Table 1 Symbols and parameters for mathematical formulations

\({\mathcal{A}}_{\mathcal{D}}\)

The totally allocated spectrum resource for the \(\mathcal{D}\)

\({\mathcal{A}}_{\mathcal{D}}^{\mathfrak{B}}\)

The \(\mathcal{D}\)’s LTE spectrum amount

\({\mathcal{A}}_{\mathcal{D}}^{\mathcal{W}}\)

The \(\mathcal{D}\)’s WiFi spectrum amount

\(\mathfrak{B}\)

The LTE base station

\({\mathbb{B}}_{\mathcal{W}}\)

The \(\mathcal{W}\)’s IoT devices generated BESs

\({\mathbb{D}}\)

The set of IoT devices

\({\mathcal{D}}_{1\le j\le n}\)

The jth IoT device

\({\mathbb{G}}_{\mathfrak{B}}\)

The second phase bargaining game

\({\mathbb{G}}_{\mathcal{W}}^{\mathfrak{B}}\)

The first phase bargaining game

\({\mathbb{G}}_{\mathcal{W}}\)

The third phase bargaining game

\({\mathcal{J}}_{\mathcal{D}}\)

The \(\mathcal{D}\)’s strategy in the \({\mathbb{G}}_{\mathcal{W}}\)

\({\mathfrak{M}}_{\mathfrak{B}}^{t}\)

Total sum of \({\mathcal{T}}_{\mathcal{D}}\) where \(\mathcal{D}\in {\mathbb{Q}}_{\mathfrak{B}}\)

\({\mathfrak{M}}_{\mathcal{W}}^{t}\)

Total sum of \({\mathcal{T}}_{\mathcal{D}}\) where \(\mathcal{D}{\in {\mathbb{B}}}_{\mathcal{W}}\)

\({\mathcal{R}}_{\mathfrak{B}}\)

The \(\mathfrak{B}\)’s part of the \({\mathcal{S}}_{\mathfrak{B}}\)

\({\mathcal{R}}_{\mathcal{W}}\)

The \(\mathcal{W}\)’s part of the \({\mathcal{S}}_{\mathfrak{B}}\)

\({\Upsilon}_{\mathcal{D}}\)

The \(\mathcal{D}\)’s strategy to share the \({\mathcal{R}}_{\mathfrak{B}}\) in the \({\mathbb{G}}_{\mathfrak{B}}\)

\({\mathcal{S}}_{\mathfrak{B}}\)

The licensed spectrum band of \(\mathfrak{B}\)

\({\mathcal{S}}_{\mathcal{W}}\)

The unlicensed spectrum band of \(\mathcal{W}\)

\({\mathcal{T}}_{\mathcal{D}}\)

The \(\mathcal{D}\)’s maximum spectrum requirement

\({\mathfrak{U}}_{\mathfrak{B}}\left(\cdot \right)\)

The \(\mathfrak{B}\)’s utility function in the \({\mathbb{G}}_{\mathcal{W}}^{\mathfrak{B}}\)

\({\mathfrak{U}}_{\mathcal{W}}\left(\cdot \right)\)

The \(\mathcal{W}\)’s utility function in the \({\mathbb{G}}_{\mathcal{W}}^{\mathfrak{B}}\)

\({U}_{\mathcal{D}}\left(\cdot \right)\)

The \(\mathcal{D}\)’s utility function in the \({\mathbb{G}}_{\mathcal{W}}^{\mathfrak{B}}\) game

\({\mathcal{U}}_{\mathcal{D}}^{BES}\left(\cdot \right)\)

The \(\mathcal{D}\)’s utility function of BES in the \({\mathbb{G}}_{\mathcal{W}}\)

\({\mathcal{U}}_{\mathcal{D}}^{QPS}\left(\cdot \right)\)

The \(\mathcal{D}\)’s utility function of QPS in the \({\mathbb{G}}_{\mathcal{W}}\)

\({\mathbb{Q}}_{\mathfrak{B}}\)

All IoT devices, which generate QPSs

\({\mathbb{Q}}_{\mathcal{W}}\)

The \(\mathcal{W}\)’s IoT devices generated QPSs

\({\mathbb{W}}\)

The set of WiFi access points

\({\mathcal{W}}_{1\le i\le m}\)

The ith WiFi access point