## 4 Joint Recommendation Selection and Visiting Time Management Problem
### Visiting Time Management  
Given the recommendation selection, each museum visitor \(i,i\in\mathbb{N}\) aims to determine his/her optimal visiting time \(t_{i}^{*}\) in order to maximize his/her perceived Quality of Experience, which is expressed in Equation (2). Therefore, the previously mentioned goal is formulated as a distributed maximization problem of each visitor's combined QoE function with respect to his/her visiting time, which is shown below. \[ \begin{array}{l}\underset{t_{i}}{\max}Q_{i}(t_{i},\mathbf{t_{-i}}),\\ s.t.\hskip 5.690551 ptt_{i}^{Min}\leq t_{i}\leq t_{i}^{Max}\\ \forall i\in\mathbb{N},\hskip 5.690551 ptx=A,B,C,\\ y=a,b,f,g,\mathbb{N}=\aleph_{a}\cup\aleph_{b}\cup\hskip 5.690551 pt\aleph_{f} \cup\hskip 5.690551 pt\aleph_{g}\end{array}  \tag{5}\]  
Considering the distributed nature of the optimization problem (5) and the selfish behavior of the visitors in terms of optimizing their perceived QoE, a game theoretic approach is adopted for determining the optimal visiting time vector \(\mathbf{t}^{*}=\{t_{1}^{*},\ldots,t_{i}^{*},\ldots,t_{N}^{*}\}\) of the museum visitors. Let us denote by \(G=\left\{\aleph,T_{i},Q_{i}\right\}\) the non-cooperative visiting time management game where \(\aleph\) is the set of players (museum visitors), \(T_{i}=\left[t_{i}^{Min},t_{i}^{Max}\right]\) is the strategy space of the \(i^{th}\) museum visitor, and \(Q_{i}\) is his/her corresponding combined QoE function. The concept of Nash equilibrium is adopted towards seeking analytically the solution of the non-cooperative visiting time management game. Towards proving the existence and uniqueness of Nash equilibrium in the non-cooperative visiting time management game, we should prove that the museum visitor's combined QoE function \(Q_{i}\) is quasi-concave with respect to \(t_{i}\). **Definition 1:**_A function \(Q_{i}\) is strictly quasi-concave if for any pair of distinct points \(t_{i}\) and \(t_{i}{}^{\prime}\) in the convex domain \(T_{i}\) and for \(0<\lambda<1\):_  
\[Q_{i}(t_{i}{}^{\prime})>Q_{i}(t_{i})\Rightarrow Q_{i}(\lambda\cdot t_{i}+(1- \lambda)\cdot t_{i}{}^{\prime})>Q_{i}(t_{i})\]  
_Based on Definition 1, any concave function is quasi-concave._  
**Theorem 1. **_Museum visitor's \(i,i\in\mathbb{N}\), \(\aleph=\aleph_{a}\cup\aleph_{b}\cup\aleph_{f}\cup\hskip 5.690551 pt\aleph_{g}\) combined QoE function is quasi-concave in the modified strategy space \(T_{i}{}^{\prime}\) corresponding to the relative visiting time ratio interval \(rt_{i}\in\left(\frac{\ln M_{y}}{A_{y}},rt_{B}\right)\), where \(rt_{B}\in\left(\frac{\ln M_{y}}{A_{y}},\frac{\ln 10^{4}M_{y}}{A_{y}}\right)\). Thus, the Nash equilibrium of the non-cooperative visiting time management game \(G=\left\{\aleph,T_{i},Q_{i}\right\}\) exists and it is unique in the corresponding strategy space._  
**Proof of Theorem 1. ** Towards proving the quasi-concavity of museum visitor's combined QoE function, we examine the sign of its second order derivative with respect to \(t_{i}\). \[\frac{\partial^{2}Q_{i}}{\partial t_{i}{}^{2}}=g(rt_{i})+h(rt_{i})-c_{x}N_{x}e^ {t_{i}}, \tag{6}\]where \(g(rt_{i})=\frac{2Q_{x}}{t_{i}^{3}}{\left(1-e^{-A_{y}rt_{i}}\right)^{M_{y}-1}} \left[-M_{y}A_{y}e^{-A_{y}rt_{i}}rt_{i}+1-e^{-A_{y}rt_{i}}\right]\quad\text{and} \quad h(rt_{i})=\frac{M_{y}A_{y}^{2}Q_{x}}{t_{i}^{3}}\left(1-e^{-A_{y}rt_{i}} \right)^{M_{y}-2}e^{-A_{y}rt_{i}}rt_{i}^{2}\left(M_{y}e^{-A_{y}rt_{i}}-1\right)\). The sign of the individual terms of Equation (6) is examined. Considering the function \(g(rt_{i})\), the Bolzano theorem is applied, which is an important specialization of the Intermediate Value Theorem [23]. For \(rt_{i}=\frac{\ln M_{y}}{A_{y}}\), we have: \(g\left(rt_{i}=\frac{\ln M_{y}}{A_{y}}\right)=-\ln M_{y}-\frac{1}{M_{y}}+1<0\) and for \(rt_{i}=\frac{\ln 10^{4}M_{y}}{A_{y}}\), we have: \(g\left(rt_{i}=\frac{\ln 10000M_{y}}{A_{y}}\right)=-\frac{\ln 10000M_{y}}{10000}- \frac{1}{10000M_{y}}+1>0\), \(\forall M_{y}\in(1,1000)\).