i}\cdot\log_{2}(1+\sum\limits_{j=1}^{m}C_{ij})-\sum \limits_{j=1}^{m}b_{j}\cdot q_{ij}-\sum\limits_{j=1}^{m}\frac{\beta_{j}}{W_{j }-Q_{j}}\] (9)For the Bayesian resource-competition game, the expected utility of user \(i\) denotes as \(\overline{u_{i}^{F}}\):  
\[ \begin{array}{l}\overline{u_{i}^{F}}(\mathbf{s}_{i}(\mathbf{h}_{i}),\mathbf{s }_{-i};\mathbf{h}_{i},\mathbf{h}_{-i})=\\ \int_{\mathbf{h}_{-i}}\left[\alpha_{i}\cdot\log_{2}(1+\sum\limits_{j=1}^{m}C_{ ij})-\sum\limits_{j=1}^{m}b_{j}\cdot q_{ij}-\sum\limits_{j=1}^{m}\frac{\beta_{j}}{ W_{j}-Q_{j}}\right]\cdot f_{-i}(\mathbf{h}_{-i})d\mathbf{h}_{-i}\end{array}  \tag{10}\]  
where \(f_{-i}(\mathbf{h}_{-i})=\prod_{k\
eq i}f_{k}(\mathbf{h}_{k})\) is the joint probability density of \(\mathbf{h}_{-i}=(\mathbf{h}_{1},\mathbf{h}_{2},\ldots,\mathbf{h}_{i-1},\mathbf{ h}_{i+1},\ldots,\mathbf{h}_{n})\) and \(d\mathbf{h}_{-i}=\prod_{k\
eq i}d\mathbf{h}_{k}\). To obtain the Bayesian Nash equilibrium in the follower subgame, the best response of user \(i\) is defined as the highest expected utility, with respect to the strategies of other players and the belief about the probability distribution of other types.