Sustainable Management of Tourist Flow Networks: A Mean Field Model

Abstract

In this article, we propose a mean field game approach for modeling the flows of excursionists within a network of tourist attractions. We prove the existence of an equilibrium within the network using a balance ordinary differential equation together with optimality conditions in terms of the value function. We also propose a bi-level formulation of the problem where we aim at achieving a sustainable-oriented control strategy in the upper level and at maximizing excursionists’ satisfaction in the lower level. Our proposed model may provide an effective management tool for local authorities who deal with the challenging problem of finding an optimal control policy to the often conflicting objectives of ensuring the maximum excursionists’ satisfaction while pursuing the highest sustainability benefits.

Appendix

The function ${{{\mathcal {V}}}}_r^{\xi ,e_{11}}$ as defined in (21) is of the form

$$\begin{aligned} \mathcal{V}_r^{\xi ,e_{11}}(t)= & {} \frac{c_{\xi ,1}}{2}\frac{(l_{e_{11}})^2}{T-t}+\frac{c_{\xi ,4}}{2}\left( {\widetilde{u}}^{\xi , e_{11}}-\frac{l_{e_{11}}}{T-t}\right) ^2(T-t)\nonumber \\{} & {} +\, \int _{t}^T\Big ( c_{\xi ,2}{{\widetilde{\varphi }}}_{e_{11}}-\frac{c_{\xi ,3}(\tanh (s-1)-1)}{m_{e_{11}}}\Big )\,\textrm{d}s, \end{aligned}$$

(52)

only if

$$\begin{aligned} \varDelta q_{e_{11}}+\frac{c_{\xi ,4}}{2}({\widetilde{u}}^{\xi , e_{11}})^2 \ge \frac{c_{\xi ,1}}{2}\frac{(l_{e_{11}})^2}{T-t}+\frac{c_{\xi ,4}}{2}\left( {\widetilde{u}}^{\xi , e_{11}}-\frac{l_{e_{11}}}{T-t}\right) ^2(T-t), \end{aligned}$$

from which we get

$$\begin{aligned} (T-t)\left( 1-\frac{c_{\xi ,4}({\widetilde{u}}^{\xi , e_{11}})^2(T-t)-2c_{\xi ,4}{\widetilde{u}}^{\xi , e_{11}}l_{e_{11}} }{2\varDelta q_{e_{11}}+c_{\xi ,4}({\widetilde{u}}^{\xi , e_{11}})^2}\right) \ge \frac{(c_{\xi ,1}+c_{\xi ,4})l_{e_{11}}^2}{2\varDelta q_{e_{11}}+c_{\xi ,4}({\widetilde{u}}^{\xi , e_{11}})^2}. \nonumber \\ \end{aligned}$$

(53)

Considering the first factor of the left hand side of (53), we get that

$$\begin{aligned} t\le T-\frac{(c_{\xi ,1}+c_{\xi ,4})l_{e_{11}}^2}{2\varDelta q_{e_{11}}+c_{\xi ,4}({\widetilde{u}}^{\xi , e_{11}})^2}\le T-h', \nonumber \\ \end{aligned}$$

with $h'>0$ independent of r, on $\rho$ and of the control. If instead, we analyze the second factor of the left hand side of (53) we have

$$\begin{aligned} t\ge T-\frac{l_{e_{11}}(-l_{e_{11}}(c_{\xi ,1}+c_{\xi ,4})+2c_{\xi ,4}{\widetilde{u}}^{\xi , e_{11}})+c_{\xi ,4}({\widetilde{u}}^{\xi , e_{11}})^2+2\varDelta q_{e_{11}}}{c_{\xi ,4}({\widetilde{u}}^{\xi , e_{11}})^2}\ge T-h'', \nonumber \\ \end{aligned}$$

(54)

with $h''$ independent of r, on $\rho$ and on the control. If in (54) $h''>0$ then ${{{\mathcal {V}}}}_r^{e_{11},\xi }$ has the shape of (52) only if $T-\max \{ h', h''\}\le t\le T-\min \{h', h'' \}$. If, instead, $h''\le 0$ then $\mathcal{V}_r^{e_{11},\xi }$ is not defined as in (52) but rather by (see (21)):

$$\begin{aligned} {{{\mathcal {V}}}}_r^{\xi ,e_{11}}(t)=\varDelta q_{e_{11}}+\frac{c_{\xi ,4}}{2}({\widetilde{u}}^{\xi , e_{11}})^2+ \int _{t}^T\Bigg ( c_{\xi ,2}{{\widetilde{\varphi }}}_{e_{11}}-\frac{c_{\xi ,3}(\tanh (s-1)-1)}{m_{e_{11}}}\Bigg )\,\textrm{d}s. \end{aligned}$$